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MIT's OpenCourseWare has released a new version of Linear Algebra, one of its most visited courses, in the innovative OCW Scholar format designed for independent learners. Taught by Professor Gilbert Strang, 18.06SC Linear Algebra addresses systems of linear equations and the properties of matrices. The concepts of linear algebra are used to solve problems in physics, economics, engineering and other disciplines. 18.06SC is the first of six OCW Scholar courses planned for release by the end of February.
Linear Algebra was one of the original 50 courses published on the MIT OpenCourseWare proof-of-concept site launched in 2002. Over the past 10 years, this course has received a total of 3.1 million visits from educators and learners around the world. Strang, who is one of the most widely known mathematicians in the world, hopes that the new, robust version — with its problem-solving videos — will help students everywhere.
"I'm very proud of this new version of 18.06," Strang says. "OCW has reached out to millions of educators and learners around the globe. With this new approach, even more people can see the beauty and usefulness of Linear Algebra." In September, Strang was named the first MathWorks Professor of Mathematics, assuming a professorship recently endowed by MathWorks, the maker of mathematical software.
OCW Scholar courses represent a new approach to OCW publication. MIT professors and students work closely with the OCW team to restructure the learning experience for independent learners, who typically have few additional resources available to them. The courses offer more materials than typical OCW courses and include new custom-created content. The OCW Scholar version of Linear Algebra includes videos of all the course lectures supplemented by lecture summaries and by 36 short videos showing how to solve specific problems.
The first five of a planned 15 OCW Scholar courses were launched by MIT OpenCourseWare in January 2011, and have collectively received more than 800,000 visits in less than a year. The initial OCW Scholar courses included Classical Mechanics, Electricity and Magnetism, Solid State Chemistry, Single Variable Calculus and Multivariable Calculus. Linear Algebra is the first of seven OCW Scholar courses that will be published in 2012. Other upcoming OCW Scholar courses include Principles of Microeconomics, Differential Equations, Introduction to Psychology, Fundamentals of Biology, Introduction to Electrical Engineering and Computer Science I, and Introduction to Computer Science and Programming. OCW Scholar courses are published on the OCW site with the support of the Stanton Foundation. |
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Algebra Essentials and Applications is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique language of mathematics, so that they are prepared for every type of problem-solving and assessment situation.
Elements of Literature Kentucky Edition has standards are separated into five categories: Reading, Writing, Speaking/Listening/Observing, Inquiry, and Technology as Communication. At the Elements of Literature Internet site, you can read texts by professional writers and learn the inside stories behind your favorite authors. You can also build your word power and analyze messages in the media.
Holt Algebra 2 provides many opportunities for you to prepare for standardized tests for daily practice. This textbook will prepare students for future Algebra courses and also consolidate the core ideas of previous mathematics courses, so that it will serve students of varied mathematics backgrounds. The contents of this course are many of the fundamental ideas and procedures necessary to be successful in subsequent mathematics courses and many careers.
The book broadly covers selections from major forms of literature fiction, poetry, drama, biographies of authors among others. Each section contains a detailed critical introduction to each form, brief biographies of the authors, and a clear, concise editorial apparatus.
Students will learn about Core Skills, Critical Thinking, Reading/Literature, Speaking/Listening, and Writing/Usage/Grammar throughout the book. Students will also read texts by professional writers and learn the inside stories behind your favorite authors. They will also learn to build word power and analyze messages in the media. As they move through Elements of Literature, they will find the best online resources at go.hrw.com.
The textbook provides the student/learner with eight categories of academic standards for reading, writing, and speaking and listening. It also helps the student/learner study the Writer's Models and learn to write short stories, editorials, and moreRead Science in Action articles to learn more about science in the real world. These articles will give you an idea of how interesting, strange, helpful, and action-packed science is. At the end of each chapter, you will find three short articles. And if your thirst is still not quenched, go to go.hrw.com for in-depth coverage.
The book has a Reading Warm-Up at the beginning of every section provides the students with the section's objectives and key terms. A Reading Strategy at the beginning of every section provides tips to help students organize and remember the information covered in the section.
Have you ever wondered...why some places are deserts while other places get so much rain? What makes certain times of the year cooler than others? Why do some rivers run dry? Maybe you live near mountains and wonder what processes created them. You will learn about this and so much more in this |
The proper approach depends on your goals. If you are good and want to get better, that requires one technique. If you are intimidated by Math and you have always struggled, that requires a different method |
More About
This Textbook
Overview
Accessible to students and flexible for instructors, COLLEGE ALGEBRA, SEVENTH EDITION, uses the dynamic link between concepts and applications to bring mathematics to life. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, work independently, and obtain greater mathematical fluency. The text also includes technology features to accommodate courses that allow the option of using graphing calculators. The authors' proven Aufmann Interactive Method allows students to try a skill as it is presented in example form. This interaction between the examples and Try Exercises serves as a checkpoint to students as they read the textbook, do their homework, or study a section. In the Sixth Edition, Review Notes are featured more prominently throughout the text to help students recognize the key prerequisite skills needed to understand new concepts.
Related Subjects
Meet the Author
Richard Aufmann is the lead author of two bestselling developmental math series and a bestselling college algebra and trigonometry series, as well as several derivative math texts. He received a BA in mathematics from the University of California, Irvine, and an MA in mathematics from California State University, Long Beach. Mr. Aufmann taught math, computer science, and physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum, and the impact of technology on curriculum development.
Vernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Richard Nation is Professor of Mathematics at Palomar College. He is the co-author of several Aufmann |
MATH IIIA: PRECALCULUS
A primer course for calculus exploring linear, polynomial, rational, exponential, logarithmic and trigonometric functions. The graphing calculator is introduced and used throughout the course to operate with real-life data and applications. Students are taught to examine a situation from numerical, graphical and analytical perspectives. The last trimester deals with analytic trigonometry: fundamental identities, solving trigonometric equations, sum and difference formulas, Law of Sines and Law of Cosines. The course ends with sequences, series, sigma notation, and multivariable systems of equations and inequalities.
MATH III: INTERMEDIATE ALGEBRA
Intermediate Algebra provides a straightforward program of study with continual reinforcement and progressive confidence building. The topics covered include a chapter on real numbers, linear equations, functions and graphs, systems of linear equations and inequalities, polynomials, factoring, simplifying expressions, radical equations and complex rational expressions. Practice of sample SAT's are offered to keep the students comfortable with this particular type of examination format. The graphing calculator is introduced and used frequently in this course. The course ends with sequences, series, sigma notation, and multivariable systems of equations and inequalities.
Projective Geometry
Projective Geometry covers a historical view of the mathematical breakthroughs symbolizing new realms of thought. A comprehensive and hands-on experience of projective transformations and in particular, the geometrical experience of infinity, gives students a real experience of synthesizing these new ideas. |
Find a Tucson Algebra 1In modern Algebra we most frequently use "x" to represent "the thing". Start any word problem with labeling the unknown, "Let x = the number of ...." It is this great art that has so greatly advanced all the modern sciences. Think of it as the art that supports the sciences! |
as a reference or quick review of the fundamentals of linear algebra, this book offers amatrix-oriented approach--with more emphasis on Euclidean n-space, problem solving, and applications, and less emphasis on abstract vector spaces. It features a variety of applications, boxed statements of important results, and a large number of numbered and unnumbered examples.Matrices, Vectors, and Systems of Linear Equations. Matrices and Linear Transformations. Determinants. Subspaces and Their Properties. Eigenvalues, Eigenvectors, and Diagonalization. Orthogonality. Vector Spaces. Complex Numbers.A professional reference for computer scientists, statisticians, and some engineers.
Table of Contents
1. Matrices, Vectors, and Systems of Linear Equations.
Matrices and Vectors. Linear Combinations, Matrix-Vector Products, and Special Matrices. Systems of Linear Equations. Gaussian Elimination. Applications of Systems of Linear Equations. The Span of a Set Vectors. Linear Dependence and Independence. Chapter 1 Review.
2. Matrices and Linear Transformations.
Matrix Multiplication. Applications of Matrix Multiplication. Invertibility and Elementary Matrices. The Inverse of a Matrix. The LU Decomposition of a Matrix. Linear Transformations and Matrices. Composition and Invertibility of Linear Transformations. Chapter 2 Review. |
Secondary Curricula
Intelligent mathematics software that adapts to meet the needs of ALL students.
Our adaptive curricula, Cognitive Tutor software, is based on over 20 years of research into how students think and learn. The software was developed around an artificial intelligence model that identifies weaknesses in each individual student's mastery of mathematical concepts. It then customizes prompts to focus on areas where the student is struggling, and sends the student to new problems that address those specific concepts. The result is a powerful learning tool with the most precise method of differentiating instruction available.
Cognitive Tutor Software
I think the biggest difference with this curriculum is definitely student attitude. They are so excited. I have so many more students that now say that math is their favorite subject and their favorite class, and that usually is not the case. |
Additions and changes to the original content are the responsibility of the instructor. 2 Holt McDougal Algebra 1 Chapter Test Form B continued 15.
CALIFORNIA STANDARDS TEST Released Test Questions Algebra II Introduction - Algebra II The following released test questions are taken from the Algebra II Standards Test.
Practicing Test-Taking Skills Offers students problems that they can use to practice the skills discussed on the preceding Building Test-Taking Skills pages. Cumulative ... |
Name and Form — Learning to Relate the Shape of a Graph to Its Corresponding Function
A
mathematical function quantifies the relationship between two related
quantities and can be used to model change. Functions and their graphs
are essential to all branches of mathematics and their applications.
Topics: trigonometry, algebra of functions, compositions and inverses of
functions, functions (trigonometric, power, polynomial, and rational),
and applications. (4 credits) Prerequisite: MATH 161 |
$1.5M Grant Helping Researchers Understand How Students Learn Algebra
April 1, 2011-Houston-
Algebra is the gateway for students seeking careers in science, technology, engineering and mathematics (STEM). While much research has been focused on the skills required to learn basic mathematics, fewer investigations have been dedicated to those necessary for learning algebra.
Thanks to a $1.5 million grant from the Institute for Education Sciences, researchers from the University of Houston and Vanderbilt University will conduct a four-year study devoted to understanding factors connected with how students learn algebra.
Principal investigators are Paul Cirino, UH associate professor of psychology; Tammy Tolar, research associate at UH's Texas Institute for Measurement, Evaluation and Statistics (TIMES) and Lynn Fuchs, Nicholas Hobbs Professor of Special Education and Human Development at Vanderbilt University.
"We are continuing to learn much about early math development and difficulty," Cirino said. "What we know about how students learn and understand algebra, however, pales in comparison. What we want to do is fill in some of the gaps in the current research literature."
Titled "Arithmetical and Cognitive Antecedents and Concomitants of Algebraic Skill," the study will track students in grades three through nine.
Among the researchers' objectives is to evaluate whether procedural skill and conceptual knowledge are separate but related abilities in relation to learning algebra.
Procedural skills are used by students when solving routine textbook problems ("Solve: 3x + 5 = 17") and can involve implementing algorithms. Conceptual knowledge is used when students must instead rely on their understanding of the underlying mathematical principles (the associative property of multiplication) to arrive at a potential solution, as when problems are presented in a novel way ("If (3p)r = 9 then pr = __").
"There has not been enough research to indicate the differences and similarities between procedural and conceptual knowledge," Cirino said. "We're interested in distinguishing between these two types of knowledge and learning more about students who might be weak in either area. This will help us understand if there are certain characteristics among these students or if there is a certain of way of teaching that can benefit them."
Another goal of the study is to evaluate the factors that help students understand more basic arithmetic and whether such factors impact how they learn algebra. Cirino, Tolar and Fuchs will observe student data from third - sixth graders. They will follow these students and their peers as they begin learning algebra.
"Both basic arithmetic and algebra share similar procedures and concepts," Cirino said. "Weak arithmetic, conceptual or procedural skills might have implications on how well students grasp algebra. If educators are aware of these kinds of predictors, it could have implications for the way basic math is taught in schools."
Cirino and his colleagues also will explore how the cognitive skills (language, memory, visual-spatial skills) relative to basic math are applied when learning algebra.
"We know that algebra is very difficult for many students," Cirino said. "Many students don't pass their algebra classes or fully understand the material. We need to understand why that is. This study will help to do that by evaluating the skills that contribute to learning algebra and whether there are different types of knowledge that require different combinations of skills."
Cirino, Tolar and Fuchs will observe two student cohorts. One is based in Texas and another is in Tennessee. The Tennessee cohort will include students in eighth and ninth grades taking algebra. Elementary and middle school data from these same students had been compiled during a previous study. The Texas cohort will include students in middle school whose arithmetic progress will be tracked through their introduction to algebra in eighth and ninth grades.
The IES is the research arm of the U.S. Department of Education. Its mission is to provide evidence regarding effective education practices and policies. Through its sponsored research efforts, the IES aims to improve educational outcomes for all students, particularly those at risk of failure. To learn more about the IES, visit |
Description:
The dedicated folks at the Mathematical Association of America (MAA) have created this handy compendium of learning capsules as part of their online digital library. This compendium contains fifteen different areas, ranging from General Tools to Antidifferentiation. These resources have been contributed and vetted by mathematics professors, learning specialists, and others actively involved in the fields of mathematics and mathematics education. Many of these resources appeared in reputable sources like the College Mathematics Journal or as part of other publications. Visitors can search these materials by title, author, subject matter, or keyword, and they can also look through the Tips for Searching area for additional assistance. |
Thanks for visiting ARIS or MathZone. We have retired ARIS and MathZone, but no worries! We've replaced them with Connect and ConnectPlus, our new generation of digital learning products with improved user experience and enhanced functionality.
Elementary & Intermediate Algebra, 3/e content in context. The authors use a three-pronged approach (I. Communication, II. Pattern Recognition, and III. Problem Solving) to present the material and stimulate critical thinking skills. Items such as Math Anxiety boxes, Check Yourself exercises, and Activities |
When you are able to conquer Algebra, you can conquer the world! Algebra 2 can't be properly understood without a thorough understanding of Algebra 1. It makes sense, but in actuality the concepts and functions of Algebra 2 are built upon Algebra 1 |
Marketplaces
departments
payment types
certifications—accuracy, precision, depth, strong student support, and abundant exercises—while exposing readers to functions in the first chapter. To ensure that students master basic skills and develop the conceptual understanding they need for the course, this text focuses on the fundamentals:
Find more:
Find more:
Highlights: R
Highlights:
This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Books a la Carte also offer a great value—this format costs significantly less than a new textbook
Highlights:
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry, Second Edition embodies Sullivan/Sullivan's hallmarks-accuracy, precision, depth, strong student support, and abundant exercises-while exposing readers to functions in the first chapter. To ensure that students master basic skills and develop the conceptual understanding they need for the course, this text focuses on the fundamentals: "preparing "for class, "practicing "their homework, and "reviewing" the concepts. After usingPrecalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry, Second Edition embodies Sullivan /Sullivan &rsquo ;s hallmarks &ndash ;accuracy , precision, depth, strong student support, and abundant exercises &ndash ;while exposing readers to functions in the first chapter. To ensure that students master basic skills and develop the conceptual understanding they need for the course, this text focuses on the fundamentals: preparing for class, practicing their homework, and - -accuracy , precision, depth, strong student support, and abundant exercises--while exposing readers to functions in the first chapter. To ensure that students master basic skills and develop the conceptual understanding they need for the course, this text focuses on the fundamentals:
Highlights:
Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Fifth Edition gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find enoughcarefully placed learning aids and review tools to help them do the math without getting distracted from their objectives. Regardless of their goals beyond the course, all students will benefit from Dugopolski's emphasis on problem solving and critical thinking,
HighlightsPack , Trigonometry: A Unit Circle Approach has evolved to meet today's course needs, building on these hallmarks by integrating projects and other interactive learning tools for use in the classroom or online.
Highlights: Pearson's MyLab & Mastering products. Pack hallmarks by integrating projects and other interactive learning tools for use in the classroom or online. New Internet-based Chapter Projects apply skills to real-world problems and are accompanied by assignable |
3 Tutorial
This chapter explains how to use Calc and its many features, in
a step-by-step, tutorial way. You are encouraged to run Calc and
work along with the examples as you read (see Starting Calc).
If you are already familiar with advanced calculators, you may wish
to skip on to the rest of this manual.
This tutorial describes the standard user interface of Calc only.
The Quick mode and Keypad mode interfaces are fairly
self-explanatory. See Embedded Mode, for a description of
the Embedded mode interface.
The easiest way to read this tutorial on-line is to have two windows on
your Emacs screen, one with Calc and one with the Info system. Press
C-x * t to set this up; the on-line tutorial will be opened in the
current window and Calc will be started in another window. From the
Info window, the command C-x * c can be used to switch to the Calc
window and C-x * o can be used to switch back to the Info window.
(If you have a printed copy of the manual you can use that instead; in
that case you only need to press C-x * c to start Calc.)
This tutorial is designed to be done in sequence. But the rest of this
manual does not assume you have gone through the tutorial. The tutorial
does not cover everything in the Calculator, but it touches on most
general areas.
You may wish to print out a copy of the Calc Summary and keep notes on
it as you learn Calc. See About This Manual, to see how to make a
printed summary. See Summary. |
General Mathematics - Units 1 & 2
Structure
Units 1 and 2. At Emmaus College General Mathematics is offered as two streams: General Maths A (Advanced) and General Maths B.
General Mathematics A (Advanced)
This course has been designed for students who intend to study higher level Mathematics at Year 12 (Specialist Maths and/or Mathematical Methods 3, 4). General Mathematics A must be taken in conjunction with Mathematical Methods.
General Mathematics B
General Mathematics B has been designed as a preparatory course of study for VCE Further Mathematics.
It may also be undertaken by those students who would like to further their study in mathematics without completing any Maths at Year 12 level.
Each unit deals with specific content and is designed to enable students to achieve a set of outcomes. Each outcome is described in terms of key knowledge and skills.
Outcomes
Outcomes define what students will know and be able to do as a result of undertaking the study.
Outcomes include a summary statement and the key knowledge and skills that underpin them. Only the summary statements have been reproduced below and must be read in conjunction with the key knowledge and skills published in the study design.
Units 1 & 2
General Mathematics provides courses of study for a broad range of students and may be implemented in a number of ways. Some students will not study Mathematics beyond Units 1 and 2, while others will intend to study Further Mathematics Units 3 and 4. Others will also be studying Mathematics Methods Units 1 and 2 or Mathematics Methods Computer Algebra System (CAS) Units 1 and 2 and intend to study Mathematical Methods Units 3 and 4, or Mathematical Methods (CAS) Units 3 and 4 and, in some cases, Specialist Mathematics Units 3 and 4 as well.
The areas of study for Unit 1 and Unit 2 of General Mathematics are 'Arithmetic', 'Data analysis and simulation', 'Algebra', 'Graphs of linear and non-linear relations', 'Decision and business mathematics' and 'Geometry and trigonometry'.
Units 1 and 2 are to be constructed to suit the range of students entering the study by selecting material from the six areas of study using the following rules:
• for each unit, material covers four or more topics selected from at least three different areas of study; • courses intended to provide preparation for study at the Units 3 and 4 level should include selection of material from areas of study which provide a suitable background for these studies; • selected material from an area of study provide a clear progression in key knowledge and key skills from Unit 1 to Unit 2.
The appropriate use of technology to support and develop the teaching and learning of mathematics is to be incorporated throughout the course. This will include the use of some of the following technologies for various areas of study or topics: graphics calculators, spreadsheets, graphing packages, dynamic geometry systems, statistical analysis systems, and computer algebra systems.
Outcome 1
On completion of each unit the student should be able to define and explain key concepts in relation to the topics from the selected areas of study, and apply a range of related mathematical routines and procedures.
Outcome 2
On completion of each unit the student should be able to apply mathematical processes in non-routine contexts, and analyse and discuss these applications of mathematics in at least three areas of study.
Outcome 3
On completion of each unit the student should be able to use technology to produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches in at least three areas of study.
Assessment
Satisfactory Completion
Demonstrated achievement of the set of outcomes specified for the unit. |
Why
is algebra and other advanced mathematics necessary? Consider
this: the computer technology that we have today, the precision-guided
missiles used in the Iraq war that saved so many of our soldier's
lives, the medical technology that helps us to live longer
and have better quality lives, the current exploration of
Mars, all are possible because of advanced math and science.
Students who have a gift in math will be able to use their
gift to enable the people of the world to accomplish even
more in the coming years. (See Raising
an Isaac Newton.)
However,
not all students are endowed with the gift of math. Other
students are gifted in art, music, business, and other very
important skills. Depending on your child's gifts and career
goals, your child may not need advanced math. Instead, he
may need more consumer and/or business math. Select your curriculum
to suit your child's needs, not the needs of someone else's
child. For more information, see Math
Curriculum Recommendations and Homeschooling
High School.
For kids who like to learn on their own and on the computer
or for parents who need extra help teaching math, check
out ALEKS.
It cost $19.95 month,
and they offer discounts for multiple students and yearly
rates.
you
prefer a textbook series, I highly recommend the Math-U-See
program. I wish this were available when I was
teaching homeschool! It espeically works well for
kids (and teaching moms) of Visual,
Kinesthetic, and Auditory learning styles.
(For your traditional Read/Write or advanced math
students, see below.) It teaches with manipulatives
for the Kinesthetic and Visual Learners and video
instructions (VHS or DVD)
& music (skip counting songs on cassette or CD) for
the Auditory & Visual Learners. The teacher's manual
includes instructions and the answers. The student workbook
includes practice pages and review. There are three practice
pages for each lesson so that you child can do one or
more depending on his particular needs to understand
the lesson. Following the practice pages, there are three
review pages which includes one or more problems from
the current lesson and from all the previous lessons
so that your child will not forget what he learned earlier
in the year. Again, your child may or may not need to
do all the pages.
Video-Text
Interactive. This six-video course covers Pre-algebra
through Algebra II. The graphics make this a real winner
for your Visual learner,
but all learners will benefit
from this course. The
program is set up to take two years, but you can begin
as soon as your child is ready for pre-algebra. Spread
the videos out over three years or more if your start
early. This is an expensive curriculum, but if you want
your child to really learn algebra and not just plug
in the numbers, this is the program to buy.
Math
87. The Saxon math program is an excellent math program
for the Read/Write
learnerand
advanced student. It's also a lot less
expensive. This is a pre-algebra course. If your
child does not need the extra practice, you could
skip to the Saxon Algebra 1/2 textbook.
The
Saxon Algebra program is an excellent math program
for the
Read/Write learner and
advanced student. This is a pre-algebra
course.
Most
math textbooks integrate geometry into their algebra program.
Video Text Interactive; however, does NOT include geometry
in their algebra curriculum. (They are working on a geometry/trigonometry
video series projected to be completed by summer 2004.) In
the meantime, you can use the following for a geometry course.
If your child is using the Saxon Algebra textbooks, he will
benefit from taking a brief detour from the textbook and completing
any of the following:
SAXON
CALCULUS.
The Saxon math program is an excellent math program for
the Read/Write
learnerand
advanced student. Topics include algebra,
geometry, trigonometry, discrete mathematics, and
mathematical analysis. A rigorous treatment of Euclidean
geometry is also presented. Conceptually oriented
problems that prepare students for college entrance
exams (such as ACT and SAT) are included in the problem
sets.
Calc
for the Clueless series by Bob Miller. Here's a
book that every calculus student should read. It's a
simple-to-understand book that was written to the student, not to teachers.
It makes calculus more appealing to everyone, even the child
hates math.
How to Work With Probability and Statistics, Grades 6-8
This
48-page workbook is an introduction to probability
and statistics created for students in grades 4 - 6;
however, I recommend it for any age. The easy-to-read
text, charts, and other visual aids will benefit Visual
learners of all ages.
Probability, Statistics & Graphing, Grades 4-5 (Basic Skills Series).
This 48-page workbook is an introduction to probability,
statistics, & graphing
created for students in grades 4 - 6; however, I recommend
it for any age. The easy-to-read text, hands-on activities,
charts, and other visual aids will benefit Visual
and Kinethetic learners of all ages.
Consumer Math Success Kit . It's hard to find a good
consumer math textbook. They tend to cover things that
are not relevant to consumer math or they are a disguise
for more traditional mathematics. After looking at several,
this one appears to be the most useful. It covers banking,
budget, credit cards, eating out, grocery shopping, heating
costs, housing, income taxes, interest, investment, property
taxes, utility bills, travel and a few other items that
may or may not be relevant. I recommend that your child
complete Larry Burkett's,
Money Matters Workbook for Teens, Ages 15-18 . in
addition to, or instead of this curriculum.
Lifepac
Electives Accounting Complete Set . Students first
get a general overview of accounting, then learn about
specifics such as debits and credits; journalizing and
posting transactions; financial statements for a proprietorship;
payroll
accounting, tax recording, and payment; business simulation
activity; and more. This is a very good program that the
student can go through on his own.
The
Accounting Game: Basic Accounting Fresh from the Lemonade
Stand.
This book provides a first step to understanding accounting
by using the world of a kid's lemonade stand. I highly
recommend it to anyone who wants to understand concepts
like assets, liabilities, earnings, inventory and notes
payable. You'll learn about advertising, borrowing money,
purchasing inventory, and selling as well as create
and understand an income statement and balance sheet,
track inventory using LIFO and FIFO, and create cash
statements and understand cash flow and liquidity. A
simplified approach that works, especially for the Visual
and Kinesthetic learner. |
More About
This Textbook
Overview
This practical and useful arithmetic text helps students—particularly those having trouble with math—learn the fundamentals of real estate math. The book's relaxed, easy-to-understand style helps anxiety-ridden students calm down and focus on the subject. Beginning with a review of basic mathematics, the text guides students through real estate calculations, applications, and prorations |
Mathematics: A Practical Odyssey - 7th edition
Summary: Users discover the many ways in which mathematics is relevant to their lives with MATHEMATICS: A PRACTICAL ODYSSEY, 7E and its accompanying online resources. They master problem-solving skills in such areas as calculating interest and understanding voting systems and come to recognize the relevance of mathematics and to appreciate its human aspect97.25 +$3.99 s/h
VeryGood
Bookbyte-OR Salem, OR
Has minor wear and/or markings. SKU:9780538495059-3-0
$101.49 +$3.99 s/h
New
textbook_rebellion2 Troy, MI
0538495057130.53148.93 +$3.99 s/h
LikeNew
Love Is the Answer Washington, DC
Hardcover. Strong binding. Clean, unmarked text. Thanks for looking!
$160.96212.86212.87 |
Extending Frontiers of Mathematics - 06 edition
ISBN13:978-0470412220 ISBN10: 0470412224 This edition has also been released as: ISBN13: 978-1597570428 ISBN10: 1597570427
Summary: In the real world of research mathematics, mathematicians do not know in advance if their assertions are true or false. Extending the Frontiers of Mathematics: Inquiries into proof and argumentation requires students to develop a mature process that will serve them throughout their professional careers, either inside or outside of mathematics. Its inquiry-based approach to the foundations of mathematics promotes exploring proofs and other advanced mathematical ideas through these fe...show moreatures: - Puzzles and patterns introduce the pedagogy. These precursors to proofs generate creativity and imagination that the author builds on later - Prove and extend or disprove and salvage, a consistent format of the text, provides a framework for approaching problems and creating mathematical proofs - Mathematical challenges are presented which build upon each other, motivate analytical skills, and foster interesting discussion123.12 +$3.99 s/h
New
StudentSolutions Stone Mountain, GA
Brand New US edition as listed. We ship promptly with good, sturdy packaging! Ask for JR with any product questions!
$125.88 +$3.99 s/h
New
PROFESSIONAL & ACADEMIC BOOKSTORE Dundee, MI
0470412224129.94 |
Summary: These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have a created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan'sAlgebra and Trigonometry: Enhanced with Graphing Utilit...show moreiesgives students a model for success in mathematics |
Mathematics as a Constructive Activity [NOOK Book]NOOK Study
NOOK for Web
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Table of Contents
Contents: Preface. Introduction in Exemplification in Mathematics. Learner-Generated Examples in Classrooms. From Examples to Example Spaces. The Development of Learners' Example Spaces. Pedagogical Tools for Developing Example Spaces. Strategies for Prompting and Using Learner-Generated Examples. Mathematics as a Constructive Activity. Append |
Math: Factoring
Factoring polynomials is one of the most important skills you will learn and use now and in your future math classes. Factoring is an essential skill that can make or break you and your grade. Join us for a "factoring summary," and learn four straight-forward things you can do to make it in factoring and your math class. |
Description
Like many instructors today, Bob Prior's teaching has evolved in recent years to address the changing world of developmental math: the class format, the classroom itself, the teachers, and most importantly, the students. Bob teaches in a variety of formats (online, face-to-face, hybrid). He sees some students in class regularly and in office hours, while he knows other students only by name and email address.
Basic Mathematics is based on Bob Prior's own varied teaching experiences and is designed to serve the needs of today's developmental math student and classroom. Bob knows that because today's students don't always have a lot of face time with their instructors, a usable, thorough, easy-to-follow text is key to their success. He draws students into the book by incorporating practice opportunities throughout the body of text. Thorough explanations and examples explain the "why" behind the mathematics and patiently develop each concept.
Basic Mathematics is presented in a user-friendly, spiral-bound format, and is available with a Student Resources DVD-ROM set that includes video lectures for each section of the text, all chapter test solutions on video, and the Student's Solutions Manual. This new streamlined format conserves natural resources, while also providing convenience and savings for students.
Features
The thorough explanations assume no mathematical background, breaking down every topic into its most basic ideas.
Prior reinforces self-assessment and active participation with a variety of features.
Preparation Exercises at the start of every chapter help students review what they need to know before beginning the chapter.
A "You Need to Know" list references materials that students should understand to successfully complete the section objectives, which are clearly stated at the beginning of each section.
Multiple You Try It Exercises follow each example, allowing students to immediately practice and apply the concepts they have just learned, while effectively bringing exercise sets into each lesson.
Think About It boxes pose thought-provoking questions that encourage discussion of the concepts and topics. The concepts covered in these boxes are revisited in the Think Again exercises that begin each exercise set.
Aninteractive format and presentation style encourage students to keep the book open and write in its pages, fostering active learning. The spiral binding provides a convenient way for students to study from their texts while working at a computer.
Student supplements are available on one DVD-ROM set, saving natural resources and passing cost savings on to students.
Author |
The course is divided into two tracks. In the Math 41 track we will study single variable calculus; the goal is to become comfortable with limits, derivatives, and integrals. In the Math 51 track we will learn the basics of linear algebra; the goal is to give you a head start with material that is likely new to you.
On the first day of class your will take a diagnostic test to determine your track for the summer. On the last day of class you will take a final exam, and we will recommend a course for you to take during the academic school year based on this final exam. Note that your summer track placement does not determine what course we will recommend!
The tracks have lectures and sections on alternate days from 1:30-2:30.
Here are the diagnostic test and solutions. The exam was out of 70 points. The average score was 43.9, the median score was 48, and the standard deviation was 13.9. The advised cutoff for the math 51 track is 47 out of 70. |
This is a guide to some of the ways in which the Worldwide Web can be used as a resource for teachers and learners of mathematics. Browsing the Internet can be very time-consuming, so this is just one person's selection of tutorials, interactive materials and mathematical tools. The selection given here, along with the accompanying notes, was collated by Pam Bishop, Assistant Director of the LTSN Mathematics, Statistics and Operational Research (OR) Network (see for the original version).
Recently the Mathematics, Statistics and OR Network has added a section to its Web site, 'Accessing Maths and Stats By Other Means' ( which includes tools for students with disability. Help with finding mathematical sites on the Internet is available via the online tutorial 'The Internet Mathematician' (
Some of these sites use special browsers or plug-ins which have to be installed as part of Netscape or Internet Explorer. If you want to take advantage of such facilities on machines in your own institution, you (or a network manager) may have to download and install the correct version for your machine or network.
Many other sites can be searched for within the Network's resource database at or the 'Internet Guide to Mathematics' at
Given the dynamic nature of the Web, URLs and site content are liable to change but the information here was correct at the time of publication |
These video lectures of Professor Gilbert Strang teaching 18.06 were recorded in Fall 1999 and do not correspond precisely to the current edition of the textbook. However, this book is still the best reference for more information on the topics covered in each lecture.
Related Resources
Okay. This is lecture five in linear algebra. And, it will complete this chapter of the book. So the last section of this chapter is two point seven that talks about permutations, which finished the previous lecture, and transposes, which also came in the previous lecture.
There's a little more to do with those guys, permutations and transposes. But then the heart of the lecture will be the beginning of what you could say is the beginning of linear algebra, the beginning of real linear algebra which is seeing a bigger picture with vector spaces -- not just vectors, but spaces of vectors and sub-spaces of those spaces. So we're a little ahead of the syllabus, which is good, because we're coming to the place where, there's a lot to do.
Okay. So, to begin with permutations.
Can I just -- so these permutations, those are matrices P and they execute row exchanges.
And we may need them. We may have a perfectly good matrix, a perfect matrix A that's invertible that we can solve A x=b, but to do it -- I've got to allow myself that extra freedom that if a zero shows up in the pivot position I move it away. I get a non-zero.
I get a proper pivot there by exchanging from a row below.
And you've seen that already, and I just want to collect the ideas together. And principle, I could even have to do that two times, or more times.
So I have to allow -- to complete the -- the theory, the possibility that I take my matrix A, I start elimination, I find out that I need row exchanges and I do it and continue and I finish. Okay.
Then all I want to do is say -- and I won't make a big project out of this -- what happens to A equal L U? So A equal L U -- this was a matrix L with ones on the diagonal and zeroes above and multipliers below, and this U we know, with zeroes down here.
That's only possible. That description of elimination assumes that we don't have a P, that we don't have any row exchanges. And now I just want to say, okay, how do I account for row exchanges? Because that doesn't. The P in this factorization is the identity matrix. The rows were in a good order, we left them there. Maybe I'll just add a little moment of reality, too, about how Matlab actually does elimination. Matlab not only checks whether that pivot is not zero, as every human would do.
It checks for is that pivot big enough, because it doesn't like very, very small pivots. Pivots close to zero are numerically bad. So actually if we ask Matlab to solve a system, it will do some elimination some row exchanges, which we don't think are necessary. Algebra doesn't say they're necessary, but accuracy -- numerical accuracy says they are. Well, we're doing algebra, so here we will say, well, what do row exchanges do, but we won't do them unless we have to.
But we may have to. And then, the result is -- it's hiding here. It's the main fact.
This is the description of elimination with row exchanges.
So A equal L U becomes P A equal L U.
So this P is the matrix that does the row exchanges, and actually it does them -- it gets the rows into the right order, into the good order where pivots will not -- where zeroes won't appear in the pivot position, where L and U will come out right as up here.
So, that's the point. Actually, I don't want to labor that point, that a permutation matrix -- and you remember what those were. I'll remind you from last time of what the main points about permutation matrices were -- and then just leave this factorization as the general case. This is -- any invertible A we get this. For almost every one, we don't need a P. But there's that handful that do need row exchanges, and if we do need them, there they are. Okay, finally, just to remember what P was. So permutations, P is the identity matrix with reordered rows.
I include in reordering the possibility that you just leave them the same. So the identity matrix is -- okay. That's, like, your basic permutation matrix -- your do-nothing permutation matrix is the identity. And then there are the ones that exchange two rows and then the ones that exchange three rows and then then ones that exchange four -- well, it gets a little -- it gets more interesting algebraically if you've got four rows, you might exchange them all in one big cycle. One to two, two to three, three to four, four to one.
Or you might have -- exchange one and two and three and four. Lots of possibilities there. In fact, how many possibilities? The answer was (n)factorial. This is n(n-1)(n-2)... (3)(2)(1).
That's the number of -- this counts the reorderings, the possible reorderings. So it counts all the n by n permutations. And all those matrices have these -- have this nice property that they're all invertible, because we can bring those rows back into the normal order. And the matrix that does that is just P -- is just the same as the transpose.
You might take a permutation matrix, multiply by its transpose and you will see how -- that the ones hit the ones and give the ones in the identity matrix. So this is a -- we'll be highly interested in matrices that have nice properties.
And one property that -- maybe I could rewrite that as P transpose P is the identity. That tells me in other words that this is the inverse of that.
Okay. We'll be interested in matrices that have P transpose P equal the identity.
There are more of them than just permutations, but my point right now is that permutations are like a little group in the middle -- in the center of these special matrices. Okay.
So now we know how many there are.
Twenty four in the case of -- there are twenty four four by four permutations, there are five factorial which is a hundred and twenty, five times twenty four would bump us up to a hundred and twenty -- so listing all the five by five permutations would be not so much fun. Okay.
So that's permutations. Now also in section two seven is some discussion of transposes.
And can I just complete that discussion.
First of all, I haven't even transposed a matrix on the board here, have I? So I'd better do it. So suppose I take a matrix like (1 2 4; 3 3 1). It's a rectangular matrix, three by two. And I want to transpose it.
So what's -- I'll use a T, also Matlab would use a prime.
And the result will be -- I'll right it here, because this was three rows and two columns, this was a three by two matrix. The transpose will be two rows and three columns, two by three.
So it's short and wider. And, of course, that row -- that column becomes a row -- that column becomes the other row.
And at the same time, that row became a column.
This row became a column. Oh, what's the general formula for the transpose? So the transpose -- you see it in numbers. What I'm going to write is the same thing in symbols. The numbers are the clearest, of course. But in symbols, if I take A transpose and I ask what number is in row I and column J of A transpose? Well, it came out of A.
It came out A by this flip across the main diagonal.
And, actually, it was the number in A which was in row J, column I.
So the row and column -- the row and column numbers just get reversed. The row number becomes the column number, the column number becomes the row number. No problem.
Okay. Now, a special -- the best matrices, we could say. In a lot of applications, symmetric matrices show up. So can I just call attention to symmetric matrices? What does that mean? What does that word symmetric mean? It means that this transposing doesn't change the matrix.
A transpose equals A. And an example.
So, let's take a matrix that's symmetric, so whatever is sitting on the diagonal -- but now what's above the diagonal, like a one, had better be there, a seven had better be here, a nine had better be there.
There's a symmetric matrix. I happened to use all positive numbers as its entries. That's not the point.
The point is that if I transpose that matrix, I get it back again. So symmetric matrices have this property A transpose equals A. I guess at this point -- I'm just asking you to notice this family of matrices that are unchanged by transposing. And they're easy to identify, of course. You know, it's not maybe so easy before we had a case where the transpose gave the inverse.
That's highly important, but not so simple to see.
This is the case where the transpose gives the same matrix back again. That's totally simple to see.
Okay. Could actually -- maybe I could even say when would we get such a matrix? For example, this -- that matrix is absolutely far from symmetric, right? The transpose isn't even the same shape -- because it's rectangular, it turns the -- lies down on its side. But let me tell you a way to get a symmetric matrix out of this.
Multiply those together. If I multiply this rectangular, shall I call it R for rectangular? So let that be R for rectangular matrix and let that be R transpose, which it is.
Then I think that if I multiply those together, I get a symmetric matrix. Can I just do it with the numbers and then ask you why, how did I know it would be symmetric? So my point is that R transpose R is always symmetric. Okay? And I'm going to do it for that particular R transpose R which was -- let's see, the column was one two four three three one. I called that one R transpose, didn't I, and I called this guy one two four three three one.
I called that R. Shall we just do that multiplication? Okay, so up here I'm getting a ten. Next to it I'm getting two, a nine, I'm getting an eleven. Next to that I'm getting four and three, a seven. Now what do I get there? This eleven came from one three times two three, right? Row one, column two.
What goes here? Row two, column one.
But no difference. One three two three or two three one three, same thing.
It's going to be an eleven. That's the symmetry.
I can continue to fill it out. What -- oh, let's get that seven. That seven will show up down here, too, and then four more numbers.
That seven will show up here because one three times four one gave the seven, but also four one times one three will give that seven. Do you see that it works? Actually, do you want to see it work also in matrix language? I mean, that's quite convincing, right? That seven is no accident. The eleven is no accident.
But just tell me how do I know if I transpose this guy -- How do I know it's symmetric? Well, I'm going to transpose it. And when I transpose it, I'm hoping I get the matrix back again.
So can I transpose R transpose R? So just -- so, why? Well, my suggestion is take the transpose.
That's the only way to show it's symmetric.
Take the transpose and see that it didn't change.
Okay, so I take the transpose of R transpose R.
Okay. How do I do that? This is our little practice on the rules for transposes.
So the rule for transposes is the order gets reversed.
Just like inverses, which we did prove, same rule for transposes and -- which we'll now use.
So the order gets reversed. It's the transpose of that that comes first, and the transpose of this that comes -- no.
Is that -- yeah. That's what I have to write, right? This is a product of two matrices and I want its transpose.
So I put the matrices in the opposite order and I transpose them. But what have I got here? What is R transpose transpose? Well, don't all speak at once.
R transpose transpose, I flipped over the diagonal, I flipped over the diagonal again, so I've got R.
And that's just my point, that if I started with this matrix, I transposed it, I got it back again. So that's the check, without using numbers, but with -- it checked in two lines that I always get symmetric matrices this way.
And actually, that's where they come from in so many practical applications. Okay.
So now I've said something today about permutations and about transposes and about symmetry and I'm ready for chapter three. Can we take a breath -- the tape won't take a breath, but the lecturer will, because to tell you about vector spaces is -- we really have to start now and think, okay, listen up.
What are vector spaces? And what are sub-spaces? Okay. So, the point is, The main operations that we do -- what do we do with vectors? We add them. We know how to add two vectors.
We multiply them by numbers, usually called scalers.
If we have a vector, we know what three V is.
If we have a vector V and W, we know what V plus W is.
Those are the two operations that we've got to be able to do.
To legitimately talk about a space of vectors, the requirement is that we should be able to add the things and multiply by numbers and that there should be some decent rules satisfied. Okay.
So let me start with examples. So I'm talking now about vector spaces. And I'm going to start with examples. Let me say again what this word space is meaning. When I say that word space, that means to me that I've got a bunch of vectors, a space of vectors. But not just any bunch of vectors. It has to be a space of vectors -- has to allow me to do the operations that vectors are for.
I have to be able to add vectors and multiply by numbers.
I have to be able to take linear combinations.
Well, where did we meet linear combinations? We met them back in, say in R^2.
So there's a vector space. What's that vector space? So R two is telling me I'm talking about real numbers and I'm talking about two real numbers.
So this is all two dimensional vectors -- real, such as -- well, I'm not going to be able to list them all. But let me put a few down.
|3; 2|, |0;0|, |pi; e|.
So on. And it's natural -- okay.
Let's see, I guess I should do algebra first. Algebra means what can I do to these vectors? I can add them. I can add that to that.
And how do I do it? A component at a time, of course. Three two added to zero zero gives me, three two. Sorry about that.
Three two added to pi e gives me three plus pi, two plus e. Oh, you know what it does.
And you know the picture that goes with it.
There's the vector three two. And often, the picture has an arrow. The vector zero zero, which is a highly important vector -- it's got, like, the most important here -- is there.
And of course there's not much of an arrow. Pi -- I'll have to remember -- pi is about three and a little more, e is about two and a little more.
So maybe there's pi e. I never drew pi e before.
It's just natural to -- this is the first component on the horizontal and this is the second component, going up the vertical. Okay.
And the whole plane is R two. So R two is, we could say, the plane.
The xy plane. That's what everybody thinks.
But the point is it's a vector space because all those vectors are in there. If I removed one of them -- Suppose I removed zero zero. Suppose I tried to take the -- considered the X Y plane with a puncture, with a point removed.
Like the origin. That would be, like, awful to take the origin away.
Why is that? Why do I need the origin there? Because I have to be allowed -- if I had these other vectors, I have to be allowed to multiply three two -- this was three two -- by anything, by any scaler, including zero. I've got to be allowed to multiply by zero and the result's got to be there.
I can't do without that point. And I have to be able to add three two to the opposite guy, minus three minus two.
And if I add those I'm back to the origin again.
No way I can do without the origin.
Every vector space has got that zero vector in it.
Okay, that's an easy vector space, because we have a natural picture of it. Okay.
Similarly easy is R^3. This would be all -- let me go up a little here. This would be -- R three would be all three dimensional vectors -- or shall I say vectors with three real components.
Okay. Let me just to be sure we're together, let me take the vector three two zero.
Is that a vector in R^2 or R^3? Definitely it's in R^3.
It's got three components. One of them happens to be zero, but that's a perfectly okay number.
So that's a vector in R^3. We don't want to mix up the -- I mean, keep these vectors straight and keep R^n straight. So what's R^n? R^n.
So this is our big example, is all vectors with n components. And I'm making these darn things column vectors. Can I try to follow that convention, that they'll be column vectors, and their components should be real numbers.
Later we'll need complex numbers and complex vectors, but much later. Okay.
So that's a vector space. Now, let's see. What do I have to tell you about vector spaces? I said the most important thing, which is that we can add any two of these and we -- still in R^2.
We can multiply by any number and we're still in R^2.
We can take any combination and we're still in R^2.
And same goes for R^n. It's -- honesty requires me to mention that these operations of adding and multiplying have to obey a few rules. Like, we can't just arbitrarily say, okay, the sum of three two and pi e is zero zero.
It's not. The sum of three two and minus three two is zero zero. So -- oh, I'm not going to -- the book, actually, lists the eight rules that the addition and multiplication have to satisfy, but they do.
They certainly satisfy it in R^n and usually it's not those eight rules that are in doubt. What's -- the question is, can we do those additions and do we stay in the space? Let me show you a case where you can't.
So suppose this is going to be not a vector space.
Suppose I take the xy plane -- so there's R^2.
That is a vector space. Now suppose I just take part of it. Just this.
Just this one -- this is one quarter of the vector space.
All the vectors with positive or at least not negative components. Can I add those safely? Yes. If I add a vector with, like, two -- three two to another vector like five six, I'm still up in this quarter, no problem with adding.
But there's a heck of a problem with multiplying by scalers, because there's a lot of scalers that will take me out of this quarter plane, like negative ones.
If I took three two and I multiplied by minus five, I'm way down here. So that's not a vector space, because it's not -- closed is the right word. It's not closed under multiplication by all real numbers. So a vector space has to be closed under multiplication and addition of vectors.
In other words, linear combinations.
It -- so, it means that if I give you a few vectors -- yeah look, here's an important -- here -- now we're getting to some really important vector spaces.
Well, R^n -- like, they are the most important.
But we will be interested in so- in vector spaces that are inside R^n. Vector spaces that follow the rules, but they -- we don't need all of -- see, there we started with R^2 here, and took part of it and messed it up. What we got was not a vector space. Now tell me a vector space that is part of R^2 and is still safely -- we can multiply, we can add and we stay in this smaller vector space. So it's going to be called a subspace. So I'm going to change this bad example to a good one. Okay.
So I'm going to start again with R^2, but I'm going to take an example -- it is a vector space, so it'll be a vector space inside R^2. And we'll call that a subspace of R^2.
Okay. What can I do? It's got something in it. Suppose it's got this vector in it. Okay.
If that vector's in my little subspace and it's a true subspace, then there's got to be some more in it, right? I have to be able to multiply that by two, and that double vector has to be included.
Have to be able to multiply by zero, that vector, or by half, or by three quarters.
All these vectors. Or by minus a half, or by minus one. I have to be able to multiply by any number. So that is going to say that I have to have that whole line. Do you see that? Once I get a vector in there -- I've got the whole line of all multiples of that vector. I can't have a vector space without extending to get those multiples in there.
Now I still have to check addition.
But that comes out okay. This line is going to work, because I could add something on the line to something else on the line and I'm still on the line.
So, example. So this is all examples of a subspace -- our example is a line in R^2 actually -- not just any line. If I took this line, would that -- so all the vectors on that line.
So that vector and that vector and this vector and this vector -- in lighter type, I'm drawing something that doesn't work. It's not a subspace.
The line in R^2 -- to be a subspace, the line in R^2 must go through the zero vector. Because -- why is this line no good? Let me do a dashed line.
Because if I multiplied that vector on the dashed line by zero, then I'm down here, I'm not on the dashed line.
Z- zero's got to be. Every subspace has got to contain zero -- because I must be allowed to multiply by zero and that will always give me the zero vector.
Okay. Now, I was going to make -- create some subspaces. Oh, while I'm in R^2, why don't we think of all the possibilities.
R two, there can't be that many.
So what are the possible subspaces of R^2? Let me list them. So I'm listing now the subspaces of R^2. And one possibility that we always allow is all of R two, the whole thing, the whole space. That counts as a subspace of itself. You always want to allow that.
Then the others are lines -- any line, meaning infinitely far in both directions through the zero.
So that's like the whole space -- that's like whole two D space. This is like one dimension.
Is this line the same as R^1 ? No. You could say it looks a lot like R^1. R^1 was just a line and this is a line.
But this is a line inside R^2. The vectors here have two components. So that's not the same as R^1, because there the vectors only have one component.
Very close, you could say, but not the same.
Okay. And now there's a third possibility. There's a third subspace that's -- of R^2 that's not the whole thing, and it's not a line.
It's even less. It's just the zero vector alone. The zero vector alone, only. I'll often call this subspace Z, just for zero. Here's a line, L. Here's a plane, all of R^2. So, do you see that the zero vector's okay? You would just -- to understand subspaces, we have to know the rules -- and knowing the rules means that we have to see that yes, the zero vector by itself, just this guy alone satisfies the rules. Why's that? Oh, it's too dumb to tell you. If I took that and added it to itself, I'm still there. If I took that and multiplied by seventeen, I'm still there.
So I've done the operations, adding and multiplying by numbers, that are required, and I didn't go outside this one point space. So that's always -- that's the littlest subspace. And the largest subspace is the whole thing and in-between come all -- whatever's in between.
Okay. So for example, what's in between for R^3? So if I'm in ordinary three dimensions, the subspace is R, all of R^3 at one extreme, the zero vector at the bottom. And then a plane, a plane through the origin. Or a line, a line through the origin. So with R^3, the subspaces were R^3, plane through the origin, line through the origin and a zero vector by itself, zero zero zero, just that single vector.
Okay, you've got the idea. But, now comes -- the reality is -- what are these -- where do these subspaces come -- how do they come out of matrices? And I want to take this matrix -- oh, let me take that matrix. So I want to create some subspaces out of that matrix. Well, one subspace is from the columns. Okay.
So this is the important subspace, the first important subspace that comes from that matrix -- I'm going to -- let me call it A again. Back to -- okay.
I'm looking at the columns of A.
Those are vectors in R^3. So the columns are in R^3.
The columns are in R^3. So I want those columns to be in my subspace. Now I can't just put two columns in my subspace and call it a subspace.
What do I have to throw in -- if I'm going to put those two columns in, what else has got to be there to have a subspace? I must be able to add those things.
So the sum of those columns -- so these columns are in R^3, and I have to be able -- I'm, you know, I want that to be in my subspace, I want that to be in my subspace, but therefore I have to be able to multiply them by anything.
Zero zero zero has got to be in my subspace.
I have to be able to add them so that four five five is in the subspace. I've got to be able to add one of these plus three of these. That'll give me some other vector. I have to be able to take all the linear combinations. So these are columns in R^3 and all there linear combinations form a subspace. What do I mean by linear combinations? I mean multiply that by something, multiply that by something and add. The two operations of linear algebra, multiplying by numbers and adding vectors.
And, if I include all the results, then I'm guaranteed to have a subspace. I've done the job.
And we'll give it a name -- the column space.
Column space. And maybe I'll call it C of A.
C for column space. There's an idea there that -- Like, the central idea for today's lecture is -- got a few vectors. Not satisfied with a few vectors, we want a space of vectors. The vectors, they're in -- these vectors in -- are in R^3 , so our space of vectors will be vectors in R^3. The key idea's -- we have to be able to take their combinations.
So tell me, geometrically, if I drew all these things -- like if I drew one two four, that would be somewhere maybe there. If I drew three three one, who knows, might be -- I don't know, I'll say there.
There's column one, there's column two.
What else -- what's in the whole column space? How do I draw the whole column space now? I take all combinations of those two vectors.
Do I get -- well, I guess I actually listed the possibilities. Do I get the whole space? Do I get a plane? I get more than a line, that's for sure. And I certainly get more than the zero vector, but I do get the zero vector included. What do I get if I combine -- take all the combinations of two vectors in R^3 ? So I've got all this stuff on -- that whole line gets filled out, that whole line gets filled out, but all in-between gets filled out -- between the two lines because I -- I allowed to add something from one line, something from the other. You see what's coming? I'm getting a plane. That's my -- and it's through the origin. Those two vectors, namely one two four and three three one, when I take all their combinations, I fill out a whole plane. Please think about that. That's the picture you have to see. You sure have to see it in R^3 , because we're going to do it in R^10, and we may take a combination of five vectors in R^10, and what will we have? God knows. It's some subspace.
We'll have five vectors. They'll all have ten components. We take their combinations. We don't have R^5 , because our vectors have ten components. And we possibly have, like, some five dimensional flat thing going through the origin for sure. Well, of course, if those five vectors were all on the line, then we would only get that line. So, you see, there are, like, other possibilities here.
It depends what -- it depends on those five vectors. Just like if our two columns had been on the same line, then the column space would have been only a line. Here it was a plane.
Okay. I'm going to stop at that point. That's the central idea of -- the great example of how to create a subspace from a matrix.
Take its columns, take their combinations, all their linear combinations and you get the column space.
And that's the central sort of -- we're looking at linear algebra at a higher level. When I look at A -- now, I want to look at Ax=b. That'll be the first thing in the next lecture. How do I understand Ax=b in this language -- in this new language of vector spaces and column spaces. And what are other subspaces? So the column space is a big one, there are others to come |
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This new title in Barron's E-Z Series contains everything students need to prepare themselves for an algebra class. Separate chapters focus on fractions, integers, ratios, proportions, expressions, equations, inequalities, graphing, statistics and probability basics, word problems, and more. Review questions and chapter reviews all have answers. The fast-growing E-Z Series presents new, updated, and improved versions of Barron's longtime popular Easy Way books. New cover designs, new interior layouts, and more graphic material than ever make these books ideal as self-teaching manuals. Teachers have discovered that E-Z titles also make excellent supplements to classroom textbooks. Skill levels range between senior high school and college-101 standards. All titles present detailed reviews of the target subject plus short quizzes and longer tests to help students assess their learning progress. All exercises and tests come with answers.
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Pre-Algebra makes sense when you approach it the E-Z way! Open this book for a clear, concise, step-by-step review of:
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The Kids biggest best book on additions. The book is specifically made for kids with a several flash card type questions that make the kids easily learn mathematics. Also will help as a great time pass book.
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This eBook reviews simultaneous equations and inequalities. We introduce simultaneous equations as systems of equations, and consider some relatively simple pairs of simultaneous equations, one pair involving a pair of linear equations, and another pair involving one linear equation and one quadratic equation. We go on to introduce the two methods of solving simultaneous equations, elimim. ...
This eBook introduces the subjects of co-ordinate geometry and graphs, ranging from finding the equations of the straight-line joining two points for which the co-ordinates are known, to calculating both the mid-point and length of a line between two known co-ordinates to plotting equations of the form y = kxn where n is even or odd for various values of k, as well as y = k√(x) where x is ..
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This eBook introduces the process of solving equations and rearranging formulas using elementary algebraic manipulation. We use examples that illustrate the process of solving an equation as well as the process of rearranging an equation, as well as set a number of challenging questions.Sixth Grade Math (For Home School or Extra Practice)"; it more thoroughly covers more fifthFifth Grade Math (For Home School or Extra Practice)"; it more thoroughly covers more fifth grade topics to help your child get a better understanding of fifth |
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Mathematical Applications and Modelling is the second in the series of the yearbooks of the Association of Mathematics Educators in Singapore. The book is unique as it addresses a focused theme on mathematics education. The objective is to illustrate the diversity within the theme and present research that translates into classroom pedagogies.The book, comprising of 17 chapters, illuminates how application and modelling tasks may help develop the capacity of students to use mathematics in their present and future lives. Several renowned international researchers in the field of mathematical modelling have published their work in the book. The chapters are comprehensive and laden with evidence-based examples for both mathematics educators and classroom teachers. The book is an invaluable contribution towards the emerging field of research in mathematical applications and modelling. It is a must-read for graduate research students and mathematics educators. less |
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Many mathematical statistics texts are heavily oriented toward a rigorous mathematical development of probability and statistics, without much attention paid to how statistics is actually used.. In contrast, Modern Mathematical Statistics with Applications, Second Edition strikes a balance between mathematical foundations and statistical practice.
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PROFESSOR STRANG: OK,
so this is the start.
10
00:00:23 --> 00:00:27
I won't be able to do it all in
one day of what I think of as
11
00:00:27 --> 00:00:35
the number one model in
applied math, in discrete
12
00:00:35 --> 00:00:37
implied math, I'll say.
13
00:00:37 --> 00:00:41
Let me review what our
four examples are.
14
00:00:41 --> 00:00:43
Just so you see
the big picture.
15
00:00:43 --> 00:00:48
So the first example was
the springs and masses.
16
00:00:48 --> 00:00:49
That was beautiful.
17
00:00:49 --> 00:00:50
It's simple.
18
00:00:50 --> 00:00:56
The masses are all in a line,
and the matrix K, the
19
00:00:56 --> 00:01:01
free-fixed and fixed-fixed and
free-free come out closely
20
00:01:01 --> 00:01:05
related to our K t b matrices.
21
00:01:05 --> 00:01:09
So that was the natural place
to start, and actually we also
22
00:01:09 --> 00:01:15
got a chance to do the most
important equation in time.
23
00:01:15 --> 00:01:15
Ku''.
24
00:01:17 --> 00:01:17
Sorry Mu''+Ku=0.
25
00:01:17 --> 00:01:20
26
00:01:20 --> 00:01:23
So that was a key example.
27
00:01:23 --> 00:01:26
Then least squares.
28
00:01:26 --> 00:01:29
Very important, I'm already
getting questions from the
29
00:01:29 --> 00:01:34
class about problems that
come up in your work,
30
00:01:34 --> 00:01:37
least square problems.
31
00:01:37 --> 00:01:45
Maybe I'll just mention that
the professional numerical
32
00:01:45 --> 00:01:50
guys don't always go
to A transpose A.
33
00:01:50 --> 00:01:53
If it's a badly conditioned
problem, in that conditioning
34
00:01:53 --> 00:01:58
is a topic that was in 1.7 and
we'll eventually come back to,
35
00:01:58 --> 00:02:02
if it's a badly conditioned
problem matrix a then, a
36
00:02:02 --> 00:02:05
transpose a kind of
makes it worse.
37
00:02:05 --> 00:02:09
So there's another way to
orthogonalize in advance.
38
00:02:09 --> 00:02:10
And if you're working with
orthogonal vectors, or
39
00:02:10 --> 00:02:17
orthonormal vectors, numerical
calculations are as
40
00:02:17 --> 00:02:20
safe as they can be.
41
00:02:20 --> 00:02:21
Yeah.
42
00:02:21 --> 00:02:24
Wall Street is more
like A transpose A.
43
00:02:24 --> 00:02:31
And the orthonormal
is the safe way.
44
00:02:31 --> 00:02:36
Alright, this is
today's lecture.
45
00:02:36 --> 00:02:39
You'll see the matrix a for
a graph, for a network.
46
00:02:39 --> 00:02:46
It's simple to construct, and
it just shows up everywhere.
47
00:02:46 --> 00:02:48
Because networks
are everywhere.
48
00:02:48 --> 00:02:56
And, just, looking ahead,
trusses are there partly
49
00:02:56 --> 00:02:58
because they're the most fun.
50
00:02:58 --> 00:02:59
You'll enjoy trusses.
51
00:02:59 --> 00:03:02
I mean, it's kind of fun to
figure out is the truss
52
00:03:02 --> 00:03:04
going to collapse or not.
53
00:03:04 --> 00:03:05
It's good.
54
00:03:05 --> 00:03:09
And actually, what's the
linear algebra in there?
55
00:03:09 --> 00:03:14
The collapsing or not will
depend on solutions to Au=0.
56
00:03:15 --> 00:03:19
Let me just recall
the equation Au=0.
57
00:03:21 --> 00:03:26
If A is our key matrix
in each example, it's
58
00:03:26 --> 00:03:28
different in each example.
59
00:03:28 --> 00:03:34
And we sort of hope that Au=0
doesn't have solutions, or that
60
00:03:34 --> 00:03:36
it has solutions we know.
61
00:03:36 --> 00:03:42
Because if Au=0 has solutions
that's the case where a
62
00:03:42 --> 00:03:48
transpose a is not invertable
and we have to do something.
63
00:03:48 --> 00:03:49
Very useful to review.
64
00:03:49 --> 00:03:56
What were the solutions to
Au=0, in the case of springs?
65
00:03:56 --> 00:04:00
Well, there were some
in the free-free case.
66
00:04:00 --> 00:04:04
The all ones vector was the
solution u or all constant, was
67
00:04:04 --> 00:04:07
the solution u in the free-free
case and that's why we
68
00:04:07 --> 00:04:09
couldn't invert it.
69
00:04:09 --> 00:04:12
But the fixed-free or the
fixed-fixed, when we have one
70
00:04:12 --> 00:04:17
support or two supports, that
removed the all ones solution.
71
00:04:17 --> 00:04:20
Good.
72
00:04:20 --> 00:04:23
These squares, we assume
there weren't any.
73
00:04:23 --> 00:04:27
We assumed because we wanted to
work directly with a transpose
74
00:04:27 --> 00:04:32
a, the normal equations, so we
assumed that the columns
75
00:04:32 --> 00:04:34
of a were independent.
76
00:04:34 --> 00:04:39
We assumed that there were no
non-zero solutions to Au=0.
77
00:04:40 --> 00:04:43
Because if there were, that
would have made a transpose a
78
00:04:43 --> 00:04:48
singular, and we would have had
to do something different.
79
00:04:48 --> 00:04:52
Here, this'll be a
lot like this one.
80
00:04:52 --> 00:04:57
Today, once you see A, you'll
spot the solutions to Au=0.
81
00:04:59 --> 00:05:03
This is A for a network.
82
00:05:03 --> 00:05:07
And the solution is going to
be that same guy, all ones.
83
00:05:07 --> 00:05:11
And that only tells us again
that we have to ground a node,
84
00:05:11 --> 00:05:14
I may use an electrical term.
85
00:05:14 --> 00:05:18
Grounding a node is like
fixing a displacement.
86
00:05:18 --> 00:05:22
Once you've fixed one of those,
say at zero, whatever, but
87
00:05:22 --> 00:05:24
zero's the natural choice.
88
00:05:24 --> 00:05:27
Once you've said one of the
potentials, one of the
89
00:05:27 --> 00:05:30
voltages is zero, then
you know all the rest.
90
00:05:30 --> 00:05:33
You can find all the rest
from our equations.
91
00:05:33 --> 00:05:37
So this is like this in having
this all ones solution.
92
00:05:37 --> 00:05:42
And as you'll see with trusses,
that could, depending on the
93
00:05:42 --> 00:05:44
truss, have more solutions.
94
00:05:44 --> 00:05:46
And if there are more
solutions that's when
95
00:05:46 --> 00:05:49
the truss collapses.
96
00:05:49 --> 00:05:54
So the trusses need more than
just a single support to
97
00:05:54 --> 00:05:58
hold up a whole truss.
98
00:05:58 --> 00:06:00
OK.
99
00:06:00 --> 00:06:01
So that's the Au=0.
100
00:06:02 --> 00:06:06
Now we're ready for
the lecture itself.
101
00:06:06 --> 00:06:09
Graphs and networks.
102
00:06:09 --> 00:06:12
OK, let me start with,
what's a graph.
103
00:06:12 --> 00:06:18
A graph is a bunch of nodes
and some or all of the
104
00:06:18 --> 00:06:20
edges between them.
105
00:06:20 --> 00:06:23
Let me take just a particular
example of a graph.
106
00:06:23 --> 00:06:26
And this of course you
spot in the book.
107
00:06:26 --> 00:06:30
Oh and everybody recognized
that, and it's probably now
108
00:06:30 --> 00:06:34
corrected, that in the
homework where it said 3.4
109
00:06:34 --> 00:06:37
it meant 2.4, of course.
110
00:06:37 --> 00:06:41
And this is Section 2.4 now.
111
00:06:41 --> 00:06:44
Let me draw a different graph.
112
00:06:44 --> 00:06:49
Maybe it'll have four nodes,
at those four edges, let
113
00:06:49 --> 00:06:51
me put in a fifth edge.
114
00:06:51 --> 00:06:54
OK, that's a graph.
115
00:06:54 --> 00:06:56
It's not a complete graph
because I didn't include
116
00:06:56 --> 00:06:58
that extra edge.
117
00:06:58 --> 00:07:03
It's not a tree because
there are some loops here.
118
00:07:03 --> 00:07:08
So complete graphs are
one extreme where all
119
00:07:08 --> 00:07:12
the edges are in.
120
00:07:12 --> 00:07:15
A tree is the other extreme,
where you have a minimum
121
00:07:15 --> 00:07:16
number of edges.
122
00:07:16 --> 00:07:19
It would only take
probably three edges.
123
00:07:19 --> 00:07:23
So just while we're looking
at it, there are a bunch of
124
00:07:23 --> 00:07:28
possible trees that would be
sort of inside this graph.
125
00:07:28 --> 00:07:32
Sub-graphs of this graph,
if I knock out those two
126
00:07:32 --> 00:07:35
edges I have a tree.
127
00:07:35 --> 00:07:39
Going out, or a tree
could be like this.
128
00:07:39 --> 00:07:41
Or a tree could be like this.
129
00:07:41 --> 00:07:48
Anyway, five edges is in this
graph, six in a complete
130
00:07:48 --> 00:07:51
graph, it would be
three edges in a tree.
131
00:07:51 --> 00:07:55
OK, and the number of
edges is always m.
132
00:07:55 --> 00:07:59
So five edges.
133
00:07:59 --> 00:08:04
And the number of nodes
is always n, for nodes.
134
00:08:04 --> 00:08:07
So a will be five by four.
135
00:08:07 --> 00:08:10
OK.
136
00:08:10 --> 00:08:15
And it's called, so we get a
special name in this world,
137
00:08:15 --> 00:08:19
it's called the incidence
matrix of the graph.
138
00:08:19 --> 00:08:21
The incidence matrix.
139
00:08:21 --> 00:08:25
Or, of course, these things
come up so often they
140
00:08:25 --> 00:08:26
have other names, too.
141
00:08:26 --> 00:08:32
But incidence matrix is
a pretty general name.
142
00:08:32 --> 00:08:35
OK, I have to number the
nodes just so we can
143
00:08:35 --> 00:08:37
create the matrix, A
one, two, three, four.
144
00:08:38 --> 00:08:40
And I have to number the edges.
145
00:08:40 --> 00:08:42
If I don't number them, I
don't know which is which.
146
00:08:42 --> 00:08:48
So let me call this edge one,
from one to two, and I'll
147
00:08:48 --> 00:08:51
draw an arrow on the edges.
148
00:08:51 --> 00:08:55
So from one to two, maybe
this'll be edge two,
149
00:08:55 --> 00:08:57
from one to three.
150
00:08:57 --> 00:08:58
This'll be edge three.
151
00:08:58 --> 00:09:01
Oh no, let me put edge three
there, would be a natural
152
00:09:01 --> 00:09:03
one, say from two to three.
153
00:09:03 --> 00:09:07
And how about edge four
there, from two to four.
154
00:09:07 --> 00:09:11
And edge five going
from three to four.
155
00:09:11 --> 00:09:18
OK, so now I have numbered,
I've identified the nodes, and
156
00:09:18 --> 00:09:20
I've identified the edges.
157
00:09:20 --> 00:09:23
And there were five
edges and four nodes.
158
00:09:23 --> 00:09:25
Usually m is bigger than n.
159
00:09:25 --> 00:09:30
We're in this, except
for trees, m will be at
160
00:09:30 --> 00:09:34
least as large as n.
161
00:09:34 --> 00:09:37
And I've put arrows on,
so you could say it's
162
00:09:37 --> 00:09:38
a directed graph.
163
00:09:38 --> 00:09:40
Because I've given a direction.
164
00:09:40 --> 00:09:45
You'll see that the directions,
those arrow directions, which
165
00:09:45 --> 00:09:51
are just to tell me which way
current should count as plus,
166
00:09:51 --> 00:09:53
if it's with the arrow, or
which way it should
167
00:09:53 --> 00:09:55
count as minus if it's
against the arrow.
168
00:09:55 --> 00:09:58
Of course, current
could go either way.
169
00:09:58 --> 00:10:01
It's just, now I have a
convention of which is
170
00:10:01 --> 00:10:03
plus and which is minus.
171
00:10:03 --> 00:10:06
OK, so now let me tell you
the incidence matrix.
172
00:10:06 --> 00:10:11
So everybody can get it right
away, how do you create
173
00:10:11 --> 00:10:12
this incidence matrix?
174
00:10:12 --> 00:10:15
A five by four.
175
00:10:15 --> 00:10:18
So it's going to have five
rows, one for every edge.
176
00:10:18 --> 00:10:22
So what's the row for edge one?
177
00:10:22 --> 00:10:25
And it's got four columns,
one for every node.
178
00:10:25 --> 00:10:27
So these are the nodes.
179
00:10:27 --> 00:10:32
Nodes one, two, three, four.
180
00:10:32 --> 00:10:36
So there's a column for every
node and a row for every edge.
181
00:10:36 --> 00:10:39
OK, edge one.
182
00:10:39 --> 00:10:43
This is just going to tell me
everything about the graph.
183
00:10:43 --> 00:10:47
So exactly what's in that
picture will be in this matrix.
184
00:10:47 --> 00:10:49
If I've erased one, I
could reproduce it's by
185
00:10:49 --> 00:10:51
knowing the other one.
186
00:10:51 --> 00:10:55
OK, edge one goes from
node one to node two.
187
00:10:55 --> 00:10:59
So it leaves node one, I'll
put a minus one there.
188
00:10:59 --> 00:11:00
In the first column.
189
00:11:00 --> 00:11:05
And a plus one in
the second column.
190
00:11:05 --> 00:11:09
Edge one doesn't touch
nodes three and four.
191
00:11:09 --> 00:11:11
So there you go,
that's edge one.
192
00:11:11 --> 00:11:15
Let me do edge two and
then you'll be able
193
00:11:15 --> 00:11:16
to fill in the rest.
194
00:11:16 --> 00:11:21
So edge two goes from one to
three, minus one, and a one.
195
00:11:21 --> 00:11:27
Edge three goes from two to
three, I'll just keep going.
196
00:11:27 --> 00:11:29
Minus one and a one.
197
00:11:29 --> 00:11:35
Edge four goes
from two to four.
198
00:11:35 --> 00:11:43
And edge five goes
from three to four.
199
00:11:43 --> 00:11:44
OK.
200
00:11:44 --> 00:11:46
Simple, right?
201
00:11:46 --> 00:11:47
Got it.
202
00:11:47 --> 00:11:50
That matrix has got all the
information that's in my
203
00:11:50 --> 00:11:56
picture, and the matrix, but
the point about matrices
204
00:11:56 --> 00:11:58
is, they do something.
205
00:11:58 --> 00:12:02
They multiply vector u
to produce something.
206
00:12:02 --> 00:12:07
They have a meaning beyond
just a record of the picture.
207
00:12:07 --> 00:12:09
So a is a great thing.
208
00:12:09 --> 00:12:11
In fact, what does it do?
209
00:12:11 --> 00:12:12
Let's see.
210
00:12:12 --> 00:12:15
So that's the matrix
a that we work with.
211
00:12:15 --> 00:12:18
Oh, first tell me about Au=0.
212
00:12:18 --> 00:12:22
Because we brought up
that subject already?
213
00:12:22 --> 00:12:26
Are those four
columns independent?
214
00:12:26 --> 00:12:30
I've got four columns, they're
sitting in five-dimensional
215
00:12:30 --> 00:12:32
space, there's plenty
of room there for four
216
00:12:32 --> 00:12:34
independent vectors.
217
00:12:34 --> 00:12:37
Are these four columns
independent vectors?
218
00:12:37 --> 00:12:38
No.
219
00:12:38 --> 00:12:39
No, they're not.
220
00:12:39 --> 00:12:42
Because what combination
of them produces the zero
221
00:12:42 --> 00:12:45
vector? .
222
00:12:45 --> 00:12:48
If I take that column plus
that, plus that, plus
223
00:12:48 --> 00:12:50
that, I'm multiplying by.
224
00:12:50 --> 00:12:55
So, A I'll just put that
up here and then I won't
225
00:12:55 --> 00:12:56
have to write it again.
226
00:12:56 --> 00:13:05
A times <1, 1, 1,
1>, is five zeroes.
227
00:13:05 --> 00:13:15
So that u, that particular u,
of all ones is, I would say, in
228
00:13:15 --> 00:13:17
the null space of the matrix?
229
00:13:17 --> 00:13:20
The null space is all
the solutions at Au=0.
230
00:13:21 --> 00:13:25
In other words, so these
four columns, tell me
231
00:13:25 --> 00:13:26
about the geometry again.
232
00:13:26 --> 00:13:31
These four columns, if I take
all their combinations, yeah.
233
00:13:31 --> 00:13:32
Think about this.
234
00:13:32 --> 00:13:36
If I take all four
combinations, all combination,
235
00:13:36 --> 00:13:38
any amount of this column, this
column, this column, that
236
00:13:38 --> 00:13:41
fourth column, those
are all vectors in
237
00:13:41 --> 00:13:44
five-dimensional space.
238
00:13:44 --> 00:13:47
Now, this isn't essential
but it's good.
239
00:13:47 --> 00:13:50
Do you have an idea
of what you'd get?
240
00:13:50 --> 00:13:55
What would you get if you took,
so this, think of four vectors,
241
00:13:55 --> 00:13:58
pointing along, take all their
combinations, that
242
00:13:58 --> 00:14:00
kind of fills in.
243
00:14:00 --> 00:14:02
Whatever fill in may mean.
244
00:14:02 --> 00:14:04
And what does it fill in?
245
00:14:04 --> 00:14:06
What do I get?
246
00:14:06 --> 00:14:08
What's your image?
247
00:14:08 --> 00:14:09
Frankly, I don't know.
248
00:14:09 --> 00:14:13
I can't visualize
five-dimensional space.
249
00:14:13 --> 00:14:14
That well.
250
00:14:14 --> 00:14:18
But still, we can use words.
251
00:14:18 --> 00:14:21
What do you think?
252
00:14:21 --> 00:14:23
You get a something subspace.
253
00:14:23 --> 00:14:28
You got a something, you
get something flat.
254
00:14:28 --> 00:14:30
I don't know if you do.
255
00:14:30 --> 00:14:32
It's pretty flat, somehow.
256
00:14:32 --> 00:14:37
Like I'm just asking you to
jump up from a case we know.
257
00:14:37 --> 00:14:42
Where we had columns in
three-dimensional space and
258
00:14:42 --> 00:14:44
we took a combination and
they gave us a plane.
259
00:14:44 --> 00:14:47
Right, when they
were dependent?
260
00:14:47 --> 00:14:52
Now, how would you visualize
the combinations in
261
00:14:52 --> 00:14:53
five-dimensional space?
262
00:14:53 --> 00:14:56
Just for the heck of it?
263
00:14:56 --> 00:15:00
It's some kind of a
subspace, I would say.
264
00:15:00 --> 00:15:03
And what's its dimension, maybe
that's what I want to ask you.
265
00:15:03 --> 00:15:04
What's the dimension?
266
00:15:04 --> 00:15:07
Do I get, like, a
four-dimensional subspace of
267
00:15:07 --> 00:15:11
five-dimensional space when I
take the combinations of these
268
00:15:11 --> 00:15:13
particular four guys?
269
00:15:13 --> 00:15:15
Yes or no?
270
00:15:15 --> 00:15:18
Do I get a four-dimensional
subspace, whatever
271
00:15:18 --> 00:15:19
that may mean?
272
00:15:19 --> 00:15:20
No.
273
00:15:20 --> 00:15:22
Right answer, I don't.
274
00:15:22 --> 00:15:23
I don't.
275
00:15:23 --> 00:15:27
Somehow the dimension of that
subspace, whatever I get, isn't
276
00:15:27 --> 00:15:32
four because this fourth guy is
not contributing anything new.
277
00:15:32 --> 00:15:35
The fourth one is a combination
of the first three.
278
00:15:35 --> 00:15:37
So I get a three-dimensional
subspace.
279
00:15:37 --> 00:15:41
The rank of this
matrix is three.
280
00:15:41 --> 00:15:49
If you allow me to introduce
that key word, rank, is the
281
00:15:49 --> 00:15:52
number of independent columns.
282
00:15:52 --> 00:15:58
It tells you how big
the matrix really is.
283
00:15:58 --> 00:16:01
You know, if the matrix, if I
pile on a whole lot of zero
284
00:16:01 --> 00:16:07
columns, or a lot of zero rows,
the matrix looks bigger.
285
00:16:07 --> 00:16:10
But of course it
isn't truly bigger.
286
00:16:10 --> 00:16:12
The heart of the matrix,
the core of the matrix
287
00:16:12 --> 00:16:15
is somehow just three.
288
00:16:15 --> 00:16:20
And actually, I tell you now
and we'll see it happen, can I
289
00:16:20 --> 00:16:25
tell you the key result in the
first half of linear algebra?
290
00:16:25 --> 00:16:27
It's this.
291
00:16:27 --> 00:16:29
That if I have three
independent columns, and by the
292
00:16:29 --> 00:16:33
way any three are independent,
it's just all four
293
00:16:33 --> 00:16:35
together are dependent.
294
00:16:35 --> 00:16:38
This has three independent
columns, then the great
295
00:16:38 --> 00:16:42
fact is, it has three
independent rows.
296
00:16:42 --> 00:16:44
That's kind of fantastic.
297
00:16:44 --> 00:16:49
Since it's such a beautiful
and remarkable and basic
298
00:16:49 --> 00:16:52
fact, look at the rows.
299
00:16:52 --> 00:16:55
That what linear
algebra is all about.
300
00:16:55 --> 00:16:59
Looking at a matrix by columns,
and then by rows, and seeing
301
00:16:59 --> 00:17:01
what are the connections.
302
00:17:01 --> 00:17:05
And the connection is, the key
connection is, that these
303
00:17:05 --> 00:17:09
five rows, now what
space are they in?
304
00:17:09 --> 00:17:15
What what space are these rows
in? four-dimensional space.
305
00:17:15 --> 00:17:17
They only have four components.
306
00:17:17 --> 00:17:24
So I had four columns in 5-D,
I have five rows in 4-D.
307
00:17:24 --> 00:17:27
But now, are those five
rows independent?
308
00:17:27 --> 00:17:29
Let me just ask that question.
309
00:17:29 --> 00:17:32
Are those five independent
rows, are they pointing in
310
00:17:32 --> 00:17:36
different directions, or could
any combination give the zero
311
00:17:36 --> 00:17:40
vector in 4-D, looking
at those five rows?
312
00:17:40 --> 00:17:43
What do you say, wait a minute.
313
00:17:43 --> 00:17:46
Five vectors, in
four-dimensional space?
314
00:17:46 --> 00:17:48
Dependent, of course.
315
00:17:48 --> 00:17:48
Right.
316
00:17:48 --> 00:17:51
So they're dependent.
317
00:17:51 --> 00:17:55
There couldn't be five
independent vectors in 4-D.
318
00:17:55 --> 00:18:00
But are there four in
this particular case?
319
00:18:00 --> 00:18:03
And here's the great fact,
no, there are three.
320
00:18:03 --> 00:18:08
If there are three independent
columns and no more, then there
321
00:18:08 --> 00:18:11
are three independent
rows and no more.
322
00:18:11 --> 00:18:15
And we'll get to see which
rows are independent.
323
00:18:15 --> 00:18:16
And which are not.
324
00:18:16 --> 00:18:20
That's a question about A
transpose, and we haven't
325
00:18:20 --> 00:18:22
got to A transpose yet.
326
00:18:22 --> 00:18:26
OK, are you OK with
that incidence matrix?
327
00:18:26 --> 00:18:36
Because this is like the
central matrix of our subject.
328
00:18:36 --> 00:18:41
We can figure out A transpose
A, that's kind of fun.
329
00:18:41 --> 00:18:46
I do a transpose a then you'll
see the core computations
330
00:18:46 --> 00:18:49
of this neat section.
331
00:18:49 --> 00:18:53
So if I do A transpose A, so
I'm going to bring in a
332
00:18:53 --> 00:18:57
transpose and you know that I'm
not just bringing it in from
333
00:18:57 --> 00:19:02
nowhere, that networks.
334
00:19:02 --> 00:19:05
The balance law is going
to involve a transpose.
335
00:19:05 --> 00:19:07
So let's just anticipate.
336
00:19:07 --> 00:19:10
What do you think a
transpose a looks like?
337
00:19:10 --> 00:19:12
Now, how am I going
to do this for you?
338
00:19:12 --> 00:19:17
May I write may I erase this
for a moment, and try to
339
00:19:17 --> 00:19:19
squeeze in a transpose here?
340
00:19:19 --> 00:19:26
So that you can multiply it by
site and see the answer, and
341
00:19:26 --> 00:19:28
then you'll see the pattern.
342
00:19:28 --> 00:19:31
That's the great
thing about math.
343
00:19:31 --> 00:19:35
You do a few examples, and
you hope that a pattern
344
00:19:35 --> 00:19:36
reveals itself.
345
00:19:36 --> 00:19:39
So let me show a transpose.
346
00:19:39 --> 00:19:43
So now I'm going to take that
column and make it a row.
347
00:19:43 --> 00:19:47
I'm going to take that column
and make it a row, it's going
348
00:19:47 --> 00:19:50
to be a little squeezed
but we can do it.
349
00:19:50 --> 00:19:56
Take that column,
.
350
00:19:56 --> 00:20:01
And the last column,
.
351
00:20:01 --> 00:20:02
OK.
352
00:20:02 --> 00:20:05
So I just wrote a
transpose here.
353
00:20:05 --> 00:20:10
And now could you help
me with A transpose A.
354
00:20:10 --> 00:20:14
Which is the key matrix
in the graph here.
355
00:20:14 --> 00:20:17
What size will it be?
356
00:20:17 --> 00:20:19
Everybody knows it's going to
be square, it's going to be
357
00:20:19 --> 00:20:23
symmetric, and just
tell me the size.
358
00:20:23 --> 00:20:23
Four by four.
359
00:20:23 --> 00:20:28
Right, we have a four by five
times a five by four, we're
360
00:20:28 --> 00:20:30
expecting this to
be four by four.
361
00:20:30 --> 00:20:33
And what's the first entry?
362
00:20:33 --> 00:20:35
Two.
363
00:20:35 --> 00:20:39
Right, take row one, dot
it with column one.
364
00:20:39 --> 00:20:44
I get two ones and then a bunch
of zeroes, so I just get a two.
365
00:20:44 --> 00:20:46
What's the next entry?
366
00:20:46 --> 00:20:48
Take row one against
column two, can you
367
00:20:48 --> 00:20:50
do that in your head?
368
00:20:50 --> 00:20:55
Row one, column two, the top
one is going to hit on a minus
369
00:20:55 --> 00:20:59
one, and I think that's
all there is, right?
370
00:20:59 --> 00:21:03
Then this one hits a zero
and those three zeroes, so.
371
00:21:03 --> 00:21:09
And then what about
the next guy here?
372
00:21:09 --> 00:21:10
A minus one.
373
00:21:10 --> 00:21:13
And the last guy?
374
00:21:13 --> 00:21:14
A zero.
375
00:21:14 --> 00:21:19
So that's row one
of A transpose A.
376
00:21:19 --> 00:21:23
Can I just look at that
for a moment before
377
00:21:23 --> 00:21:24
I fill in the rest?
378
00:21:24 --> 00:21:30
And then, when you fill in the
rest it'll confirm the idea.
379
00:21:30 --> 00:21:31
Why do I have a zero there?
380
00:21:31 --> 00:21:38
Why did a zero appear
in the 1, 4 position?
381
00:21:38 --> 00:21:41
If I look back at the graph,
what is it about nodes one
382
00:21:41 --> 00:21:44
and four that told
me ahead of time?
383
00:21:44 --> 00:21:49
You're going to get a zero
in that A transpose A.
384
00:21:49 --> 00:21:52
Everybody see what nodes
one and four are?
385
00:21:52 --> 00:21:54
Yeah, say it again.
386
00:21:54 --> 00:21:56
Not connected.
387
00:21:56 --> 00:21:58
No edge.
388
00:21:58 --> 00:22:00
Here there was an edge
from node one to two.
389
00:22:00 --> 00:22:03
Here is an edge from
node one to three.
390
00:22:03 --> 00:22:05
Those both produce
the minus ones.
391
00:22:05 --> 00:22:11
And on the diagonal came
the two to balance it.
392
00:22:11 --> 00:22:12
What does that two represent?
393
00:22:12 --> 00:22:16
That two represents the
number of edges that
394
00:22:16 --> 00:22:17
do go into node one.
395
00:22:17 --> 00:22:20
See, that row is all
about node one.
396
00:22:20 --> 00:22:24
So they're two edges into it,
and then an edge out, and an
397
00:22:24 --> 00:22:29
edge out, and the edge
out and the no edge.
398
00:22:29 --> 00:22:30
OK.
399
00:22:30 --> 00:22:34
So, now I know it's going to be
a symmetric matrix, so I could
400
00:22:34 --> 00:22:36
speed up and fill those in.
401
00:22:36 --> 00:22:38
What's the next entry here?
402
00:22:38 --> 00:22:41
What's the guy on
this diagonal?
403
00:22:41 --> 00:22:46
So that's row two against
column two, so I have a one
404
00:22:46 --> 00:22:50
there, a one there, a one
there, that makes a three.
405
00:22:50 --> 00:22:53
Why a three?
406
00:22:53 --> 00:22:55
Because there are,
yeah, you got it.
407
00:22:55 --> 00:23:04
There are three edges
into node number two.
408
00:23:04 --> 00:23:07
Three edges into node number
two, and now I'm going to have
409
00:23:07 --> 00:23:11
some minus ones off the
diagonal for those edges.
410
00:23:11 --> 00:23:16
So what are these entries
going to be here?
411
00:23:16 --> 00:23:18
They're both minus ones.
412
00:23:18 --> 00:23:22
Edge two is connected to
all three other nodes.
413
00:23:22 --> 00:23:25
So I'm going to see a minus one
and a minus one there, and
414
00:23:25 --> 00:23:27
it's going to be symmetric.
415
00:23:27 --> 00:23:32
And I'm nearly there.
416
00:23:32 --> 00:23:36
Of course, I'm describing a
pattern that you're just seeing
417
00:23:36 --> 00:23:42
unfold, but I'm doing it that
way so that you'll feel hey, I
418
00:23:42 --> 00:23:46
can write down a transpose a,
or check it quite quickly,
419
00:23:46 --> 00:23:51
without doing this complete
matrix multiplication.
420
00:23:51 --> 00:23:53
So what number goes there?
421
00:23:53 --> 00:23:57
That's to do with node three,
and I see node three connected
422
00:23:57 --> 00:24:02
to all three other nodes, and
so what do you expect there?
423
00:24:02 --> 00:24:04
Minus one there, and a
minus one there, and
424
00:24:04 --> 00:24:06
what do you expect here?
425
00:24:06 --> 00:24:07
Two.
426
00:24:07 --> 00:24:13
And so now I have my matrix.
427
00:24:13 --> 00:24:16
The a transpose a matrix.
428
00:24:16 --> 00:24:18
And that's square
and it's symmetric.
429
00:24:18 --> 00:24:22
Now I ask you, is it
positive definite?
430
00:24:22 --> 00:24:23
Or is it only semi-definite?
431
00:24:23 --> 00:24:27
Right, we know that A transpose
A is always positive
432
00:24:27 --> 00:24:29
definite in the best case.
433
00:24:29 --> 00:24:34
But only positive semi-definite
if it's singular, if there's
434
00:24:34 --> 00:24:38
some vector in its null space,
if a transpose a times
435
00:24:38 --> 00:24:40
some vector gives zero.
436
00:24:40 --> 00:24:42
If some combination of
those columns gives
437
00:24:42 --> 00:24:45
me the zero column.
438
00:24:45 --> 00:24:47
Which is it?
439
00:24:47 --> 00:24:52
Have I got a singular matrix
or an invertible matrix here?
440
00:24:52 --> 00:24:54
Singular.
441
00:24:54 --> 00:24:56
Why singular?
442
00:24:56 --> 00:24:59
Because a had some
solutions to Au=0.
443
00:25:00 --> 00:25:06
So if Au equaled zero, then I
could multiply both sides by a
444
00:25:06 --> 00:25:10
transpose, that same u, A
transpose times zero will still
445
00:25:10 --> 00:25:14
be zero, it might be a
different size zero,
446
00:25:14 --> 00:25:17
but it'll be zero.
447
00:25:17 --> 00:25:19
And what's the u, then?
448
00:25:19 --> 00:25:21
It's the all ones vector.
449
00:25:21 --> 00:25:27
What am I saying about the
columns of A transpose A?
450
00:25:27 --> 00:25:31
They're dependent.. they add
up because it's the <1, 1,
451
00:25:31 --> 00:25:34
1, 1> vector that's guilty.
452
00:25:34 --> 00:25:37
Every row adds to zero.
453
00:25:37 --> 00:25:40
Every row adds to zero.
454
00:25:40 --> 00:25:45
Let me just say for a moment,
introduce two notation
455
00:25:45 --> 00:25:50
for the diagonal matrix.
456
00:25:50 --> 00:25:54
D, that's the diagonal matrix,
two, three, three, two.
457
00:25:54 --> 00:25:57
And then I'll put in a
minus sign, and this
458
00:25:57 --> 00:26:02
is and I'll call it W.
459
00:26:02 --> 00:26:05
So you can pick out what
D and W are, but let
460
00:26:05 --> 00:26:07
me do it for sure.
461
00:26:07 --> 00:26:13
So D, the degree matrix,
OK this is this is like
462
00:26:13 --> 00:26:15
fun because I'm not
doing anything yet.
463
00:26:15 --> 00:26:17
I'm just giving names here.
464
00:26:17 --> 00:26:19
Two, three, three, two.
465
00:26:19 --> 00:26:22
The degree of a node, the
degree means how many
466
00:26:22 --> 00:26:25
edges go from it.
467
00:26:25 --> 00:26:26
How many edges touch it.
468
00:26:26 --> 00:26:33
And W is also a great
matrix, it's called
469
00:26:33 --> 00:26:42
the adjacency matrix.
470
00:26:42 --> 00:26:46
It's also beautiful.
471
00:26:46 --> 00:26:50
Now it'll have plus ones
because I wanted minus W, so it
472
00:26:50 --> 00:26:56
has, these nodes are not
adjacent to themselves but it's
473
00:26:56 --> 00:26:59
got this one and this one and
this one this one and that
474
00:26:59 --> 00:27:01
one, and that's a zero.
475
00:27:01 --> 00:27:07
So there are five, the
adjacency matrix tells me
476
00:27:07 --> 00:27:09
which nodes are connected
to which other nodes.
477
00:27:09 --> 00:27:13
And of course the connections
are going both ways.
478
00:27:13 --> 00:27:17
So I see five ones
from five edges.
479
00:27:17 --> 00:27:24
And I see five more ones below
the diagonal, because the edges
480
00:27:24 --> 00:27:27
are connecting both ways.
481
00:27:27 --> 00:27:30
Ones connected to three, and
three is connected to one.
482
00:27:30 --> 00:27:33
One is not connected to
four, and four is not
483
00:27:33 --> 00:27:34
connected to one.
484
00:27:34 --> 00:27:37
One is not connected to itself.
485
00:27:37 --> 00:27:38
By an edge.
486
00:27:38 --> 00:27:43
If we allowed, like, little
self loops, then a one could
487
00:27:43 --> 00:27:44
appear on the diagram.
488
00:27:44 --> 00:27:45
But we don't.
489
00:27:45 --> 00:27:48
OK, so that's D and W.
490
00:27:48 --> 00:27:49
Here are the key matrices.
491
00:27:49 --> 00:27:54
This is actually, I venture
to say that any afternoon at
492
00:27:54 --> 00:27:59
MIT there's a seminar that
involves these matrices.
493
00:27:59 --> 00:28:02
One name for this is
the graph, Laplacian.
494
00:28:02 --> 00:28:09
From Laplace's equation and
we'll see pretty soon where
495
00:28:09 --> 00:28:11
that name's coming from.
496
00:28:11 --> 00:28:12
But it's there.
497
00:28:12 --> 00:28:19
And should I think, I think I
should, just about networks.
498
00:28:19 --> 00:28:22
Like where, does the
networks come from?
499
00:28:22 --> 00:28:25
I think we've got
networks all around us.
500
00:28:25 --> 00:28:26
Right?
501
00:28:26 --> 00:28:33
Electrical networks are the
simplest, maybe in some ways
502
00:28:33 --> 00:28:36
the simplest to visualize.
503
00:28:36 --> 00:28:54
So that's the example, that's
the language I'll use.
504
00:28:54 --> 00:28:59
Now, I get a network, I use
the word network when there's
505
00:28:59 --> 00:29:01
a c_1, c_2, c_3, c_4, c_5.
506
00:29:03 --> 00:29:04
Those extra numbers.
507
00:29:04 --> 00:29:10
I've got the A, and now the
network comes from the c part.
508
00:29:10 --> 00:29:15
That diagonal matrix, and if
I'm talking electricity,
509
00:29:15 --> 00:29:17
these could be resistors.
510
00:29:17 --> 00:29:19
Status springs,
they're resistors.
511
00:29:19 --> 00:29:27
So it's the conductance in
those five resistors, are
512
00:29:27 --> 00:29:27
c_1, c_2, c_3, c_4, and c_5.
513
00:29:27 --> 00:29:31
514
00:29:31 --> 00:29:33
So I'm ready for that.
515
00:29:33 --> 00:29:36
Ready for the C matrix,
because we got the a matrix.
516
00:29:36 --> 00:29:44
And we've got A transpose A,
but the the applications
517
00:29:44 --> 00:29:47
throw in a c matrix also.
518
00:29:47 --> 00:29:51
What are other applications, I
was saying, like this one is
519
00:29:51 --> 00:29:56
the one, I'll use the word
current, for flow in the edges,
520
00:29:56 --> 00:29:59
or I'll use the word flow.
521
00:29:59 --> 00:30:04
A network of oil, or natural
gas, or water pipes would be
522
00:30:04 --> 00:30:13
just that, and then the
electrical people study.
523
00:30:13 --> 00:30:19
Professor Vergasian in Course
6 studies the electric grid.
524
00:30:19 --> 00:30:22
The US electric grid, or the
western, off in the western
525
00:30:22 --> 00:30:24
half of the US electric grid.
526
00:30:24 --> 00:30:27
So that's got a whole
lot of things.
527
00:30:27 --> 00:30:29
Pumping stations.
528
00:30:29 --> 00:30:30
You see it?
529
00:30:30 --> 00:30:35
Actually, the world wide web,
the internet, is a giant
530
00:30:35 --> 00:30:39
network that people would
love to understand.
531
00:30:39 --> 00:30:41
And the phone company would
love to understand those
532
00:30:41 --> 00:30:43
networks of phone calls.
533
00:30:43 --> 00:30:47
I mean, those are really,
that's what, giant businesses
534
00:30:47 --> 00:30:52
are are dependent on
understanding and
535
00:30:52 --> 00:30:55
maintaining networks.
536
00:30:55 --> 00:30:57
OK, so I'm going
to use resistors.
537
00:30:57 --> 00:31:00
Of course, I'm staying linear.
538
00:31:00 --> 00:31:04
And I'm staying steady state.
539
00:31:04 --> 00:31:06
So by staying linear there
aren't any transistors
540
00:31:06 --> 00:31:08
in this net.
541
00:31:08 --> 00:31:10
By staying steady state,
there aren't any
542
00:31:10 --> 00:31:12
capacitors or inductors.
543
00:31:12 --> 00:31:16
Those guys would be linear
elements, but they
544
00:31:16 --> 00:31:20
would be coming in a
time-dependent problem.
545
00:31:20 --> 00:31:22
A UTT problem.
546
00:31:22 --> 00:31:29
And I'm just staying now with
Ku=f, I'm trying to create K.
547
00:31:29 --> 00:31:32
The stiffness matrix, which
maybe here we might call
548
00:31:32 --> 00:31:35
the conductance matrix.
549
00:31:35 --> 00:31:38
OK, so ready for
the picture now?
550
00:31:38 --> 00:31:42
That these come into?
551
00:31:42 --> 00:31:47
You know what the picture looks
like, it's going to have the
552
00:31:47 --> 00:31:54
usual four, we'll start with
these potentials u at the
553
00:31:54 --> 00:32:01
nodes, potentials at nodes, so
those will be u_1,
554
00:32:01 --> 00:32:02
u_2, u_3, u_4.
555
00:32:02 --> 00:32:05
556
00:32:05 --> 00:32:11
Voltages, if I'm really
speaking, those units
557
00:32:11 --> 00:32:18
would be volts, and now
comes the matrix A.
558
00:32:18 --> 00:32:22
And now I get, what
do I get from A?
559
00:32:22 --> 00:32:24
What do I get from A?
560
00:32:24 --> 00:32:25
Key question.
561
00:32:25 --> 00:32:29
If I multiply A times u, and
you know that's coming, right?
562
00:32:29 --> 00:32:35
If I multiply A times u, so
I'll erase A transpose now,
563
00:32:35 --> 00:32:37
because we've got that.
564
00:32:37 --> 00:32:41
So there's A, and now I'll
make space to multiply
565
00:32:41 --> 00:32:45
by u, alright?
566
00:32:45 --> 00:32:48
So now I want to look at Au.
567
00:32:49 --> 00:32:53
So A multiplies a bunch
of potentials, a
568
00:32:53 --> 00:32:55
bunch of voltages.
569
00:32:55 --> 00:32:57
And let's just do this
multiplication and see
570
00:32:57 --> 00:32:59
what it produces.
571
00:32:59 --> 00:33:02
This is the great thing
about matrices, they
572
00:33:02 --> 00:33:05
produce something.
573
00:33:05 --> 00:33:08
OK, what's the first
component of Au?
574
00:33:10 --> 00:33:14
Of course, Au is going
to be five by five.
575
00:33:14 --> 00:33:17
It's going to be
associated with edges.
576
00:33:17 --> 00:33:20
Right, u's associated with
nodes, a u with edges.
577
00:33:20 --> 00:33:22
Just, the pattern is so nice.
578
00:33:22 --> 00:33:26
Alright, what's the
first component?
579
00:33:26 --> 00:33:30
Just do that multiplication
and what do you get? u_2-u_1.
580
00:33:30 --> 00:33:33
581
00:33:33 --> 00:33:36
What do you get in the
second component?
582
00:33:36 --> 00:33:38
Do that multiplication
and you get u_3-u_1.
583
00:33:41 --> 00:33:42
The third one will be u_3-u_2.
584
00:33:44 --> 00:33:48
The fourth one would be
u_4-u_2, and the fifth
585
00:33:48 --> 00:33:49
one will be u_4-u_3.
586
00:33:49 --> 00:33:56
587
00:33:56 --> 00:34:03
Just like our first
difference matrices.
588
00:34:03 --> 00:34:11
But this one deals with, I
mean, our first difference
589
00:34:11 --> 00:34:15
matrices were exactly like
this when the graph
590
00:34:15 --> 00:34:17
was all in a line.
591
00:34:17 --> 00:34:21
The big step now is that the
graph is not in a line, not
592
00:34:21 --> 00:34:24
even necessarily in a plane.
593
00:34:24 --> 00:34:29
Could be in, it's a bunch
of points, and edges.
594
00:34:29 --> 00:34:33
Actually, the position of those
points, we don't have to
595
00:34:33 --> 00:34:34
know are they in a plane.
596
00:34:34 --> 00:34:37
I think of them as
nodes and edges.
597
00:34:37 --> 00:34:44
OK, what's the
natural name for Au?
598
00:34:44 --> 00:34:46
I would call those potential
differences, right?
599
00:34:46 --> 00:34:48
Voltage differences.
600
00:34:48 --> 00:34:50
So that's what we see here and
those will be e. e_1, e_2, e_3,
601
00:34:50 --> 00:35:00
e_4, e_5 will be potential
or voltage differences.
602
00:35:00 --> 00:35:02
Voltage drops, you might say.
603
00:35:02 --> 00:35:05
Potential differences,
voltage drops.
604
00:35:05 --> 00:35:10
Oh well, now.
605
00:35:10 --> 00:35:15
When I say voltage drops,
that's because, as we noted
606
00:35:15 --> 00:35:21
before, the current goes from a
higher to a lower potential.
607
00:35:21 --> 00:35:24
It goes in the
direction of the drop.
608
00:35:24 --> 00:35:31
And I think that what we need
now is minus Au, for e.
609
00:35:31 --> 00:35:35
So I think we need a minus
sign and it's quite common
610
00:35:35 --> 00:35:36
to have the minus sign.
611
00:35:36 --> 00:35:40
We saw it already
with least squared.
612
00:35:40 --> 00:35:48
And let me say also,
so this is e.
613
00:35:48 --> 00:35:52
I'll abbreviate those always
five e's I just wrote
614
00:35:52 --> 00:35:53
down, five of them.
615
00:35:53 --> 00:35:55
So you would remember
there are five.
616
00:35:55 --> 00:35:57
We're talking about
the currents.
617
00:35:57 --> 00:36:01
We're talking about,
this is the e in e=IR.
618
00:36:03 --> 00:36:06
The voltage drop.
619
00:36:06 --> 00:36:08
That makes some current go.
620
00:36:08 --> 00:36:14
Now, also, just as with least
squares, so it was great that
621
00:36:14 --> 00:36:18
we saw it before, there could
be a source term here.
622
00:36:18 --> 00:36:24
So I'm completing the picture
here, allowing the source term.
623
00:36:24 --> 00:36:27
And we'll come back to what
does that mean, physically.
624
00:36:27 --> 00:36:32
But at that point could
enter b, and b is really
625
00:36:32 --> 00:36:35
standing for batteries.
626
00:36:35 --> 00:36:40
I work hard to make the
language match the initials.
627
00:36:40 --> 00:36:41
These letters.
628
00:36:41 --> 00:36:45
OK, now what?
629
00:36:45 --> 00:36:51
That step just involved
A, nothing physical.
630
00:36:51 --> 00:36:57
Now comes the step that
involves A, so w will be Ce.
631
00:36:57 --> 00:37:02
And these will be the
currents on the edges.
632
00:37:02 --> 00:37:09
And that's the law, then, with
a matrix C, of course C is
633
00:37:09 --> 00:37:12
our old friend c_1 to c_5.
634
00:37:14 --> 00:37:17
And tell me first, the name.
635
00:37:17 --> 00:37:19
Whose law is this?
636
00:37:19 --> 00:37:21
That the current is
proportional to
637
00:37:21 --> 00:37:22
the voltage drop?
638
00:37:22 --> 00:37:24
Ohm.
639
00:37:24 --> 00:37:28
So this is Ohm's law.
640
00:37:28 --> 00:37:30
Instead of Hooke's
law, it's Ohm's law.
641
00:37:30 --> 00:37:34
And I've written it with
conductances, not resistances.
642
00:37:34 --> 00:37:42
So resistances are 1/R, the
usual R in e=IR, would be, I'm
643
00:37:42 --> 00:37:48
more looking at it as i current
equals Ce, instead of e=IR.
644
00:37:49 --> 00:37:55
So I'm flipping the the the
resistance, or the impedance
645
00:37:55 --> 00:37:57
to give the conductance.
646
00:37:57 --> 00:38:05
OK, and now finally can
you tell me what the last
647
00:38:05 --> 00:38:08
step is going to be?
648
00:38:08 --> 00:38:15
If life is good, well you might
wonder whether life is good,
649
00:38:15 --> 00:38:21
reading the papers, but
it's still good here.
650
00:38:21 --> 00:38:23
OK, what matrix shows up there?
651
00:38:23 --> 00:38:25
Everybody knows it.
652
00:38:25 --> 00:38:26
A transpose.
653
00:38:26 --> 00:38:31
So the final equation, the
balance equation, will
654
00:38:31 --> 00:38:35
be, let me write it so I
don't catch it up here.
655
00:38:35 --> 00:38:39
Will be A transpose
w equals whatever.
656
00:38:39 --> 00:38:42
Will be the balance equation.
657
00:38:42 --> 00:38:45
The current balance, it's the
balance of currents, balance
658
00:38:45 --> 00:38:48
of charge, whatever
you like to say.
659
00:38:48 --> 00:38:51
At each node, it's the
balance at the nodes.
660
00:38:51 --> 00:38:55
Because when we're up
on this line, we're
661
00:38:55 --> 00:38:56
in the node picture.
662
00:38:56 --> 00:38:58
We have four equations
here, right?
663
00:38:58 --> 00:39:01
We're talking about
at each node.
664
00:39:01 --> 00:39:05
Here we're talking
about on each edge.
665
00:39:05 --> 00:39:06
There is so critical.
666
00:39:06 --> 00:39:09
These two variables.
667
00:39:09 --> 00:39:13
Which we're seeing physically
as node variables
668
00:39:13 --> 00:39:17
and edge variables.
669
00:39:17 --> 00:39:21
That pair of variables
just shows up everywhere.
670
00:39:21 --> 00:39:28
In displacements and stresses,
it's fundamental in elasticity.
671
00:39:28 --> 00:39:35
And oh, there are just so
many in optimization,
672
00:39:35 --> 00:39:36
it's everywhere.
673
00:39:36 --> 00:39:39
And a big part of this course
is to see it everywhere.
674
00:39:39 --> 00:39:46
OK, why don't I, just so
you see the main picture.
675
00:39:46 --> 00:39:51
We're going to have the A
transpose C A matrix that I'm
676
00:39:51 --> 00:39:54
going to maybe call K again.
677
00:39:54 --> 00:39:58
And now of course there
could be current sources.
678
00:39:58 --> 00:40:04
Just the way there could be
forces that we had to balance.
679
00:40:04 --> 00:40:09
There could be, not always
but there could be, current
680
00:40:09 --> 00:40:10
sources from outside.
681
00:40:10 --> 00:40:12
External current sources.
682
00:40:12 --> 00:40:15
So these are external
voltage sources.
683
00:40:15 --> 00:40:17
These are external
current sources.
684
00:40:17 --> 00:40:22
So in a way, we now have
combined our first two
685
00:40:22 --> 00:40:26
examples, our springs
and masses only had
686
00:40:26 --> 00:40:28
forces external.
687
00:40:28 --> 00:40:33
Our least squares problem
had an external b.
688
00:40:33 --> 00:40:34
Measurement.
689
00:40:34 --> 00:40:36
This picture is the whole deal.
690
00:40:36 --> 00:40:40
It's gotta b and f, and
actually I could put in
691
00:40:40 --> 00:40:46
even a little more.
692
00:40:46 --> 00:40:55
Sources like, well, we already
kind of caught on to the fact
693
00:40:55 --> 00:41:00
that we'd better ground the
node or A transpose C A as it
694
00:41:00 --> 00:41:03
stands, A transpose C A as
it stands will be singular.
695
00:41:03 --> 00:41:06
You know, it's the matrix,
there's A transpose A
696
00:41:06 --> 00:41:09
and the C in the middle
isn't going to help any.
697
00:41:09 --> 00:41:10
That's singular.
698
00:41:10 --> 00:41:17
If we wanted to be able to
compute voltages, we've
699
00:41:17 --> 00:41:19
got to set one of them.
700
00:41:19 --> 00:41:23
It's like setting one
temperature, it's like deciding
701
00:41:23 --> 00:41:24
where is absolute zero.
702
00:41:24 --> 00:41:28
Let's put absolute zero
down here. u_4=0.
703
00:41:29 --> 00:41:32
Grounded the node.
704
00:41:32 --> 00:41:36
OK, so we've fixed a potential.
705
00:41:36 --> 00:41:40
So here's a boundary
condition coming in u_4=0.
706
00:41:40 --> 00:41:43
707
00:41:43 --> 00:41:46
That's another source term,
another thing coming, you could
708
00:41:46 --> 00:41:50
say sort of from outside
the A transpose C A.
709
00:41:50 --> 00:41:54
We could fix another voltage
at, I mean, I'm thinking now
710
00:41:54 --> 00:41:58
about what's the picture.
711
00:41:58 --> 00:42:04
What's the whole problem?
712
00:42:04 --> 00:42:09
So the problem could have
batteries, in the edges.
713
00:42:09 --> 00:42:12
It could have current
sources into the nodes.
714
00:42:12 --> 00:42:19
It could fix u_1 at
some voltage like ten.
715
00:42:19 --> 00:42:20
Our problem could fix
716
00:42:20 --> 00:42:22
- we must fix one of them.
717
00:42:22 --> 00:42:28
Otherwise our matrix
isn't as singular.
718
00:42:28 --> 00:42:31
But once we've set up the
matrix, and when we fix
719
00:42:31 --> 00:42:35
u_4=0 by the way, what
happens to our matrix?
720
00:42:35 --> 00:42:39
Let me take u_4=0, so
this is a key step here.
721
00:42:39 --> 00:42:42
When I set u_4=0,
I now know u_4.
722
00:42:43 --> 00:42:45
It's not an unknown any more.
723
00:42:45 --> 00:42:52
So I've removed u_4
from the problem.
724
00:42:52 --> 00:42:55
And then it'll be also
removed from A transpose A.
725
00:42:55 --> 00:42:58
So this, is you could say,
like a reduced A, or
726
00:42:58 --> 00:43:00
a grounded matrix A.
727
00:43:00 --> 00:43:03
It's now five by three.
728
00:43:03 --> 00:43:05
And A transpose A,
what shape will the a
729
00:43:05 --> 00:43:08
transpose a matrix be?
730
00:43:08 --> 00:43:10
It'll be three by three, right?
731
00:43:10 --> 00:43:13
I now have five by
three, three by five.
732
00:43:13 --> 00:43:16
Multiplying five by three
gives me three by three.
733
00:43:16 --> 00:43:21
This column is gone,
and that row is gone.
734
00:43:21 --> 00:43:24
Because the row came from A
transpose and the column
735
00:43:24 --> 00:43:27
came from A, and we've
just thrown them away.
736
00:43:27 --> 00:43:29
By grounding that node.
737
00:43:29 --> 00:43:37
Now give me the key fact about
that A transpose A matrix?
738
00:43:37 --> 00:43:39
What what do you see there?
739
00:43:39 --> 00:43:44
Now, you see a reduced, a
grounded A transpose A.
740
00:43:44 --> 00:43:46
What kind of a
matrix have I got?
741
00:43:46 --> 00:43:47
Positive def.
742
00:43:47 --> 00:43:48
Good.
743
00:43:48 --> 00:43:49
Positive definite.
744
00:43:49 --> 00:43:54
It's now not singular any
more, its determinant is
745
00:43:54 --> 00:43:56
some positive number.
746
00:43:56 --> 00:43:59
And everything is positive,
its eigenvalues are all
747
00:43:59 --> 00:44:02
positive, everything's
good about that matrix.
748
00:44:02 --> 00:44:08
OK, and I guess what I was
starting to say here, if I
749
00:44:08 --> 00:44:13
wanted to fix, this would
be a natural problem.
750
00:44:13 --> 00:44:17
Fix the top voltage
at one, say.
751
00:44:17 --> 00:44:22
Fix u_1=1 and see how
much current flows.
752
00:44:22 --> 00:44:25
That would be a
natural question.
753
00:44:25 --> 00:44:28
What's the system resistance
between the top node and the
754
00:44:28 --> 00:44:33
bottom, if I'm given, or
the system conductance.
755
00:44:33 --> 00:44:40
If I'm given a c_1, a c_2, a
c_3, a c_4, and a c_5, I could
756
00:44:40 --> 00:44:44
say I could fix that voltage at
one, I could fix this at zero.
757
00:44:44 --> 00:44:47
Maybe one of the homework
problems asks you for
758
00:44:47 --> 00:44:48
something like this.
759
00:44:48 --> 00:44:52
And then you find
all the currents.
760
00:44:52 --> 00:44:54
And the voltages, you
solve the problem.
761
00:44:54 --> 00:44:58
And you know what
the currents are.
762
00:44:58 --> 00:45:02
You know the total current that
leaves node one, enters node
763
00:45:02 --> 00:45:08
four when the voltages drop
by one, between, right?
764
00:45:08 --> 00:45:11
So current can flow down
here, cross over here,
765
00:45:11 --> 00:45:13
down here whatever.
766
00:45:13 --> 00:45:19
Somehow all these five numbers
are going to play a part
767
00:45:19 --> 00:45:21
in that system resistance.
768
00:45:21 --> 00:45:24
So that would be an
interesting number to know.
769
00:45:24 --> 00:45:29
Out of those five numbers,
somehow five c's, there's a
770
00:45:29 --> 00:45:32
system resistance between
that node and that node.
771
00:45:32 --> 00:45:35
And we can find it by setting
this to be one, this to be
772
00:45:35 --> 00:45:40
zero, having the reduced matrix
- oh, well what will happen?
773
00:45:40 --> 00:45:43
How many unknowns well I have?
774
00:45:43 --> 00:45:45
Just do this mental experiment.
775
00:45:45 --> 00:45:51
Suppose I introduce u_1
to be one, for example.
776
00:45:51 --> 00:45:54
This is just one type
of possible problem.
777
00:45:54 --> 00:46:03
If I take u_1 to be one, what
happens to my matrix A?
778
00:46:03 --> 00:46:07
It loses its first column, too.
u_1 is not unknown any more.
779
00:46:07 --> 00:46:12
u_1 will not be unknown.
780
00:46:12 --> 00:46:16
And that value one is somehow
going to move to the
781
00:46:16 --> 00:46:17
right-hand side, right?
782
00:46:17 --> 00:46:21
People have asked me after
class, well what happens if a
783
00:46:21 --> 00:46:24
boundary condition isn't zero?
784
00:46:24 --> 00:46:28
Suppose we have this fixed
springs and we pull this
785
00:46:28 --> 00:46:32
spring down to make
its displacement 12.
786
00:46:32 --> 00:46:34
Well, somehow that 12 is going
to show up on the right
787
00:46:34 --> 00:46:36
side of the equation.
788
00:46:36 --> 00:46:39
It's a source, it's
an external term.
789
00:46:39 --> 00:46:43
OK, so if we had u_1
equals whatever, this
790
00:46:43 --> 00:46:44
u_1 would disappear.
791
00:46:44 --> 00:46:47
I would only have a
two by two problem.
792
00:46:47 --> 00:46:49
Because I would only have
two, I now have only
793
00:46:49 --> 00:46:52
two unknown u's, right?
794
00:46:52 --> 00:46:55
So that's where
sources can come.
795
00:46:55 --> 00:47:01
And can I just complete the
picture of the source stuff?
796
00:47:01 --> 00:47:08
We could fix, we could.
797
00:47:08 --> 00:47:10
Look, here's what
I'm going to say.
798
00:47:10 --> 00:47:12
External stuff.
799
00:47:12 --> 00:47:15
Sources can come into here.
800
00:47:15 --> 00:47:17
They can come into here.
801
00:47:17 --> 00:47:20
They can come into here, so of
course everybody says why
802
00:47:20 --> 00:47:22
shouldn't they come in here?
803
00:47:22 --> 00:47:23
And the answer is we
could send them here.
804
00:47:23 --> 00:47:33
So we could fix, we
could fix some w's.
805
00:47:33 --> 00:47:36
Of course, you understand
we can't do everything.
806
00:47:36 --> 00:47:41
I mean, there's a limit to how
much we can put on the system.
807
00:47:41 --> 00:47:44
We want to have some
unknowns left.
808
00:47:44 --> 00:47:46
Some matrix still, but anyway.
809
00:47:46 --> 00:47:50
I like this picture now,
it's more complete.
810
00:47:50 --> 00:47:57
That you now see the node
variables and node equations,
811
00:47:57 --> 00:48:00
the edge variables, e and w.
812
00:48:00 --> 00:48:01
The currents.
813
00:48:01 --> 00:48:08
These guys are the big ones. w
and u are what I think of as
814
00:48:08 --> 00:48:14
the crucial unknowns. e is sort
of on the way. f is the source.
815
00:48:14 --> 00:48:17
But now we have the
possibility of sources
816
00:48:17 --> 00:48:20
at all four positions.
817
00:48:20 --> 00:48:24
OK, let's see.
818
00:48:24 --> 00:48:31
If I wrote out, If I looked at
A transpose C A, would you
819
00:48:31 --> 00:48:34
like to tell me, yeah.
820
00:48:34 --> 00:48:35
Have we got?
821
00:48:35 --> 00:48:37
No, we don't.
822
00:48:37 --> 00:48:40
I was going to say, what's a
typical row of A transpose C A,
823
00:48:40 --> 00:48:43
can I just say it in words?
824
00:48:43 --> 00:48:45
It'll be too quick
to really catch.
825
00:48:45 --> 00:48:49
So without the C,
this is what we had.
826
00:48:49 --> 00:48:53
So what do you think that two
becomes if there's an A
827
00:48:53 --> 00:48:56
transpose C A, if there's
a C in the middle.
828
00:48:56 --> 00:48:58
Have you got the pattern yet?
829
00:48:58 --> 00:49:02
That two was there
because of two edges.
830
00:49:02 --> 00:49:05
Edges one and two,
it happened to be.
831
00:49:05 --> 00:49:08
So instead of the two, I'm
going to see c_1+c_2.
832
00:49:11 --> 00:49:11
Right.
833
00:49:11 --> 00:49:14
When those were ones,
I got the two.
834
00:49:14 --> 00:49:18
So this will be c_1+c_2,
this'll be a minus c_1, and
835
00:49:18 --> 00:49:21
that'll be a minus c_1,
when we do it out.
836
00:49:21 --> 00:49:22
And you could do it
out for yourself.
837
00:49:22 --> 00:49:27
Just tell me what
would show up there.
838
00:49:27 --> 00:49:30
In A transpose C A, so
I'm talking now about
839
00:49:30 --> 00:49:33
A transpose C A.
840
00:49:33 --> 00:49:37
So instead of one plus one
plus one, what do I have?
841
00:49:37 --> 00:49:40
What am I going to have, and
you really want to multiply
842
00:49:40 --> 00:49:44
it out, because it's so
nice to see it happen.
843
00:49:44 --> 00:49:45
What do I have?
844
00:49:45 --> 00:49:50
I'm looking at node two, I'm
seeing three edges out of it.
845
00:49:50 --> 00:49:53
And instead of one, one,
one, I'll have c_1+c_3+c_4.
846
00:49:56 --> 00:50:00
c_1+c_3+c_4 will
be sitting here.
847
00:50:00 --> 00:50:02
And minus c_1 will be here, and
minus c_3 will be here, and
848
00:50:02 --> 00:50:06
minus c_4 will be there.
849
00:50:06 --> 00:50:08
The pattern's just nice.
850
00:50:08 --> 00:50:14
So if you can read this part of
the section, I'll have more
851
00:50:14 --> 00:50:20
to say Friday about the a
transpose w, the balance.
852
00:50:20 --> 00:50:23
That critical point
we didn't do yet.
853
00:50:23 --> 00:50:27
But the main thing,
you've got it. |
Fundamentals of Algebra and Trigonometry - 9th edition
Summary: This classic in the series of highly respected Swokowski/Cole mathematics texts retains the elements that have made it so popular with instructors and students alike: it is clearly written, the time-tested exercise sets feature a variety of applications, its exposition is clear, its uncluttered layout is appealing, and the difficulty level of problems is appropriate and consistent. Now this Ninth Edition of Fundamentals of Algebra and Trigonometry has been impr...show moreoved in three important ways. First, discussions have been rewritten to enable students to more easily understand the mathematical concepts presented. Second, exercises have been added that require students to estimate, approximate, interpret a result, write a summary, create a model, explore, or find a generalization. Third, graphing calculators have been incorporated to a greater extent through the addition of examples and exercises as well as the inclusion of a cross-referenced appendix on the use of the TI-82/83. All of this has been accomplished without compromising the mathematical integrity that is the hallmark of this text. ...show less
Ships next business day! May NOT include supplemental materials such as CDs and access codes. May include some highlighting or writing.
$158237336.54 |
Problems With a Point is a site developed for mathematics students and teachers in grades 6-12. The site contains practice problems on various topics that designed to help students understand mathematical concepts and...
George Mason University?s Department of Computer Science is responsible for this great elementary calculus site. Each topic presented on this site contains learning objectives, definitions, examples, diagrams, and e...
The University of Akron has created these excellent algebra tutorials that review some of the main topics in the discipline. There are ten lessons, which focus on topics like radicals and exponents, basic algebra,...
Murray Bourne developed the Interactive Mathematics site while working as a mathematics lecturer at Ngee Ann Polytechnic in Singapore. The site contains numerous mathematics tutorials and resources for students and... |
Elem & Intermediate Algebra For Coll Students
Elementary and Intermediate Algebra
Elementary and Intermediate Algebra for College Students
Elementary and Intermediate Algebra for College Students
Elementary and Intermediate Algebra for College Students Value Package (includes Student Study Pack)
Student's Solutions Manual for Elementary and Intermediate Algebra for College Students
Worksheets for Classroom or Lab Practice for Elementary and Intermediate Algebra for College Students
Summary
TheAngel Seriescontinues to offer proven pedagogy, sound exercise sets and superior user support. An emphasis on the practical applications of algebra motivates readers and encourages them to see algebra as an important part of their daily lives. The user friendly writing style uses short, clear sentences and easy to understand language, and the outstanding pedagogical program makes the material easy to follow and comprehend. The new editions continue to place a strong emphasis on problem solving.Real Numbers; Solving Linear Equations; Formulas and Applications of Algebra; Graphing Linear Equations; Exponents and Polynomials; Factoring; Rational Expressions and Equations; Functions and Their Graphs; Systems of Linear Equations; Inequities in One and Two Variables; Roots, Radicals, and Complex Numbers; Quadratic Functions; Exponential and Logarithmic Functions; Conic Sections; and Sequence, Series, and the Binominal Theorem.For any professional needing to apply algebra to their work. |
Algebra for Students DVD Series
In Algebra for Students, students will learn about the power of algebra as a tool for representing, analyzing and generalizing situations, and will explore several functions, including linear, quadratic and exponential. Real-world applications of algebra are shown in multiple forms through tables, graphs and equations, and common errors and misconceptions are highlighted. Students will also learn how to translate verbal expressions to algebraic expressions while considering the reasonableness of solutions within the context of the situation. Teacher's guides are included and available online |
Textbooks The Heart of Mathematics: An invitation to
effective thinking, Burger & Starbird What is Mathematics Really? Reuben
Hersh
Purposes
To develop an understanding of the nature of
mathematics and its relevance in daily life.
Overview
In this course we will examine the question "What is
mathematics and what is it good for?". We will learn
by exploring mathematics that is not frequently studied in high school
or undergraduate mathematics courses. We will come to the
(perhaps
surprising) conclusion that mathematics is not primarily about
computing
or measuring, but rather about a style of thought. The important
applications of mathematics are more about making life decisions and
solving
problems than paying mortgage, finding the area of fabric, or
determining
the speed of a cannonball. We will also directly explore the
philosophy
and history of mathematics. Why does mathematics exist at all?
Reading
We have two very different books for this
course. The Heart of Mathematics is a fun coffee-table
type book. The book is about as easy to read as a light magazine.
The main point of this book is that mathematics helps us to think
about our daily life and the world around us. This will be the
source
of our mathematical content and most of the projects. Homework
will
primarily come from this book. The exams will cover material in
this book. What is Mathematics Really? is much more a
philosophy book than a mathematics book. Occasionally
the reading is rough, but the mathematics is almost always very simple.
The references and discussions may feel obscure at times.
Hersh
addresses this eventually, saying something like "If a reference is
unfamiliar, it's probably not important." We will discuss Hersh's
book for 30-45 minutes occasionally. For those days you are
required to bring reading reactions to class. These reading
reactions must include reactions (items you particularly identified
with, disagreed with, or do not understand but would like to discuss)
to at least five topics in the reading. They must be written in
intelligible English. Each one will be evaluated out of 5 points,
with points deducted for fewer than five points being
addressed.
Course Content
We will begin the course by reading Chapter One of Heart
of Mathematics. We will also read Chapters 1 - 5 and 13 of What
is Mathematics, Really? The remaining course content will be
determined based on student preference indicated on forms distributed
on the first day. The most popular sections will be discussed in
class January 31 - March 11 and April 4 - 15. Other sections will
be assigned to paired students as projects. I will also present
Hersh's brief summary of calculus to give a different perspective on
material that is commonly presented in mathematics courses.
Grading
Your grade in this course will be based upon your
performance on these items:
In-class exams
15% each
Reading
Reactions 10%
Homework
15%
Colloquium
Report 10%
Project
15%
Final Exam
20%
Colloquium Report
Attend one of the department colloquium talks.
Write a report. In the report, describe the content of
the talk (you do not need to explain all the details, but it is
necessary to include the main points that the speaker was attempting to
convey).
In addition to your description of the talk, also write how this
talk added to your understanding of the nature of mathematics.
Projects
Each student is responsible for completing a project
as part of a pair. A project will consist of reading a section of
one of our textbooks and completing all the exercises in the chosen
part (there may be supplemental exercises for projects chosen from
Hersh). Projects will be presented during the last two weeks of
class. Writeups are due on the last day of class.
Homework Exercises
There will be homework exercises assigned from each
section that we cover in class. The first homework
assignment will be finalised after we decide on the course content.
Following that each homework assignment will be announced on the
day that the previous assignment is due. You are encouraged to
consult with me outside of class on any questions toward completing
the homework. You are also encouraged to work together on
homework
assignments, but each must write up their own well-written solutions (a
simple rule - discuss homework, but do not look at each other's
writing).
Each question will be counted in the following manner:
0 – missing or copied question
1 – question copied
2 – partial question
3 – completed question (with some solution)
4 – completed question correctly and well-written
Each entire homework set will then be graded on a 90-80-70-60% (decile)
scale. Late items will not be accepted. Homework will be
returned on the following class day.
In-class Exams
In class exams will check your understanding of the
mathematical content of the course. They will have questions
directly from the "Solidifying Ideas" questions pertaining to the
sections we have discussed.
Final Exam
Half of the final exam will be in the same form as
the in-class exams. A quarter of the exam will require
you to state the main idea of several projects other than your own.
The last quarter of the exam will require you to summarise your
understanding of Hersh's book and to explain your reaction to his
ideas.
Feedback
Occasionally you will be given
anonymous feedback forms. Please use them to share any thoughts
or concerns for how the course is running. Remember, the sooner
you tell me your concerns, the more I can do about them. I have
also created a web-site
which accepts anonymous comments.
If we have not yet discussed this in class, please encourage me to
create a class code. This site may also be accessed via our
course
page on a link entitled anonymous
feedback. Of
course, you are always welcome to approach me outside of class to
discuss these issues as well.
Religious Holidays
It is my policy to give students who miss class
because of observance of religious holidays the opportunity to make up
missed work. You are responsible for notifying me no
later than January 28 of plans to observe the holiday.
Schedule
January 19 Course Introduction
January 21 Burger & Starbird - Fun and Games
January 24 Fun and Games / Course Plan and Projects Assigned
January 26 Hersh Preface
January 28 2.4
7.6 21, 23, 26, (29 or 30), (one of 31,
33, 35).
Calculus 1. Why does 6 / 3 = 2?
Why does 3 / 0 have no answer? Why does 0 / 0 have any
answer?
2. Following Hersh's discussion, consider a
stone traveling by the equation
h(t) = -16t^2 + 10t + 5, compute the speed at time t by computing the
distance
change in a moment H near time t divided that by moment H.
3. What is the slope of a curve at a point where it reaches
a maximum or
a minimum? |
Today's Developmental Math students enter college needing more than just the math, and this has directly impacted the instructor's role in the classroom. Instructors have to teach to different learning styles, within multiple teaching environments, and to a student population that is mostly unfamiliar with how to be a successful college student. Authors Andrea Hendricks and Pauline Chow have noticed this growing trend in their combined 30+ years of teaching at their respective community colleges, both in their face-to-face and online courses. As a result, they set out to create course materials that help today's students not only learn the mathematical concepts but also build life skills for future success. Understanding the time constraints for instructors, these authors have worked to integrate success strategies into both the print and digital materials, so that there is no sacrifice of time spent on the math. Furthermore, Andrea and Pauline have taken the time to write purposeful examples and exercises that are student-centered, relevant to today's students, and guide students to practice critical thinking skills. Beginning and Intermediate Algebra and its supplemental materials, coupled with ALEKS or Connect Math Hosted by ALEKS, allow for both full-time and part-time instructors to teach more than just the math in any teaching environment without an overwhelming amount of preparation time or even classroom time. |
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Counterexamples in Calculus
Sergiy Klymchuk
As a robust repertoire of examples is essential for students to learn the practice of mathematics, so a mental library of counterexamples is critical for students to grasp the logic of mathematics. Counterexamples are tools that reveal incorrect beliefs. Without such tools, learners' natural misconceptions gradually harden into convictions that seriously impede further learning. This slim volume brings the power of counterexamples to bear on one of the largest and most important courses in the mathematics curriculum. — Professor Lynn Arthur Steen, St. Olaf College, Co-author of Counterexamples in Topology
Counterexamples in Calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Methods of producing these incorrect statements vary. At times the converse of a well-known theorem is presented. In other instances crucial conditions are omitted or altered or incorrect definitions are employed. Incorrect statements are grouped topically with sections devoted to: Functions, Limits, Continuity, Differential Calculus and Integral Calculus.
Counterexamples aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. In that light it may well be useful for: high school teachers and university faculty as a teaching resource; high school and college students as a learning resource;and as a professional development resource for calculus instructors.
In the electronic version page numbers that link to the suggested solutions can be found at the end of each problem statement. |
Complex Analysis
Complex analysis forms a basis for not only advanced mathematical topics (including differential equations, number theory, operator theory and others) but also for special functions of mathematical and quantum physics - subjects used to understand the world in which we live. The course covers fundamental knowledge in the theory of analytical functions with applications to definite integration and culminates with study of harmonic and special functions.
MATH3242 cannot be counted for credit with Math2420.
Available in 2014
Callaghan Campus
Semester 2
Previously offered in 2013, 2012, 2011, 2010
Objectives
On successful completion of this course, students will be able to:
1. use analytical functions and conformal mappings; 2. compute definite integrals using residue calculus; 3. appreciate the existance of special functions and their use in a range of contexts. |
Students
will use mathematical analysis, scientific inquiry, and engineering design,
as appropriate, to pose questions, seek answers, and develop questions.
Standard 2: Information
Systems
Students
will access, generate, process, and transfer information using appropriate
technologies.
Standard 3: Mathematics
Students
will understand mathematics and become mathematically confident by communicating
and reasoning mathematically, by applying mathematics in real world settings,
and by solving problems through the integrated study of number systems,
geometry, algebra, analysis, probability, and trigonometry.
Benchmark
standards: Content standards
Standard
1: Analysis, Inquiry, and Design MathematicalAnalysis:
# 1. Abstraction and symbolic representation are used to communicate mathematically.
#3. Critical thinking skills are used in the solution of mathematical
problems.
Standard
2: Information Systems
#1. Information technology is used to retrieve, process, and communicate
information and as a tool to enhance learning.
Standard
3: Mathematics
#1
Mathematical Reasoning: Students use mathematical reasoning to analyze
mathematical situations, make conjectures, gather evidence, and construct
an argument.
#3
Operation: Students use mathematical operations and relationships
among them to understand mathematics.
#4
ModelingIMultiple Representation: Students use mathematical modeling/multiple
representations to provide a means of presenting, interpreting, communicating
and connecting mathematical information and relationships.
7
Patterns/Functions: Students use patterns and functions to develop
mathematical power, appreciate the beauty of mathematics, and construct
generalizations that describe patterns simply and efficiently.
Performance
Standards
Standard
-1:
Students will
use special mathematical notation and symbolism to communicate in
mathematics and to compare and describe quantities , express relationships,
and relate mathematics to their immediate environments.
Students will
explore and solve problems generated from school, home, and community
situations, using concrete objects or manipulative materials when
possible.
Standard
2:
Students will
use a variety of equipment and software packages to enter, process,
display, and communicate information in different forms using tables,
text, pictures, and sound.
Students will
access needed information from printed media, electronic data bases,
and community resources.
Standard
3:
Students will
use models, facts, and relationships to draw conclusions about mathematics
and explain their thinking.
Students will
justify their answers and solution processes.
Students will
use logical reasoning to reach simple conclusions.
Students will
add, subtract, multiply and divide whole numbers.
Students will
develop strategies for selecting the appropriate computational and
operational method in problem-solving strategies.
Students will
know single digit addition, subtraction, multiplication, and division
facts.
Students will
construct tables, charts and graphs to display and analyze real-world
data.
Students will
use multiple representations as tools to explain the operation of
everyday procedures.
Students will
recognize, describe, extend, and create a wide variety of patterns.
Students will
represent and describe mathematical relationships.
Students will
interpret graphs.
Content
standards or outcomes
Students will
calculate mathematical operations to determine mean, median, mode
and range.
Students will
use EXCEL program to graph/chart their data and findings. * Students
will use a word processing program to express their findings through
written statements.
Students, working
in groups, will create a daily menu in Power Point of "Healthy"
eating from the McDonalds nutrition chart based on USRDA of calories
and fat will compare and contrast their findings with other groups
information.
Procedure
This activity will take approximately 2 weeks,(10 one hour periods).
Day
2. Teacher models McDonalds chart worksheet demonstrating how to record
the mean, median, mode and range.
Students, working in groups, will find the mean, median, mode and range
from the chart under specific headings. i.e.: Sandwiches, Breakfast
foods, Desserts etc..
Day
3. Teacher demonstrates how to use EXCEL program and how to set up graphs
and charts to display data.
Teacher demonstrates how to relate mean, median, mode and range specifically
to the EXCEL program.
Students collaboratively create graphs and charts to show information.
Day
4. Teacher models writing statements in WORD interpreting the data from
graph. Teacher and students work together to complete graphs and write
statements to support.their findings.
Day
5. Teacher discusses final project outline which is to create a "Healthy"
menu for eating at McDonalds based on USRDA allowance calories and fat.
Teacher gives Power Point presentation serving as model for final project.
Students will work together to mathematically evaluate criterion for
menu final project.
Day
6. Teachers and students will work together to complete final touches
on graphs and charts as well as menu's.
Day
7. Students will present their findings to class using computer generated
graphs, charts, as well as Power Point presentations. They will also
use their written work to aid in their presentations.
Day
8. Teacher facilitates intergroup comparison and contrasting of their
findings. Students will print out their work of all findings and share
with all other groups their results.
Day
9. Students will perform self and group reflections on activity.
Day
10. - Teacher sits with students and groups to review reflections and
rubrics. |
Organized by topic, the Graphing Calculator Study Card guides students through the keystrokes needed to most efficiently use their graphing calculator. This study card includes instruction on using and entering formulas, and instructions on how to construct various types of graphs such as histograms, scatterplots, and box plots |
Description: This unit consists of four lessons, presented below:
Lesson 1: Review of 1-step Equations. This lesson falls in the middle of an algebra skills unit. Topics addressed prior to this lesson include: order of operations, translating expressions and algebraic equations into written form, evaluating expressions for a given value, and operations with integers. Once students have reviewed (from sixth grade curriculum) how to solve one step equations algebraically, they will begin solving multi-step equations, equations that involve combining like terms and equations with variables on both sides of the equal sign.
Lesson 2: Solving Algebraic Equations. Multistep equations are being introduced in this lesson. The intent of the lesson is to address past challenges regarding (a) the notion of doing something on both sides an equation as relational, (b) what to do first in a multistep process, and (c) thinking and working in reverse with the order of operations. Modeling, cooperative groups, manipulatives, technology, scaffolding, discussion, and text are used in this lesson.
Lesson 3: Multi-step equations with Combining Like Terms. This lesson falls in the middle of an algebra skills unit. Topics addressed prior to this lesson include: order of operations, translating expressions and algebraic equations into written form, evaluating expressions for a given value, and operations with integers, as well as, a review on solving one step equations and multi-step equations in the form ax ± b = c . Once students have mastered equations that involve combining like terms (taught in this lesson), they will be introduced to equations with variables on both sides of the equal sign.
Lesson 4: Equations with variables on both sides. This lesson falls towards the end of an algebra skills unit. Topics addressed prior to this lesson include: order of operations, translating expressions and algebraic equations into written form, evaluating expressions for a given value, and operations with integers. Students have reviewed (from sixth grade curriculum) how to solve one step equations algebraically, have learned how to solve multi-step equations in the form ax ± b = c, equations that involve combining like terms and now, equations with variables on both sides of the equal sign. |
2013-2014 Mathematics Courses
Mathematics has been an undeniably effective tool in humanity's ongoing effort to understand the nature of the world around us, yet the mantra of high-school students is all too familiar: What is math good for anyway?When am I ever going to use this stuff? What serves to explain the puzzling incongruity between the indisputable success story of mathematics and students' sense of the subject's worthlessness? Part of the explanation resides in the observation that all too many mathematics courses are taught in a manner that entirely removes the subject matter from its proper historical, social, and cultural context—naturally leaving students with the distinct impression that mathematics is a dead subject, one utterly devoid of meaningfulness and beauty. In reality, mathematics is one of the oldest intellectual pursuits, its history a fascinating story filled with great drama, extraordinary individuals, and astounding achievements. This seminar focuses on the role played by mathematics in the emergence of civilization and follows their joint evolution over nearly 5,000 years to the 21st century. We will explore some of the great achievements of mathematics and examine the full story behind those glorious achievements. The ever-evolving role of mathematics in society and the ever-intertwined threads of mathematics, philosophy, religion, and culture provide the leitmotif of the course. Specific topics to be explored include the early history of mathematics, logic and the notion of proof, the production and consumption of data, the analysis of conflict and strategy, and the concept of infinity. Readings will be drawn from a wide variety of sources (textbooks, essays, articles, plays, and fictional writings), connecting us to the thoughts and philosophies of a diverse set of scholars; a partial list includes Pythagoras, Euclid, Galileo, René Descartes, Isaac Newton, Immanuel Kant, Lewis Carroll, John Von Neumann, John Nash, Kurt Gödel, Bertrand Russell, Jorge Luis Borges, Kenneth Arrow, and Tom Stoppard.
An Introduction to Statistical Methods and Analysis
An introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental ideas of statistical analysis used to gain insight into diverse areas of human interest. The use, abuse, and misuse of statistics will be the central focus of the course. Topics of exploration will include the core statistical topics in the areas of experimental study design, sampling theory, data analysis, and statistical inference. Applications will be drawn from current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical software will be introduced and used extensively in this course, but no prior experience with the software is assumed. This seminar is an invaluable course for anybody planning to pursue graduate work and/or research in the natural sciences or social sciences. No college-level mathematical knowledge is required.
Calculus II: Modeling With Differential Equations
An infectious disease spreads through a community: What is the most effective action to stop an epidemic? Populations of fish swell and decline periodically: Should we change the level of fishing allowed this year to have a better fish population next year? Foxes snack on rabbits: In the long term, will we end up with too many foxes or too many rabbits? Calculus can help us answer these questions. We can make a mathematical model of each situation, composed of equations involving derivatives (called differential equations). These models can tell us what happens to a system over time which, in turn, gives us predictive power. Additionally, we can alter models to reflect different scenarios (e.g., instituting a quarantine, changing hunting quotas) and then see how these scenarios play out. The topics of study in Calculus II include power series, integration, and numerical approximation, all of which can be applied to solve differential equations. Our work will be done both by hand and by computer. Conveniently, learning the basics of constructing and solving differential equations (our first topic of the semester) includes a review of Calculus I concepts Prerequisite: Calculus I (differential calculus in either a high-school or college setting).
Calculus I: The Study of Motion and Change
Calculus is the study of rates of change of functions (the derivative), accumulated areas under curves (the integral), and how these two ideas are (surprisingly!) related. The concepts and techniques involved apply to medicine, economics, engineering, physics, chemistry, biology, ecology, geology, and many other fields. Such applications appear throughout the course, but we will focus on understanding concepts deeply and approach functions from graphical, numeric, symbolic, and descriptive points of view Facility with high-school algebra and basic geometry are prerequisites for this course. Prior exposure to trigonometry and/or precalculus is highly recommended. No previous calculus experience is necessary or desired.
Discrete Mathematics: A Gateway to Advanced Mathematics
This seminar is an introduction to the world of elegant mathematics, beyond that encountered in high school, under the guise of an introductory survey course in discrete mathematics. We will touch on the tips of many icebergs! The subject of discrete mathematics houses the intersection of mathematics and computer science; it is an active area of research that includes combinatorics, graph theory, geometry, and optimization. The topics in this course are selected to give an idea of the types of thinking used in a variety of discrete mathematics research areas. Learning the facts and techniques of discrete mathematics is inextricably intertwined with reasoning and communicating about discrete mathematics. Thus, at the same time as surveying discrete mathematics, this course is an introduction to rigorous reasoning and to writing convincing arguments. These skills are necessary in all of mathematics and computer science and very applicable to law and philosophy. Conference work will explore additional mathematical topics. The seminar is essential for students planning advanced study in mathematics and highly recommended for students studying computer science, law, or philosophy or who seek to enhance their logical thinking and problem-solving skills. Prerequisite: Prior study of Calculus or equivalent preparation.
Topics in Multivariable and Vector Calculus
Compared to the familiar single-variable territory of Calculus I and II, multivariable calculus is a foreign land. Imagine, if you will, that instead of a function taking a single input and producing a single output, we either use one input and get multiple outputs (vector functions) or use several inputs and get one output (multivariable functions). And yes, there are even functions that have several inputs and multiple outputs! In this new realm, we will investigate lines and planes, curves and surfaces, and multidimensional generalizations of these objects, with a focus on those functions that can be visualized in three dimensions. For both vector and multivariable functions, we will address the basic questions of calculus: How do we measure rates of change? How do we find areas and volumes? How can we interpret derivatives and integrals both geometrically and for practical purposes? Fascinatingly, each of these questions has more than one answer. We will examine gradients and directional derivatives, maxima and minima and saddle points, double and triple integrals, integrals taken along curves, and more—as time permits. This seminar is essential for students intending to pursue engineering, physics, mathematics, graduate study in economics, or rocket science and is recommended for students pursuing chemistry or computer science. Prerequisites: Calculus I and Calculus II.
Topology: The Nature of Shape and Space
Topology, a modernized version of geometry, is the study of the fundamental, underlying properties of shapes and spaces. In geometry, we ask: How big is it? How long is it? But in topology, we ask: Is it connected? Is it compact? Does it have holes? To a topologist there is no difference between a square and a circle and no difference between a coffee cup and a donut because, in each case, one can be transformed smoothly into the other without breaking or tearing the mathematical essence of the object. This course will serve as an introduction to this fascinating and important branch of mathematics. Conference work will be allocated to clarifying course ideas and exploring additional mathematical topics. Successful completion of a yearlong study of Calculus is a prerequisite and completion of an intermediate-level course (e.g., Discrete Mathematics, Linear Algebra, Multivariable Calculus, or Number Theory) is strongly recommended. |
University preparation: mathematics
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This course prepares students for future study of mathematics at university level by developing an understanding and application of algebra, geometry and calculus. Students will practise effective techniques for studying mathematics as well as learning to read and write mathematics clearly and intelligibly. Students will also develop an appreciation of the patterns which arise in mathematics and how they are applied in a number of different subject areas.
Before considering a University Preparation Course please review important information to determine if you are eligible for the Mature Age Entry Scheme and how to go about qualifying and applying |
The course revises and extends Matriculation and STPM topics such as differentiation and integration and includes topics such as complex numbers and differential equations, which may be new to many students.
Topics covered include parametric equations, functions, polar coordinates, vectors, and complex numbers. Students will learn how to define functions, and plot the graphs, using the Cartesian as well as polar coordinates; solve problems involving complex numbers and vectors.
Additional topics include limits and continuity, differentiation techniques and its applications, integration techniques including improper integrals. Upon completion, the students would have acquired some quite powerful tools of analysis.
This is also an introductory course on differential equations. Topic includes first order ordinary differential equations (ODEs). Students will learn how to classify and solve first order ODEs.
SSH1033 – Mathematical Methods II
This course continues and extends the techniques introduced in Mathematical Methods I, with further differential equations and calculus of multivariable functions.
Topics include linear second order ODEs with constant coefficients, functions of several variables, partial differentiation and multiple integrations. Students will learn how to classify and solve second order linear ODEs with constant coefficients using the method of undetermined coefficients and variation of parameters. They will also learn to determine the domain and range, techniques of graph sketching, and limit & continuity, find (partial) derivatives and evaluate (double and triple) integrals, pertaining to a function of two and three variables. The use of cylindrical and spherical coordinates is also highlighted. Applications include finding the volume, mass, centre of gravity, and moment of inertia of a solid.
SSH1523 – Linear Algebra
The course begins with the study of matrices and determinant. Starting with simple matrix operations, elementary row operation and inverses, and determinant of matrices. Solve the linear system using matrix inverse, Crammer's rule, Gauss and Gauss—Jordan elimination method. Next, the focus is on the vector spaces, subspace, linear independence, spanning sets, bases, coordinate vector and change of basis, orthogonal bases, and the Gram-Schmidt process. There follows a discussion of linear transformation and matrices, as well as the kernel and range. Finally, find the eigenvalues and eigenvectors and use them in diagonalization problem.
SSH1703 – Differential Equations
An introductory first course in differential equations. Topics include first order ordinary differential equations (ODEs), linear second order ODEs with constant coefficients, the Laplace transform and its inverse, Fourier series, and elementary partial differential equations (PDEs). Students will learn how to classify and solve first order ODEs, solve second order linear ODEs with constant coefficients using the method of undetermined coefficients and variation of parameters, use the technique of Laplace transforms to solve ODEs with specified initial or boundary conditions, and use the technique of separation of variables to solve initial-boundary value problems involving heat and wave equations and boundary value problems involving Laplace equation.
SSH3503 – Complex Variables
This course introduces calculus of functions of a single complex variable. Topics covered include the algebra and geometry of complex numbers, complex differentiation, complex integration, complex series including Taylor and Laurent series, the theory of residues with applications to the evaluation of complex and real integrals, and conformal mapping with applications in solving boundary value problems of science and engineering.
SSP1143 – Mechanics
This course mainly discusses motion of a body or a system. Beginning with the basic and derived physical quantities and vector as mathematical tool, various types of motion such linear, free-fall, projectile, circular, rotational and simple harmonic motions are described. Other topics such as equilibrium, elasticity, gravitation and fluids mechanics illustrate the application of a body in motion under the influence of a force.
SSP1153 – Electricity and Magnetism
The course examines the force of electromagnetism, which encompasses both electricity and magnetism. It includes the exploration of some electromagnetic phenomena. It begins by examining the nature of electric charge and then a discussion of interaction of electric charges at rest. It then study about charges in motion particularly electric circuit. lt continues into the study of magnetic interaction how moving charges and currents responds to magnetic field. The principle of electromagnetic induction and how resistors, inductors and capacitors behave in ac circuits is discussed. The understanding the electrical energy-conversion devices such as motors, generators and transformers are also discussed. Finally the study of the four fundamental equations that completely described both electricity and magnetism.
SSP1163 – Sound, Wave and Optics
The course starts with introduction to the concept sound, how it is produced, its characteristics, intensity & quality as well as the interference of sound which will be applied to modern sound devices. Finally, emphasize on optics on its dual properties. These will be inseminated in the phenomenon of interference and diffraction of light and its modern-day applications. In general, the course provides the basic concepts of sound and optics
SSP1223 – Modern Physics
The course begins with a brief discussion on the nature of science in the quest of better understandings of the natural phenomena – highlighting the dilemmas and failures of classical physics in the face of some landmark experiments and discoveries, which gave the impetus to new ideas and paradigm shift into the modern physics. Finally, formalities of quantum mechanics is introduced by discussing the 1-D time independent Schrodinger equation (TISE), applied to an idealised infinite square potential well.
SSP181121 – Physics Practical II completion, the students should have the ability to handle the instrumentations and relate the experiments to the theories learned in Physics II, perform experimental analysis on the laboratory works and write technical reports.
SSP185161 Physics Practical I (Sem II) completion, the students should have the ability to handle the instrumentations and relate the experiments to the theories learned in Physics II, perform experimental analysis on the laboratory works and write technical reports.
SSP2113 – Thermodynamics
The course starts with discussions on basic concepts of thermodynamics, thermodynamic properties of materials and thermodynamic processes. The next topics will emphasize on energy transfer and energy analysis of systems and processes using the explained first and second laws of thermodynamics. The principles of gas power and refrigeration cycles are also briefly highlighted. In general, the course provides on the basic concepts of thermodynamics and it applications in conservation and utilisation of energy as well as in automobile industry.
SSP2213 – Nuclear Physics
The course introduces to some major concepts and theories of nuclear physics. The course begins with understanding the basic knowledge of the constituents of nucleus and the properties of nuclear forces. The next topic of the course is introducing the radiation sources and the types of ionizing radiations. Nuclear decay process and the properties of ionizing radiations will be discussed in this topic. The interactions of nuclear radiations with mater and mechanism of nuclear reaction are also covered in this subject. The next topic is providing the students knowledge with some basic concept on radioactivity including radioactive decay law, radioactive decay series and radioactive equilibriums. Some nuclear models such as liquid drop model, shell model and optical model of the nucleus will be introduced at the end of the subject. In general, the course provides a basic concept of interaction processes of nuclear radiation in order to widening the appreciation of nuclear physics to the students.
SSP2313 – Basic Electronics
The course starts with introduction to electronic components, circuit building and basic measurement of signal. Various circuit theory analysis such as Superposition principle, mesh current analysis, The venin and Norton theorem are taught. DC and AC circuit analysis and the use of semiconductor devices such as diodes and transistors are discussed. The hybrid h and phi small signal models for transistor are emphasized. Next the small signal amplifiers, power amplifiers, differential amplifier are constructed for better understanding and practical experience. In general, the course provides good balance between theoretical and practical works on electronic circuits and its everyday applications.
SSP2333 – Computer Programming
This course begins with a comprehensive introduction to computer, role of computer in physics, and operating system. Computer programming involving choices of computer languages and programming concept is also discussed. In the laboratory, the student experience working with a Linux desktop, client-server working environment, and all the necessary tools for terminal-server programming works. Throughout the course students are guided to build computer programs from simple to complex, all about solving various physics problem, based on the Java programming language. Students are exposed to methods for writing command-line based programs and tools utilizing widgets for building application with graphical user interface.
SSP2413 – Solid State Physics
Introduces basic concepts in solid state physics, with emphasis on crystal structures. The roles of phonons and electrons in a solid are discussed, using various models. Upon completion, students should be able to explain basic concepts used in solid state physics and techniques used in determining crystal structures. Students should also be able to discuss thermal properties of solids and the behaviour of electrons in solids, using various models.
SSP2613 – Mathematical Physics
The main aim of the course is to provide physics students with mathematical treatment of a range of fundamental topics in physics. The course content consists of vector analysis, vector calculus, complex variable, matrices, ordinary and partial differential equations, and Fourier series. The course thus consolidates and integrates Mathematics and Physics, and helps to overcome some of the difficulties which associated with the interface between the two subjects.
SSP2811 – Physics Practical III2821 – Physics Practical IV3113 – Electromagnetism
Introduces the vectorial and calculus approaches in understanding various laws and principles of electromagnetism-and time independent Maxwell's equations. The course will also describe the time varying electromagnetic fields and it physical principles in various applications.3143 - Relativity
This course presents the essential concepts of general relativity theory. The emphasis is on the physical understanding of the theory and the mathematical development is kept simple. Space-time diagrams are used extensively in the explanation of the theory.
SSP3153 – Inferential Statistics
This course is designed to expose student to understand the most fundamental components of nature using the quark model. Some topics of interest would be the structure, definition, flavor and the combination of quarks to form other particles. Classifications of particles and their interactions into a number of easily identifiable categories, and a number of empirical rules will also be studied. Interactions between particles will be dealt with in terms of the four types of forces and the exchange of particles between them. Also included in the course will be the conservation theory of various interactions in terms of lepton number, parity, charge conjugate and time reversal. At the end of the course, the student will be exposed to the understanding of unification theory of forces which incorporate the mechanics of the strong, weak, and electromagnetic interactions into a single theory.
SSP3163 – Energy and Environment Physical
The course starts with a brief introduction on the processes and issues in environmental physics which include the global warming. The principal topics are the physics of the built environment, energy for living, environmental health, revealing the planet, the biosphere, the global climate, climate change. The alternative source of energy such as nuclear energy, wind as well as water will also be touch. This course provides an essentially physics principles that underlie environmental issues and shows how they contribute to the interdisciplinary field of environmental science as a whole. This is very important for the students to be aware of especially when they engage in the industry.
SSP3323 – Advanced Electronics
The course begins with discussion of operational amplifier (OPAMP) and its applications such as summing and differential amplifiers, differentiator/integrator, and active filters. Sensors and amplification of signals are introduced. Basic concepts and principles of digital circuits; number codes and number system, Boolean algebra, logic gates, Karnaugh maps, IC specification and interfacing, encoding and decoding, flip-flops, counters, shift registers and digital arithmetic circuits are also discussed. Finally analogue to digital and digital to analogue conversion are covered. In general, the course will be conducted by lectures and hands-on to provide students with sound basic concepts and practical experience in advanced analogue and digital electronics.
SSP3333 – Computation Physics
This course begins with a comparative discussion about analytical and numerical methods of studying physical phenomena. The design of program codes and equivalent pseudo codes are discussed. Numerical methods for investigation of elementary mechanics problems such as projectile, oscillatory, and planetary motions, and the chaos of non-linear pendulum are introduced. Calculation of potential surface, electric and magnetic fields, and visualization of the respective calculated data are also covered. Wave phenomena are investigated numerically. Methods for investigation of random system and Monte Carlo simulation are also studied. Finally the course ends with an introduction to molecular dynamic simulation method and how to animate visualization of simulated system.
SSP3343 – Instrumentation and Data Acquisition
The course consists of two parts: The first part begins with a review of basic elements in measurement systems; sensing element, signal conditioning, signal processing and signal presentation. The instrument's classification, errors in measurement, static and dynamic characteristics of instrument and calibration will also be introduced. Next, the physical quantity measurement which includes displacement, velocity and acceleration for translational and rotational motion, force and torque, low, medium and high pressure, temperature and other physical quantities, such as flow, level, humidity and electrical quantities are discussed. The second part of this course introduces basic concepts and techniques for interfacing a microcontroller to external devices for data collection and process control and developing the related software required. This includes transferring and converting analogue variables into the digital form needed for processing. It is aimed at students interested in data acquisition and real-time control systems. In general, the course provides on the general concepts of measurement system technology and physical quantities measurement technique.
SSP3433 – Quality Control3523 – Modern Optics
This course exposes the students to the variety of optics. Elementary optics, ray optics, optical instruments, source and detector, interference and diffraction, image processing, laser, polarization and electromagnetic effects, fibre optics and integrated optics are describe and discuss. At the end of the course, students should be able to understand and apply the concepts to solve the problems related to the optical phenomena. Students should have the ability to apply and using standard optical components including laser and fibre optics components. The students should also be able to explain the functions of various components in optical systems in various applications & be thankful to the Creator of light.
SSP3613 – Quantum Mechanics I
This course introduces students to new phenomena leading to quantum mechanics. It will discuss quantum phenomena such as black body radiations, photoelectric effects; Particle-wave duality, wave packets, Schrödinger equations; Observable expectation values; Quantum operator and postulations of quantum mechanics. It will examine and solve problems for one dimensional time independent Schrödinger equations for infinite and finite square potential well, potential barrier; Harmonic oscillator; Hydrogen atom using momentum operator. Basic concepts in quantum mechanics are described and the uses of the quantum mechanical approach in solving contemporary quantum mechanical problems are explained. New phenomena in quantum mechanics, which makes it different from classical mechanics.
SSP3811 – Physics Practical V
Students taking Practical Physics V will conduct two six weeks Mini Projects. They perform open-ended experiments, and produce not more than five pages formal technical report of their work. The project can be done either individually or in pairs. At the end of semester the student will present a short seminar which describes the project, the analysis and the findings. Upon completion, the student will be supervised on essentially one-to-one basis by project supervisor, but they will also be expected to develop their ability to work independently.
SSP3821 – Physics Practical VI
Students taking Practical Physics VI will conduct two six weeks Mini Projects on physics based ICT. The sudents are required to develop ICT projects to solve problems related to physics. The students will be supervised by a supervisor, but they are encoranged to work independently. At the end of semester the student will present a short seminar which describes the project, the analysis and the findings. |
Develop Algebraic Thinking 3-5 - MAT-925Use manipulatives, real-world problem-solving, and captivating activities to engage your students in developmentally appropriate algebraic thinking. This ONLINE course will provide you with numerous opportunities to reflect on current research and pedagogy related to algebraic thinking in the intermediate grades. Make direct application to your own classroom through the design and development of lessons that explore growth patterns, tables, variables, and coordinate graphs. All of the readings and activities are built upon the Common Core standards. Teachers may complete this course with or without students |
This activity allows the user to step through plotting a two-variable function, and to judge whether a given starting point is a prisoner or an escapee, in order to understand complex numbers and Juli... More: lessons, discussions, ratings, reviews,...
This tool is designed for those students who best learn by drilling/quizzing. It randomly presents examples of equalities and inequalities (such as "If a=b then ac = bc") and gives you three choices ... More: lessons, discussions, ratings, reviews,...
These flashcards feature the properties of equalities and inequalities, including transitive, reflexive, trichotomy, and more. Given an equation, name the property it exemplifies, or vice versa. Once ... More: lessons, discussions, ratings, reviews,...
The user reviews some basic definitions and properties of real numbers. After viewing explanations and examples of the properties, users can interactively test their understanding of the properties o |
2014-04-16T13:10:28ZNumerical solution of Hodgkin-Huxley's partial differential system for nerve conduction
Numerical solution of Hodgkin-Huxley's partial differential system for nerve conduction
Morton, John Baird
A numerical solution to Hodgkin and Huxley's partial differential
system for the propagated action potential is presented. In
addition a three dimensional demonstration of the absolute refractory
period is given. Lastly, theoretical evidence supporting
Rushton's hypothesis is presented.
Graduation date: 1967
1967-05-04T00:00:00ZA FORTRAN to ALGOL translator
A FORTRAN to ALGOL translator
Hill, Edward Burlingame
FORTRAN is readily feasible to translation into ALGOL since
they share many common features. Most of the features that are
unique to FORTRAN can be translated by restricting them somewhat.
The translator will handle explicit declarations of each item
in a block, compensate for the differences in various operators,
compensate for the different storage techniques and provide a
simple input /output scheme.
Running test cases through the FORTRAN programs and
through the translated ALGOL programs indicated that the ALGOL
programs take longer to execute than their FORTRAN counterparts.
Graduation date: 1969
1968-10-16T00:00:00ZThe extension problem for functions invariant under a group
The extension problem for functions invariant under a group
Chang, Bai-Ching for extending f invariantly to
the whole space, and thus generalizing the classical Tietze extension
theorem.
Graduation date: 1967
1967-05-05T00:00:00ZA study of unique factorization in quadratic integral domains
A study of unique factorization in quadratic integral domains
Van Enkevort, Ronald Lee
This thesis studies the question of unique factorization in
quadratic integral domains. In the first chapter many general
theorems and definitions from algebraic number theory are introduced.
The second chapter considers an integral domain in which
unique factorization holds. The necessary theorems to prove
unique factorization are developed. The third chapter concerns an
integral domain in which unique factorization fails. That it fails
is proved and then ideals are introduced to indicate how unique
factorization would be restored in terms of ideals.
Graduation date: 1967
1966-07-18T00:00:00Z |
Course Detail
Links
Sets and Proofs
Instructor
Staff
An introduction to proof techniques (including quantifiers and induction), elementary set theory, abstract functions, infinite cardinalities, and properties of sets of real numbers; followed by an introduction to topics chosen from topology, analysis, dynamical systems, or set theory, among others. Emphasis throughout is on developing abilities in writing proofs.
Students entering 2012 and after: satisfies the Mathematical and Quantitative Thought distribution requirement. |
MATHEMATICS A Textbook for class IX (NCERT) RECOMMENDED BOOKS: MATHEMATICS for class IX- R.S.Aggarwal / R.D.Sharma Month Course Content April CH-1: Number System CH 3: Coordinate Geometry Activity 1: To represent an irrational number on a number line
MATHEMATICS-XII 1. Solve the assignments of chapter 1 to 5 of NCERT in the notebook. ... From R.D. Sharma ,inverse trigonometric functions in your notebook. PHYSICS-XII ... Complete assignments on various topics given in class. BUSINESS STUDIES-XII 1. |
Contents description
Symmetry plays an important role in chemistry and physics, both at the
macroscopic level
(tensor of crystals such as
susceptibility and elasticity) and the microscopic level
(quantum numbers, selection rules). Group theory is an abstract setting
capturing
the symmetry in a very efficient manner, which helps to make
computations more efficient. We focus on abstract gorup theory, deal with
representations of groups by means of permutations and by means of matrices,
and deal with some applications in chemistry and physics. We shall also
devote attention to computational aspects, using the software package GAP.
Examination
The requirements for an examination are:
1. solving eight exercises from the reader (covering each of the six chapters),
2. handing in a report on a final assignment, and 3.
passing a short oral exam. |
solve equations, and simplify expressions and inequalities. You will know how to represent functions and linear equations on a graph. You will be able to calculate equations of two or three variables. You will learn how to operate with matrices and simplify polynomials and radical expressions. This course will demonstrate how to graph a quadratic function, solve conic sections, polynomial functions, rational expressions, exponential and logarithmic functions. You will understand how to solve geometric sequences and series. You will learn about the Binomial theorom. You will be more aware of permutations, combinations, probabilities and trigonometric functions such as angles, sines, cosines, circular functions and inverse functions.
Modules in Advanced Algebraic Concepts and Applications in Mathematics
Abubakar AlhassanGhana
Course Module: Module 1: Equations and inequalitiesCourse Topic: Solve equations and simplify expressionsComment: the course is good but i have finished learning and needs exam to write for my certificate 2013-04-28 09:04:00
Jan DeurwaarderBotswana
Course Module: Module 2: How to graph functions and linear equations Course Topic: Functions and linear equations Comment: Looked at and went through Module 1 and 2 of Adv Algebraic concepts and applications in Maths - however my progress states that each (sub)section is incomplete. What haven't I covered?? 2013-01-16 09:01:49
Jan DeurwaarderBotswana
Course Module: Module 2: How to graph functions and linear equations Course Topic: Functions and linear equations Comment: Module 2 A function is NOT an equation. This whole module is a mathematical stew without clear concepts and definition. It will create misconceptions in further learning. We do not want learners to produce free hand graphs - hence in demonstrations a graph board / ruler to be used - no FREE hand sketching. 2013-01-16 09:01:30
Jan DeurwaarderBotswana
Course Module: Module 2: How to graph functions and linear equations Course Topic: Functions and linear equations Comment: Module 1. Too scanty and passive. Each subtopic requires an activity for the learner. More attention to accurate presentation is needed e.g. subtopic absolute value states "every real number is either positive or negative' which is incorrect - zero is neither. The video example on absolute value gives unrelated statements - in simplification we place = sign at end line or start line. The video on solving the inequality introduces error when both sides are divided by -4, is is NOT longer equivalent to the original statement! 2013-01-16 08:01:03 |
How hard is Linear algebra vs. Calculus
I'm taking second semester calc in fall 2013 but that would fit the prereq. for linear algebra
I'm thinking of taking both at once, I'm not some sort of math wiz but I can get a B in calc if I study hard enough
Do you think it is a bad idea to take both at the same time? Like I said, I'm not a math wiz and I wouldn't have any other classes...and plus I really enjoy math to begin with
Also.... another calculus question....How hard is Calc 1 compared to Calc2? and compared to Calc3? What's the hardest out of all 3 semesters of calculus? I've spoken to kids at school and they all say completely different things
I also go to a community college so I don't know if that would make the classes easier ...
Re: How hard is Linear algebra vs. Calculus
Well,
I've only taken single-variable calculus, and I'm currently self-studying linear algebra, so I can answer just a part of your questions. Maybe someone who is more experienced will have a better answer, but this is what I think:
I wouldn't take both linear algebra and calculus at the same time because they're completely different, and learning them simultaneously will be harder and probably confusing.
Calculus 2 has to do mainly with integration (well at my college anyways). I'd say it's not much harder than Calculus 1 and probably the hardest thing is making that leap from derivatives to integrals at the beginning. I'm not sure about Calculus 3; all I know is that it has to do with the calculus of vector functions and functions in multiple variables (I think it's like that at most colleges).
I find linear algebra so far to be easier than Cal. 1, but that's probably because I'm only halfway through. I think learning about matrices, determinants and vectors is easier, but beyond that I'm not sure yet. |
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn about derivations of algebraic formulas and how to create input/output numerical tables, as well as about graphical representation and exploration. Grades 5-9. 30 minutes on DVD . |
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Mission Statement
The mission of the Mathematics Department at San Jose State University is to offer
undergraduate degrees and graduate degrees in Mathematics that prepare students to
pursue continuing graduate study, to work in industry, and to teach in secondary schools
or community colleges. We strive to teach our students to communicate mathematical
ideas effectively and to use basic computational skills, mathematical models and technology
to solve practical problems. The Mathematics Department offers a variety of courses
that serve students in other departments, notably engineering, science, and business,
to help them use mathematics and quantitative analysis effectively in their chosen
field. The Mathematics Department strives to teach well, maintain high standards for
student performance, and keep its curriculum up to date. The Department also takes
pride in a faculty that is active in scholarship and research, which includes the
encouragement and supervision of student research in Center for Applied Mathematics,
Computation and Statistics (CAMCOS) projects and graduate student theses.
Outcomes - BA Mathematics & BA Mathematics - Preparation for Teaching
Goal 1 The Ability to Use and Construct Logical Arguments The ability to reason logically to conclusions, including the ability to use precise
definitions and to use various forms of logical argument.
Specific Learning Objectives to be assessed:
Ability to give direct proofs
Ability to give proofs by contradiction
Ability to give proofs by mathematical induction
Ability to apply definitions to give proofs
Ability to give proofs and disproofs involving quantified statements
Goal 2 3 The Ability to Perform Standard Mathematical Computations
Specific Learning Objectives to be assessed:
Ability to evaluate limits
Ability to calculate derivatives and integrals
Ability to apply properties of algebraic and transcendental functions
Goal 4 4. Ability to understand that there are limits
to numerical accuracy
Outcomes - BS Applied and Computational Mathematics
Goal 1 2 The Ability to Perform Standard Mathematical Computations
Specific Learning Objectives to be assessed:
Ability to evaluate limits
Ability to calculate derivatives and integrals
Ability to determine regions of convergence 4. Ability to apply properties of algebraic
and transcendental functions
Goal 3
Ability to understand that there are limits to numerical accuracy
Goal 4 The Ability to Use Mathematical Models to Solve Practical Problems
Specific Learning Objectives to be assessed:
Ability to extract relevant information from a practical problem and give a mathematical
formulation of the problem
Ability to use numerical results to validate (or modify) a model and to understand
the limitation of a model
Ability to clearly describe models, including an analysis of the strengths and weaknesses
of models and their relationship to the underlying problem |
The super-condensed guide to college algebra combines the authority of Schaum's with quick-study approach to help busy students achieve better grades. This new edition is enhanced with a free-access online diagnostic test: an extensive set of review questions that pinpoint weaknesses and ensure full mastery of the subject.About the Book Schaum's Easy Outlines of College Algebraprovides busy students with a powerful tool to review the subject rapidly. By paring textbook subject matter down to the essentials, this handy guide makes every minute of study time count.Packed with quick study tips and at-a-glance tables and diagrams, this book is perfect for test preparation, pre-exam review, and last-minute cram situations. It combines the academic authority that the Schaum's name is known for,...
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SEA-LINK BOOKS - SELLER'S NOTES -- PLEASE READ BEFORE PLACING ORDER! This is Schaum's Outline of Theory and Problems of College Physics. Includes 984 fully worked problems. Softcover book, in very nice condition. May have very minor shelf/handling wear. FAST SHIPPING! Please ask any questions you may have before ordering, and be sure you are ordering the right item. You are responsible to read ALL the item description, including the Seller's Notes (some items may differ somewhat from the title |
Algebra 1
9780078651137
ISBN:
0078651131
Pub Date: 2005 Publisher: Glencoe/McGraw-Hill School Pub Co
Summary: A flexible program with the solid content students need "Glencoe Algebra 1" strengthens student understanding and provides the tools students need to succeed--from the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests.
Holliday, Berchie is the author of Algebra 1, published 2005 under ISBN 9780078651137 and 0078651131. Eight hundred ei...ghty two Algebra 1 textbooks are available for sale on ValoreBooks.com, seven hundred seventy five used from the cheapest price of $9.09, or buy new starting at $45.00.[read moreNOTE! This is the Virginia Edition with some additional state-specific material, otherwise the text is identical to the national edition. Book is heavily worn. Cover corners [more]
NOTE! This is the Virginia Edition with some additional state-specific material, otherwise the text is identical to the national edition. Book is heavily worn. Cover corners bumped. No markings noted on pages. Multiple copies available. Your purchase benefits world-wide relief efforts of Mennonite Central Committee.[less |
Pre-Algebra Guide
A Top Seller! Well paying careers demand skills like problem solving, reasoning, decision making, and applying solid strategies etc. and Algebra provides you with a wonderful grounding in those skills – not to mention that it can prepare you for a wide range of opportunities. This is a COMPLETE Pre-Algebra guide to well over 325 [...]
Pre-Geometry Guide
Every day we use principles of geometry to help guide decision making and now the keys to this important subject can be at your fingertips! Geometry sharpens our reasoning, logic and problem solving skills and is one of those subjects we need to know not only to keep up, but to get ahead in the [...]
Pre-Calculus Guide
Calculus may not seem very important to you but the lessons and skills you learn will be with for your whole lifetime! Calculus is the mathematical study of continuous change. It helps you practice and develop your logic/reasoning skills. It throws challenging problems your way which make you think. Although you may never use calculus [...] |
Precalculus
Precalculus A Right Triangle Approach
Summary
Ratti and McWaters have combined years of lecture notes and firsthand experience with students to bring readers a book series that teaches at the same level and in the style as the best math instructors. An extensive array of exercises and learning aids further complements the instruction readers would receive in class and during office hours. Basic Concepts of Algebra, Equations and Inequalities, The Coordinate Plane, Polynomial and Rational Functions, Exponential and Logarithmic Functions, Trigonometric Functions Angles and Their Measure, Trigonometric Identities, Applications of Trigonometric Functions, Systems of Equations and Inequalities, Matrices and Determinants, Conic Sections, Further Topics in Algebra For readers interested in precalculus.
Author Biography
J.S. Ratti has been teaching mathematics at all levels for over 35 years. He is currently a full professor of mathematics and director of the "Center for Mathematical Services" at the University of South Florida. Professor Ratti is the author of numerous research papers in analysis, graph theory, and probability. He has won several awards for excellence in undergraduate teaching at University of South Florida and known as the coauthor of a successful finite mathematics textbook.
Marcus McWaters is currently the chair of the Mathematics Department at the University of South Florida, a position he has held for the last eleven years. Since receiving his PhD in mathematics from the University of Florida, he has taught all levels of undergraduate and graduate courses, with class sizes ranging from 3 to 250. As chair, he has worked intensively to structure a course delivery system for lower level courses that would improve the low retention rate these courses experience across the country. When not involved with mathematics or administrative activity, he enjoys playing racquetball, spending time with his two daughters, and traveling the world with his wife. |
Instructor: Abhinav Kumar
Office: Room 2-169
Email: my_first_name [at] math . mit .edu
Lecture Times and Location: TR 1 - 2:30 pm, 2-132
Office Hours: W 5-6, F 3-4
COURSE DESCRIPTION
Topic: Computational number theory
Description: We will discuss algorithms for
fundamental structures in number theory. There will
be some emphasis on actual calculations with a
computer algebra system such as sage. Topics to be
covered include : Euclidean algorithm, rational
reconstruction, Chinese remainder theorem, finite
fields and fast modular arithmetic, solution of
congruences, and randomized algorithms for primality
testing, factoring integers and polynomials over
finite fields. Applications in cryptography and
coding theory will also be explored. Some more
advanced topics such as Fast Fourier transforms, LLL
lattice-basis reduction, p-adic numbers and AKS
deterministic primality testing may also be
covered.
Since this is a seminar course, the students will be
doing most of the lecturing, with some help from me.
RECENT UPDATES
[02.10.2012] Here is a link to the textbook on Victor Shoup's webpage (we're using the 2nd edition). |
Elementary Algebra - Student Solutions Manual - 8th edition
Summary: Go beyond the answers--see what it takes to get there and improve your grade! This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives you the information you need to truly understand how these problems are solved |
This lesson from Illuminations teaches students to use a computer algebra system to determine the square root of 2 to a given number of decimal places. Students will learn how utilizing technology makes an algorithm...
This lesson from Illuminations asks students to look at different classes of polynomial functions by exploring the graphs of the functions. Students should already have a grasp of linear functions, quadratic functions,...
This math unit from Illuminations introduces students to the concepts of cryptology and coding. It includes two lessons, which cover the Caesar Cipher and the Vignere Cipher. Students will learn to encode and decode...
How financial institutions use the monthly mortgage payment and mortgage amortization formulas can be a confusing concept to grasp. This lesson asks students to find a current interest mortgage rate for their city and...
With this lesson, students will use tables of fees from a few different cell phone providers to create an algebraic expression that reflects billing for services. The example helps students apply algebraic functions to... |
5th edition of Model Building in Mathematical Programming discusses the general principles of model building in mathematical programming and demonstrates how they can be applied by using several simplified but practical problems from widely different contexts. Suggested formulations and solutions are given together with some computational experience to give the reader a feel for the computational difficulty of solving that particular type of model. Furthermore, this book illustrates the scope and limitations of mathematical programming, and shows how it can be applied to real situations. By emphasizing the importance of the building and interpreting of models rather than the solution process, the author attempts to fill a gap left by the many works which concentrate on the algorithmic side of the subject. |
You can use the catalog function to display the command onboard function manual.
The onboard function manual includes an explanation of the command, syntax information,
One or more examples of how to use the command, and the function key sequence.
*Example
•Syntax Help
When you input a command, syntax help of the command will appear on the screen.*Syntax help appears in the RUN-MAT mode(for the arithmetic calculation mode only), the PRGM mode, and the STAT mode.
*Example
•eActivity Guide
eActivity Guide is a function that helps you navigate through scientific function calculation operations.You can perform calculations simply by following the instructions that appear on the display.
1. Running a guide
Running a guide will sequentially display key operations and comments.
The students can become familiar with calculator operations simply by performing input as instructed.
*Example
*Select the strip and then press [EXE].
Press the key indicated by the key indicator that appears in the upper or lower right corner of the screen.
2. Creating a guide
Key operations are recorded in each eActivity strip. You can also add comments about key operations.
**You can get various eActivity Guide files here.
**fx-9860G Slim is equipped with the same features of fx-9860G. See here. |
Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
GeoGebra Geometry Workshop Instructions for Presenter Overview
This workshop is designed to introduce Middle School Math teachers to the wonders of GeoGebra. GeoGebra is a dynamic, interactive software that will bring the Mathematics classroom alive. With the help of GeoGebra, students will become involved, interested, and excited to learn the many concepts of Algebra and Geometry. The activities done in this workshop are designed for 6th, 7th, and 8th grade students. Presenters Materials
The presenter should be prepared with this document "Instructions for Presenter" to guide him/herself through the workshop, copies of the "GeoGebra Geometry Handout" for the participants (to be handed out at the end of the workshop), laptop, projector, and the following files: o Dynamic Worksheet – Isometries: Reflection (html), Rotation (html) (Located in folder called GeoGebra Files) o Dynamic Worksheet – Angles: Parallel Lines (html) (Located in folder called GeoGebra Files) o Power Point Presentation – "GeoGebra_Workshop_Geometry" (Located in the GeoGebra Workshop Folder) Each participant should have a laptop computer with GeoGebra downloaded before the workshop.
Table of Contents (Total Time: 90 minutes)
Discussion of teaching strategies (10-15 min) – page 2 Presentation of Dynamic Worksheet – Reflection and Rotation (10 min) – page 2 Introduction and Overview of Geogebra (10 min) – pages 2,3 Recreate Dynamic Worksheet – Reflection (10 min) – pages 3,4 Recreate Dynamic Worksheet – Rotation (15 min) – page 4 Presentation of Dynamic Worksheet – Angles (5 min) – page 4 Recreate Dynamic Worksheet – Angles (20 min) – pages 4,5 Bonus Material – pages 6-9
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
GeoGebra Geometry Workshop
The GeoGebra Geometry workshop will begin by setting up the following scenario and asking the following question: You're trying to get your students to understand the properties of isometries, (reflections, rotations, translations, and dilations), how do you usually show this to your students? Discussion: (Estimated time: 10 – 15 minutes) Participants may suggest showing the students a figure on a piece of paper and folding the paper to see the line of reflection and where the reflected image would be. For a rotation they may show a clock and investigate how the hands rotate, or they may draw a point of rotation on the board and show an object being rotated about that point. Keep a mental or physical list of suggestions and ask some probing questions such as: What happens to the reflected image when the original image moves closer or farther from the line of reflection? How would you go about showing reflection in a coordinate plane? What happens to the rotated image when the original is moved?
Presentation of Dynamic Worksheet: (Estimated time: 10 min) Files: Reflection (html), Rotation (html) Participants will be exposed to the use of a pre-made dynamic worksheet that will show an example of a reflection and a rotation. The teachers will be able to see how dynamic geometry software is so helpful when we want to make a change in the original construction and observe the consequences of that change. With this example we can observe how the reflected image changes as the original image is manipulated. Dynamic geometry software allows the student to see the material in a more entertaining and engaging way than with paper and pencil. The student becomes more interested in the material because of this. Once we have their interest and attention learning flourishes. Now is a good time for the presenter and the participants to have a discussion about the benefits of using this software as opposed to an overhead, whiteboard, or chalkboard. Introduction and Overview of GeoGebra: (Estimated time: 10 minutes) There are a couple of basics that need to be shown before constructing anything in GeoGebra. We will go over the following to get the teachers acclimated to how to navigate through the software. View menu: show axis, grid, and algebra window Toolbar: Click the arrow on the bottom right side of the button to see a list of the options within that menu.
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Pointer Button: Move mode, think of this as home base. If you want to get out of a mode click the pointer button. On the bottom left hand side of the screen you can always see what mode you"re in.
(This will be just an introduction to some of the options in GeoGebra, the other options will be discussed throughout the investigations in the workshop)
Investigation of Isometries
Goal 1: Create reflected image (Estimated time: 15 minutes) Open Blank GeoGebra File Step 1: Create Polygon Ask, "What would I need to construct first in order to show the students a reflection?" We need an image or a figure. Ask "Which button has the option Polygon?" After choosing the polygon tool, we will put down 5 points. Show and emphasize that you need to click on the first point to close the figure. Step 2: Construct a line of reflection Ask, "What will I need next to reflect this image?" The participants should tell you that you need a line of reflection (that will be a line of symmetry). Have the participants find the line tool and construct a line of reflection. Step 3: Reflect the original polygon Now to reflect the image over the given line we would need to find and select the reflection tool, the select the polygon (notice how it is bolded when the pointer moves over it) and then select the line of reflection. Step 4: Observations At this point the participants can play with their construction for a few minutes. Ask what they observe and to describe their observations. Can they move and change the reflected image, why or why not? Why are the points in the reflected image labeled like they are? Can you move the line of symmetry, how? Step 5: Use the properties menu Right click on the original image and select the properties. Demonstrate the many different properties that can be changed (color, thickness, show label, etc…). Show how you can stay in the properties menu to change the properties of any of the objects in the construction.
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Step 6: Extension (If time permits) From here we can connect the corresponding points with segments. We could find the intersection points of this segment and the line of symmetry, then by showing the segment"s lengths between corresponding points and showing the length from one of the points to the line of symmetry and observe the relationship between those distances.
Goal 2: Create rotated image (Estimated time: 15 minutes) Open Blank GeoGebra File Step 1: Create polygon Step 2: Rotate polygon Ask the participants to find the rotation mode. Before using this mode read the description to see what we would need in order to rotate. We need a point to rotate around, we could use one of the vertex points of the polygon or another point we construct. Place a point anywhere on the drawing pad, then choose the rotate tool and click on the polygon and the point. At this moment GeoGebra will ask what angle of rotation you would like, you can type in a degree, name of an angle, or slider. Step 3: Create a slider for the angle of rotation Find and select the slider tool and click somewhere in the drawing pad. Select the angle option for the slider and we can change the name to anything we want, let"s say "a". We can change the range of values and the incremental change in the angle. Now choose the rotation tool. Click on the polygon and the point of rotation, when the screen appears for the angle, type in "a" and click "ok". Go to the pointer mode and slide the point on the slider to change the angle of rotation. Observe what happens when the slider moves.
Investigating Angle Relationships
Goal 1: Two parallel lines cut by a transversal (Estimated time: 20 minutes) Open Blank GeoGebra File Step 1: Construct a line and a line parallel to that line Step 2: Construct a line that, transversal, which intersects the two parallel lines Step 3: Use the Angle tool to measure the angles, more points may have to be constructed on the lines so we have three points to pick for each angle.
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Step 4: Move points and lines around and observe which angles will always stay congruent when two parallel lines are cut by a transversal. Make conjectures about your observations. Here is an example of two images from a GeoGebra file that followed the steps above.
Goal 2: Investigating Triangle Sum (Estimated time: 25 minutes) Open Blank GeoGebra File Step 1: Construct triangle ABC, using the segment tools Step 2: Measure the three interior angles of the triangle Step 3: Calculate the sum of the angles by inserting text and typing in "sum =" + This is assuming that the three interior angles are named , , and . Step 4: Drag the vertices around and observe the individual angle measures change while the sum remains constant, 180 degrees. Proving the observation Step 4: Create a polygon for Triangle ABC Step 5: The participants will prove this using a visual model. A line parallel to one side of the triangle through the opposite vertex will be constructed. Then the triangle will be rotated about the midpoints of the other sides. Step 6: Find the midpoints of the other two sides, and then set up two angle sliders to rotate the original triangle about the midpoints. Move the sliders to see if you can get the three angles to form a straight angle (180 degrees), proving the observation we saw earlier. 5
Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Here is an example of two images from a GeoGebra file that follows steps 5 and 6 above. We can see in the picture on the right that the three interior angles of the triangle form a straight angle, 180 degrees. This should be a sufficient visual proof of the triangle sum theorem.
Bonus Material (If time permits)
If time allows we will like to present some worksheets to show some of the other options that can be done using GeoGebra. Some of the worksheets will include but not be limited to: Median/Centroid Investigation
After the construction we will look at how the two segments on each median relate to each other. The sum, difference, and product always seem to change. However, the quotient seems to be the same.
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Let"s see what happens when we change the orientation of the triangle.
Even with the different orientation of the triangle we notice that the quotient is still constant, and it"s still 2! What conjecture can we make about the lengths of the segments on each median?
ASS Triangle Congruency Test Investigation
The triangle was constructed using
LK ,
KI , and , making sure the angle is not included in the given segments. We can see that there are two triangles that can be formed, however, is there a time when one triangle is formed? If so, what condition should be met?
Can we use GeoGebra to prove or disprove any other possible triangle congruency tests (AAA, ASA, SAS, SSS, SAA)?
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Investigating Symmetry in pictures/images
Another great use of GeoGebra is to use it to check for line symmetry. A nice way to do that is to get a picture and import it into GeoGebra. Then attempt to construct what you think may be a line of symmetry, using GeoGebra we can test to see if it really is a line of symmetry. We will first do this by inserting a picture to the geometry window of GeoGebra, to do this we need to go to the following menu:
After choosing the correct tool we need to click anywhere in the geometry window and browse our hard drive for a picture we would like to insert.
We can then resize the picture by fixing its corner points. We can do this by constructing two points and making those new points the corner points of the image.
Then by right clicking on the image we can choose the properties of the image. In the properties window we can change the placement of 3 of the corners, we are only changing the position of 2 corners and leaving the third one (also the fourth one) free. Here we change Corner 1 to point A and Corner 2 to point B. Then by moving point A or B we can resize the image. Once we are satisfied with the size of the image we want to set it as a Background image, this can also be done in the properties window.
Now it is time to construct a line that we think may be a line of symmetry, then construct a point on one side of the line on the figure and reflect that point by about the line, then put the trace on both points. Then pick the original point and move it along the picture and see if the reflected point traces on or off the picture. For example:
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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco
Appears to be a line of symmetry
Does not appear to be a line of symmetry
Some of the Sunshine State Standards being addressed in this workshop are:
Strand C Geometry and Spatial Sense Benchmark MA.C.1.3.1 The student understands the basic properties of, and relationships pertaining to, regular and irregular geometric shapes in two and three dimensions. Standard 1 The student describes, draws, identifies, and analyzes two- and three-dimensional shapes. Benchmark MA.C.2.3.1 The student understands the geometric concepts of symmetry, reflections, congruency, similarity, perpendicularity, parallelism, and transformations, including flips, slides, turns, and enlargements Standard 2 The student visualizes and illustrates ways in which shapes can be combined, subdivided, and changed. Benchmark MA.C.3.3.1 The student represents and applies geometric properties and relationships to solve real-world and mathematical problems. Standard 3 The student uses coordinate geometry to locate objects in both two and three dimensions and to describe objects.
Note: The Geogebra Geometry Handout will be given to the participants at the end of the workshop as an overview of the steps that were taken throughout the activities. Note: The PowerPoint slide show (GeoGebra Workshop (Geometry)) will be used as an outline of the workshop with objectives, goals, tasks, and Sunshine State Standards |
Mathematics?: An Elementary Approach to Ideas and Methods
The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ...Show synopsisThe teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but not real understanding or greater intellectual independence. The new edition of this classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. "Lucid . . . easily understandable".--Albert Einstein. 301 linecuts.Hide synopsis
Description:Fair. Good study copy, shows heavy wear, text has markings, a...Fair. Good study copy, shows heavy wear, text has markings, a good study is Mathematics?: An Elementary Approach to Ideas and Methods
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning ofmathematics |
1. Download the algebra worksheet on your computer. 2. Follow the directions on the sheet and use the websites below to complete the algebra problems. 3. Email your work to your teacher when you have finished. |
Precalculus courses often serve two distinct purposes: preparing students for calculus
and improving the quantitative literacy of students going into a wide variety of fields.
Nationally, a large number of students who take precalculus do not go on to take calculus
so it is important for us to think about the experiences we should provide for such students.
Of the students who do not go into calculus, many will go into fields or take courses which involve
gathering, organizing, and interpreting data. We propose to talk about a version of precalculus in which
data interpretation is introduced in the classes and homework, and practical experience with gathering
and interpreting data is provided by a series of labs. We expect to demonstrate as many of the materials
as time allows. (Received Sept. 19, 1998)
This survey addresses whether or not students enter South Carolina (SC) higher education institutions with the appropriate
background to be successful in college mathematics. Recent releases of SC performance from National Assessment of Educational Progress,
the College Board, and so forth give parents and the general public some indication of student preparation. We decided to access a source different
from these formal evaluations. We asked the people who know best--the college faculty (from all state-supported schools) who
provide instruction in the initial college mathematics course. This report presents their answer to whether or not our students are prepared. The
survey contains interesting information about a number of important issues: trends in precollege preparation; calculus in high school; preparation in
arithmetic, geometry, albegra, trig, probability/statistics; study skills; problem solving; student attitudes; calculators and more. We feel that
what we discovered about quantitative literacy in our state will be of general interest. (Received September 23, 1998)
Rhode Island College has no official "quantitative literacy" course and is unlikely to be able to establish
one. However, this talk will describe how some QL elements (such as citizenship education, recognizing plausibility,
making inferences from quantitative information, working with large numbers, detecting fallacies) are being incorporated
into a variety of lower division courses, including algebra, "technical" math, QBA, and basic statistics. An emphasis on multistep
problems, or those with multiple methods of solution, which this contributor feels are too often underemphsized, will be described.
(Received September 23, 1998)
Mathematics Across the Curriculum is an NSF-funded program (DUE 9354652) whose aim is to improve the quantitative ltieracy of students
and promote their appreciation of mathematics by integrating quantitative components into a variety of courses in the university curriculum.
It was conceived at the University of Nevada-Reno, along the lines of a successful Writing Across the Curriculum model. Our goal is to enhance
the quantitative content of courses where it exists and introduce it where it does not, thereby increasing students' exposure to applications
of mathematics in a variety of courses and making quantitative learning a "shared responsibility" across campus. In this talk we will describe
some of the features of the project that affect quantitative literacy. In particular we have implemented math components in political science,
humanities, anthropology, psychology, and art, among others. A Math Across the Curriculum web site has been established that allows users to
get detailed information and to download documents and resources. The URL is (Received September 23, 1998)
A discussion of approaches to stimulating student interest and willingness to invest time in understanding mathematics. Fifteen class sections
of a one-semester quantitative literacy mathematics course are evaluated. Clear among the various results is that student participation in a
seminar-type project presentation outside of the classroom very early in the semester increases student dedication to the course and course work.
The programming and use of calculators (an aplication of the symbolic logic taught in the course), to simplify tedium without eliminating conceptual
mastery, are also investigated.
Additionally, the effect of investigating with students the accepted meanings of "teacher" and "classroom" in mathematics are examined. Early analysis
indicates the method is motivational for those students for whom the ritualized rigor of mathematics is anathema (about 50% of most class sections).
The results, including comparisons of student self-perception with the same/different instructor and same course material, are discussed.
(Received September 24, 1998)
Survival in the 21st century will require fundamental quantitative skills that all college and university students should possess. In this talk, we
will begin by briefly surveying the rationale for developing a quantitative reasoning (QR) course for liberal arts students and discussing the issues
that arise in teaching such a course. This discussion will draw on the presenters' experience in developing such a course at the University of Colorado
and writing a textbook (Addison-Wesley) to support the course. The remainder of the talk will consist of classroom activities and presentations that are
used in the course. The topics that will be surveyed in these activities include (as time permits): an in-class simulation with dice of the spread of a
disease (among workshop participants!) and its modeling; supply and depletion problems as they arise in finance, engineering, drug use, and environmental
sciences; mathematics and music; the uses and abuses of percentages; mathematics in the news; the mathematics of voting. The talk is designed for instructors
at two-year and four-year colleges who are contemplating teaching QR courses, as well as for instructors who are already teaching QR courses. (Received September 24, 1998)
This Web-based Intermediate Algebra MInicourse is desinged for entering freshmen who have failed the California State University Entry Level Mathematics
(ELM) Examination with a score that falls within the upper quartile of all students who failed the test. Typyically, students who fail the ELM must take a course
in Intermediate Algebra The assumption in developing this minicourse is that students who come close to passing a placement examination need less insturctional
time than do others who fail the same examination. Because the target audience consists of students still in high school, there is a strong rationale for a course
that requires minimal time on campus. The creation of a Web-based remedial mathematics course helps the University meet two of its major objectives: (1) Decrease
the amount of remediation on campus and (2) develop strategies that allow and encourage incoming freshmen to resolve their deficiences while still in high school.
(Received September 24, 1998)
At Hope College, with the partial support of the National Science Foundation, a team consisting of two mathematicians, a geologist, and a biologist, is developing
three new general education courses. The mathematics course (Mathematics in Public Discourse) is a co- or pre-requisite for the science course (The Atmoshpere
and Environmental Change and Populations in Changing Environments). The science courses will use and build on the mathematics the students have learned.
The courses connected by common content themes (the use of functions, graphs, and statistical analysis of data to interpret the physical world); common pedagogy (exploratory labs,
investigative worksheets, cooperative learning); and common technology (the IT-83 and DBL). These courses were piloted Spring 98 with approximately 20 students. During Fall 98,
there are 150 students enrolled in the mathematics course and 40 students enrolled in the biology course (the atmospheric science course will be taught Spring semester). The assessment
done during the pilot study indicated that both attitudes and mathematical ability improved. (Received September 27, 1998)
VCU offers a course "Contemporary Applications of Mathematics: that is required of all non-science majors. Its goal is to improve students' ability: to study quantitative situations; to
use quantitative skills in actual siatuions; to learn through reading and communication with others; and to explain mathematics in writing and orally. Topics include optimal routing; fair
division; election theory; linear and exponential growth; interpreting and explaining data. As well as taking tests, each student in each section makes weekly entries in a log; studies
mathematics as a member of a team and orally describes this mathematics to others; writes two three-page
typed papers; studies topics independently to prepare for a "poster-session". The course was developed and piloted with Gwen Turbeville of J. Sargeant Reynolds Community College.
The challenge has been to devleop materials and approaches to make it possible to teach this course to large numbers of students. Wtih support from the NSF-supported Virginia
Collaborative for Excellence in the Preparations of Teachers (VCEPT), currently 25 different instructors (faculty, part-time instructors and GTAs) annually enjoy teaching this
course to 2,000 students. Its features are being adopted by other VCEPT institutions.
(Received September 28, 1998)
The presentation will report on an NSF-funded project for implementing a 6-hour quantitative literacy core course at Sam Houston State University. The quantitative literacy course
integrates mathematics and computer science content to extend student skills in interpreting and analyzing problems, applying problem solving strategies, identifying appropriate quantitative tools,
and applying those tools in the construction and communication of solutions. Students are expected to exhibit teamwork and a sense of scientific inquiry through student projects and lab activities.
Freshman students from mathematically weak and technologically disadvantaged backgrounds have been encouraged to take the course, giving them a jump-start in their collegiate experience. Curriculum
materials have been customized and extended from exemplary materials in mathematics and computer science. Topics include: internet services, word processing, presentation and graphics software,
spreadsheets, functions, trigonometry, descriptive statistics, finance, and linear programming. Effects on student retention, attitudes, and comparisons with current general education classes in
mathematics and computer science will be presented. Samples of student work will be included.
(Received September 28, 1998)
The use of investigations, prljects, reading questions, and web sites can be used to help students take responsibility for their learning in a precalculus course. This has resulted in a course
that is more in-depth and one that is quite different from a traditional course. The inviestigations and projects allow students to be actively engaged in the matieral they are studying. The
investigations are more elaborate and more in-depth than exercises. They usually involved discussion questions and are completed by studnets working in groups. The projects are longer and more
involved than the investigations. Students are also encouraged to be creative when writing up their projects. To allow enough class time to work on investigations and projects, it is expected
that studentslearn much of the basic material on their own. To help them with this, a number of things are done. The text is written for students to read and they are expected to read it. To
guide them in this, reading questions are interspersed throughout the reading. Using these questions helps students pick out the important concepts and checks them on their understanding of these
concepts. Also, the instructor's class notes, assignments, and solutions to assignments are placed on the web for student access.
(Received September 30, 1998)
Our approach combines computer-based assignments with learning communities. The core course is "Introduction to Sociology". Students learn and apply statistical concepts to sociological questions
using the STATA analysis package and social survey data. There are three sections per term, one is a learning community section. This section has a joint enrollment with either freshman composition
or intro to statistics. Faculty cooperate, but do not team-teach. Among our findings are: (1) students in the English/Sociology Community perform at a higher level; (2) students in the Statistics/Sociology
Learning Community performed poorly; this may be a result of problems in integrating the two courses; (3) when the classes involved qualitative ways of conceptualizing of statistical concepts there was accelerated learning;
(4)when students indentify with each other, they teach each other, and transform initial anxiety into confidence; (5) the focus on the logic of writing improves problem solving abilities. We note that this computer-based approach
requires additional time for students and faculty and that a Learning Community approach requires active faculty management. This effort was funded by the Long Island Consortium for Interconnected Learning.
(Received October 2, 1998)
In the fall of 1997 Domincan University faculty completely revised the general education requirements for the College of Arts and Sciences. Part of that revision was the addition of another math course beyond the
previous requirement of Intermediate Algebra. Courses already offered that would satsify the requirement included College Algebra (and several others). One of the selling points of the proposal, drafted by our department,
was the creation of a new general math course that would satisfy the requirement. The intent was to create a course more relevant to students who did not plan to take any additional mathematics. In my presentation I will
discuss the contents of the course. In it I try to focus on developing consumers more than creators of mathematical content. I will also talk about the difficulties involved in getting such a course embraced by the university.
Not only did we struggle to get the additional mathematics requirement in the new curriculum, but now that we have it we are still struggling to get advisors to place students into the new course. Most of
the enrollments las spring were still for College Algebra, even though many of the students in that course would have been much better served by the new course.
(Received 2, 1998)
Representatives of a dozen colleges in the northeast have been meeting for the last two years to pursue a common
interest in implementing quantitative reasoning courses and quantitative support on their several campuses.
Members of the Math Departments of two of the schools, Trinity and Skidmore, will report on their efforts to move
beyond the first tier of foundation courses to develop second-tier quantitative courses and modules for non-math courses.
Judith Moran, Director of the Math Center at Trinity, will describe QL courses incorprating Hartford data which were
created under Trinity's Urban Curricular Initiative, as well as quantitative modules and laboratories designed
for science and social science courses. Gove Effinger, Director of Skidmore's QL program, will describe several
second-tier courses created at Skidmore, including Statistical Controversies, Mathematics and the Art of M.C. Escher,
and The Theory of Epidemics. Materials from both schools, as well as information about programs at the other colleges
in the Northeast QL group, will be available.
(Received October 2, 1998)
Under a major NSF grant: The Integration of Science and Mathematics into a Humanities Curriculum, science and mathematics
faculty at Trinity College have created laboratories for a variety of humanities and social science courses. Helen Lang, Chair of
Trinity's Philosophy Department, and PI of the grant, will report on the overall project and the range of courses affected and
laboratories created. Judith Moran, Director of the Math Center at Trinity and a consultant on the grant, will describe in
more detail two of the laboraties: on Mayan mathematics and Mayan astronomy, that she has created for a Latin American history course.
(Received October 2, 1998)
The presentation will describe our quantitative literacy course at Appalachian State University which we believe catches the
spirit of the "Crossroads" report. It includes a computer lab and lots of hands-on work in areas of mathematics that sutdents see as useful
- Finance, Statistics, Trigonometry, and Linear Programming. Students solve problems using the computer and write reports on
their solutions. Our course won the award given by the Annenberg Foundation for quantitative literacy courses that use technology
well. You can check out some of what we are doing at
(Received October 5, 1998)
The Quantitative Reasoning Program at DePauw University was developed in the early 1980's as part of a three-part competency program for all students.
Students may enter the program at either of two levels. Placement issues and descriptions of the evolution of the developmental course(s) will be discussed.
In addition, a list of the second level "Q" courses, from many different departments, will be used to illustrate the broad base of this program.
One of the strongest features of the program is the faculty development component. Participation in an intensive faculty workshop is required in order for
faculty members andone or more of their courses to be part of the quantitative reasoning program. A description of the workshop will be included in this presentation.
The Quantitative Reasoning Center, part of the Academic Resource Center, is the supporting arm of the program for students. Peer consultants are recruited,
trained, and supervised by the Center Director. A look at how all this works will be included if time permits.
(Received October 7, 1998)
In this paper I will talk about some pedagogical issues related to piloting a quantitative literacy course at a liberal arts institution and some projects
I've had my students work on. These projects tend to be intersting and useful (to the student) and hence are generally well received.
Some of the issues that will be addressed are:
Which majors are better served by a quantitative literacy course?
What is the underlying philosophy of instruction (e.g. skills versus appreciation)?
What should the prerequisite be?
What are the goals and objectives of the course? How do they affect the course content?
What assessment criteria would be appropriate for such a course?
What is a working definition of a quantitatively literate person?Why would anybody want to pilot a course in quantitative literacy?
On the lighter side, I will also share some of the humorous responses I got from students who forgot to "determine the reasonableness" of their answers.
(Received October 14, 1998) |
The O-Level Mathematics syllabus builds on the Standard Mathematics syllabus. The N(A)-Level Mathematics syllabus is a subset of O-Level Mathematics, except that it revisits some of the topics in - Standard Mathematics syllabus.
GCE Ordinary Level (Syllabus 4016) AIMS The syllabus is intended to provide students with the fundamental mathematical knowledge and skills. The general aims of the mathematics syllabuses are to enable students to: 1.
O Level Additional Mathematics syllabus are assumed in the syllabus below and will not be tested directly, but it may be required indirectly in response to questions on other topics. The assumed knowledge for O Level Additional Mathematics is appended after this section.
Mathematics as well as other maths-related subjects. See fora full list of the qualifications ... • 4029 O Level Mathematics (Syllabus D) (Mauritius) Candidates can combine syllabus 4029 in an examination session with any other CIE syllabus, except:
Mathematics Teaching Syllabus, National Curriculum Development Centre. Ministry of Education and Sports The Republic of Uganda MATHEMATICS TEACHING ... one for each level of secondary school, i.e. S.1 to S.4. S.1 has 21 topics, S.2 has 17 topics, ...
The percentage uniform mark is stated at syllabuslevel only. It is not the same as the 'raw' mark obtained by the candidate, since it depends on the position of the grade thresholds (which may vary from one series
from the corresponding syllabuslevel at Junior Certificate is assumed. It is also envisaged that at all levels students will engage with a dynamic geometry software package. In particular, at Foundation level and Ordinary level
As a result, the Alevel mathematics syllabus was strengthened with many topics, which had been dropped in the 90s, put back in. The result was a national crisis in 20017, ... Thus Alevelmaths teachers should have a degree in maths ...
The Cambridge O Levelsyllabus also allows teaching to be placed in a localised context, making it relevant in varying regions. ... for further study in Mathematics as well as other maths-related subjects. See fora full list of the qualifications you can take.
to the ALevelsyllabus and additional specimen questions on Vectors, Complex Numbers and Continuous Random Variables; Past question papers, examiner reports and mark schemes will be available following each examination session.
O Level Mathematics (syllabus D) (4024) Will students be given a formula sheet to help them in the exam or do they need to ... O levelMaths question paper weighting is 50% per paper; IGCSE Maths is 35% for the first paper, 65% for the second.
the ALevelsyllabus and additional specimen questions on Vectors, Complex Numbers and Continuous Random Variables; Past question papers will be available following each examination session. All the above are available from CIE Publications.
concepts; the inability to deliver the prescribed syllabusfor each class/grade level within the expected time frame; ... Maths/Problem-solving Infant Yr. 1 (Pre-number) zero to ten -Recognition of number names to 10 -Number value to 10
TEACHING SYLLABUSFOR MATHEMATICS ( JUNIOR HIGH SCHOOL 1 – 3 ) Enquiries and comments on this syllabus should be addressed to: ... competencies developed at the J.H.S. level are necessary requirements for effective study in mathematics, science, ...
Subject to the requirements of the particular ALevelsyllabus, an award in an AS Levelsyllabus may be used toward the award of an ALevel in the same subject in a subsequent session within a 13 month period . These syllabuses are shown with the asterisk in 'Subjects
GCE, GCSE, and Entry Level is designed to ensure equal access to qualifications for all students (in compliance with the Equality Act 2010) ... Email bulletins from The Maths Team at Edexcel are a fantastic way to be kept informed of all the developments that are taking place in |
Modify Your Results
In this textbook there are many problems that ask you to look at situations in new and different way. This chapter offers some strategies to approach these problems. Although some of the problems in this chapter are fictitious, they give you a chance to practice skills that you will use throughout the book and throughout life.
The algebra you find in this book won't look quite like the algebra you may have seen in older textbooks. The mathematics we learn and teach in school has to change continually to reflect changes in our world. Our workplaces are changing, and technology is present everywhere, fundamentally changing the work we do. There are some new topics that are now possible to explore with technology, and some standard topics that can be approached in new ways.
A procedure that you do over and over, each time building on the previous stage, is recursive. You'll see recursion used in many different ways throughout this book. In this lesson you'll draw fractal design using a recursive procedure. After you draw the design, you'll work with fractions to examine its parts |
Middle schoolers will identify the parts of an algebraic expression by telling the number of terms in one expression. Then they will collect like terms and solve three problems. Next, they will complete a table involving three sets of algebra tile displays, evaluating each with a given value of x. In all they will solve 4 application problems. Answer key is not included.
Students investigate the achievements of the Chinese in mathematics. In this algebra lesson plan, students convert Chinese number and decimal numbers and solve a 3x3 systems of equation. They relate art to mathematics alsoMiddle schoolers create a method for finding the area of a fame for a picture and then transfer their shared methods into algebraic expressions. They develop the algebraic language to communicate and solve problems effectively and use variables and symbols to write equations. Pupils use the computer to examine graphic representation of their equations.
Learners understand that algebra is a branch of mathematics that uses symbols or letters to represent unknown numbers in problems. They also understand the definition for an algebraic expression. Make sure to click on the Download the Activity bear so that you can access a top-notch task document that walks learners through the evaluation of algebraic expressions. It can be used as part of your lesson or sent home as reinforcement.
The instructor demonstrates how to plug-in given variables to rewrite an algebraic expression as a numerical expression. Follow the order of operation rules to simplify the numerical expression once you have plugged in those given values.
Students attempt to make a transition from arithmetical to algebraic thinking by extending from problems that have single-solution responses. Values organized into tables and graphs are used to move toward symbolic representations.
How does algebra relate to real life? High schoolers will construct and test computer chips to determine the function of each, and discuss real life applications of the Boolean Algebraic concept. Included are discussion questions, gender equity considerations, web links, a clear procedure, and all necessary worksheets. The format is a slide show.
This easy-to-understand video demonstrates how to find the value of multi-step algebraic expressions with substitution. It reviews order of operations to help learners avoid common mistakes when solving. The sample problems show all four operations and use different values for substitution. As an added bonus, the resource includes a Spanish version of the video.
Students assess how algebra, telescopes, space exploration and optics are so important in astronomy. They encounter studies on the Hubble Space Telescope, Hubble Deep Field and how algebra helps to determine the effects of contamination on Hubble's optics. Students are introduced to the development of the Next Generation Space Telescope.
Middle schoolers will read word problems, create questions, and come up with solutions. They are taught to use previous knowledge combined with current content to solve problems logically and algebraically.
Students add and subtract polynomials. In this algebra lesson, students use algebra tiles to perform operation of polynomials. They model their problems and understand how to combine like terms using tiles.
Students discover the applications for Transistors. In this technology lesson, students work with transistor circuits to apply the Boolean method when solving algebra problems. There is a diagram included in this worksheet.
In this expansions worksheet, students use area to solve expansions of brackets problems. There are 4 of this type, each problem having less guidance than the previous. The second activity asks students to solve magic square puzzles using algebra. In the third activity, students use a grid to determine the answers when raising (x+1) to different powers. There are 10 problems on this worksheet.
How do algebra learners know the value of an algebraic expression that contains a variable? This video shows your number crunchers how to evaluate an algebraic expression by substituting a given number for the variable. In fact, it models an algebraic expression where the value of the variable changes and demonstrates how this changes the value of the expression. The video is part of a series on reading, writing, and evaluating algebraic expressions. |
Young scholars read an article on how calculus is used in the real world. In this calculus lesson, students draw a correlation between the Battle of Trafalgar and calculus. The purpose of this article is the show everyday uses for calculus in the real world.
In this implicit differentiation worksheet, students compute the determinant of the Jacobian matrix and solve equations by implicit differentiation. This two-page worksheet contains definitions, examples, and explanations. It contains approximately eight multi-step problems.
In this Calculus worksheet, students are provided with practice problems for their exam. Topics covered include derivatives, area bounded by a curve, local maximum, instantaneous rate of change, and the volume of a solid of revolution. The four page document contains seventeen multiple choice questions. Answers are not included.
In this Calculus worksheet, students are provided with questions that are reflective of the content of their exam. Topics covered include derivatives, volume of a solid of rotation, local maximum and minimum, and integration. The one page worksheet contains seventeen multiple choice questions. Answers are not provided.
In this trigonometry instructional activity, students differentiate the different trigonometric identities. They derive the sine, cosine, tangent and inverses of these trig identities. There are 18 questions with an answer key.
In this calculus activity, students use integration to solve word problems they differentiate between integration and anti derivatives, and between definite and indefinite integrals. There are 3 questions with an answer key.
Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled �Proof of Derivatives of Ln(x) and e^x,� has a clearer explanation of this proof.
Using a specific example, Sal shows how to find the equation of a tangent line to a given function at a specific point. Specifically, he solves the problem of finding the tangent line to the function f(x) = xex at x = 1. This problem provides a review of the product rule, slope-intercept form of a line, and steps for finding the equation of a line. It also, provides a nice visual understanding of the problem by graphing both the original equation and the found tangent line.
Twelfth graders investigate the capabilities of the TI-89. In this calculus lesson, 12th graders explore the parametric equation for a circle, for arc length of curves, and for trajectories. Students investigate the symbolic and graphical representation of vectors. Students use polar functions of investigate the area bounded by a curve. Students investigate a 3D graphing application.
In this calculus worksheet, students solve 10 different problems that include determining the first derivative in each. First, they apply properties of logarithmic functions to expand the right side of each equation. Then, students differentiate both sides with respect to x,using the chain rule on the left side and the product rule on the right. In addition, they multiply both sides by y and substitute.
In this capacitance learning exercise, students solve 19 problems about capacitance, voltage, electric charge and Ohm's Law. They use calculus to solve some of the problems and they are given equations used to solve different capacitance problems.
Directions are written to solve a related rate problem step by step. There are five example problems to practice solving for related rates. Use of the Chain Rule and/or implicit differentiation is one of the key steps to solving these word problems |
Finite Mathematics - 2nd edition
Summary: The Second Edition of this engaging text for the one-semester finite mathematics course continues to use intriguing, real-world applications to capture the interest of business, economics, life science, and social science majors. This practical approach to mathematics, along with the integration of graphing calculators and Excel spreadsheet explorations, exposes students to the tools they will encounter in future careers.
Summaries and reviews appear freque...show morently throughout the text to support students' mastery of mathematical concepts. A wealth of pedagogy includes the following distinctive features: detailed Worked-out Examples with Annotations help students through more challenging concepts; Practice Problems are offered to help students check their understanding of concepts presented in the examples; Section Summaries briefly restate essential formulas and key concepts; Chapter Summary with Hints and Suggestions unify chapter themes, give specific reminders, and reference problems in the review exercises suitable for a practice test; and Cumulative Review Exercises appear at the end of groups of chapters to reinforce previously learned concepts and skills.
Graphing Calculator Examples and Exercises located throughout the text explore new topics, guide students through "messy" calculations, or show technology pitfalls. These are optional and may be omitted without disrupting the flow or cohesion of the text.
Application Previews place mathematics in a real-world context and motivate students' interest in the material. Some examples of the diversity of applications covered include sports, genetic engineering, spread of disease, gambling, business, and environmental issues.
Annotations beside many formulas and solution steps emphasize the importance of being able to "read mathematics" by restating much of the mathematics in words.
0618372210 WE HAVE NUMEROUS COPIES, -HARDCOVER, small tear (less than 1/4") on top edge of spine repaired with tape, light to moderate wear to cover/edges/corners, generally clean inside, student na...show moremes and school markings on book and inside, minimal writing/highlighting with no detracting writing inside book, strong, solid binding ...show less
$138245 |
The Probability & Statistics Tutor: Learning By Example DVD Series teaches students through step-by-step example problems that progressively become more difficult. This DVD covers Mathematical Expectation in Probability and Statistics, including what Mathematical Expectation is and why it is important. Grades 9-12. 25 minutes on DVD. |
Pre-Algebra DVD with Books
Pre-Algebra for Distance Learning
Pre-Algebra (2nd edition) eases the transition from arithmetic to algebra. Algebraic expressions and linear equations are applied throughout a thorough review of operations on integers, fractions, decimals, percentages, and radicals. Students explore relations and functions using equations, tables, and graphs. Chapters on statistics and geometry extend foundational concepts in preparation for high school courses. Problem solving and real-life uses of math are featured in each chapter. Dominion through Math exercises regularly illustrate how mathematics can be used to manage God's creation to His glory.
>> If you plan on using this distance learning kit before May 1, 2014, please contact Customer Service at 800.845.5731 to place your order.
Printed Teacher's Editions are not included with this kit. A complete copy of the Teacher's Edition (formerly TE on CD) and the Teacher's Toolkit CD as well as files for printing materials from the Teacher Support Materials CD are available by logging into your account (Available May 1, 2014). If you desire a printed Teacher Edition, please contact Customer Service for special pricing.
All DVD and Hard Drive Distance Learning kits now include a Parent Guide as well as a Student Worksheets Packet which have all of the teacher support materials and answer keys needed for the subject(s). Learn more.
About the Instructor
Mr. Bill Harmon, BS
Bill Harmon has loved science for as long as he can remember. After completing his B.S. in Chemistry, he returned to Florida where he gained experience teaching a variety of subjects: science, math, Latin, and computer courses. Now he works as a chemist in the Safety Services Office at BJU, teaches Distance Learning Physics and Algebra, and teaches Chemistry at Bob Jones Academy. He is currently pursuing an M.Ed. in Secondary Education. He and his wife Mary Ann have two children, Brian and Janette. His favorite Bible verse is II Timothy 3:14. |
Introduction to computer algebra systems (MAPLE), symbolic computation, and the mathematical algorithms employed in such computation, with examples and applications to topics in undergraduate mathematics. |
This little book is especially concerned with those portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (
Calculus Deconstructed is a thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof. This book is appropriate for a beginning Honors Calculus course assuming high school calculus or a "bridge course" using basica analysis to motivate and illustrate mathematical rigor. It can serve as a combination textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in context of a coherent logical structure, building on famili
Designed to accompany the eighth edition,which continues to evolve to fulfill the needs of a changing market by providing flexible solutions to teaching and learning needs of all kinds. The new edition retains the strengths of earlier editions such as Anton's trademark clarity of exposition, sound mathematics, excellent exercises and examples, and appropriate level. Anton also incorporates new ideas that have withstood the objective scrutiny of many skilled and thoughtful instructors and their students.
The sequel to How to Ace Calculus, How to Ace the Rest of Calculus provides humorous and highly readable explanations of the key topics of second and third semester calculus—such as sequences and series, polor coordinates, and multivariable calculus—without the technical details and fine print that would be found in a formal text.
Stewart's CALCULUS: CONCEPTS AND CONTEXTS, THIRD EDITION offers a streamlined approach to teaching calculus, focusing on major concepts and supporting those with precise definitions, patient explanations, and carefully graded problems. CALCULUS: CONCEPTS AND CONTEXTS is highly regarded because it has successfully brought peace to departments that were split between reform and traditional approaches to teaching calculus. Not only does the text help reconcile the two schools of thought by skillfully merging the best of traditional calculus with the best of the reform movement, it does so w
Logic forms the basis of mathematics and is a fundamental part of any mathematics course. This book provides students with a clear and accessible introduction to this important subject, using the concept of model as the main focus and covering a wide area of logic. The chapters of the book cover propositional calculus, boolean algebras, predicate calculus and completelness theorems with answeres to all of the exercises and the end of the volume. This is an ideal introduction to mathematics and logic for the advanced undergraduate student. |
MA 125 Intermediate Algebra Marsh, KimballClass Assessment: Students should try to read (if time permits) the topics to be covered in class and be prepared to work examples and ask questions in class.
Grading: The final grade will be determined by the following: Midterm Test (120 points) and Final Test (120 points) = 240 points total. The midterm test will be over all material covered in the first 3 sessions. The final test is comprehensive. The Midterm and Final tests will be open-notes. A basic calculator should be acquired for use during the tests and the class. The following will be used to assign letter grades: A (216 – 240 points), B (192 – 215 points), C (168 – 191 points), D (144 – 167 points), F (<144 points)
Classroom Rules of Conduct: Class participation and questions are encouraged. Students are expected to come to all classes and to be on time. Since 8 weeks is a VERY short time to complete an Intermediate Algebra course, we will cover material at what many students consider a very rapid rate.
Course Topic/Dates/Assignments:
Assignments are not graded. However, to further your understanding of Intermediate Algebra, it is recommended that you try the problems (especially those with answers in the back of the book) in each section.
Week 1 Chapter 1: Review of the Real Number System
1.1 Basic Concepts
1.2 Operations on Real Numbers
1.3 Exponents, Roots, and Order of Operations
1.4 Properties of Real Numbers
Week 3
Chapter 3: Graphs, Linear Equations, and Functions
3.1 The Rectangular Coordinate System
3.2 The Slope of a Line
3.3 Linear Equations in Two Variables
Summary Exercises on Slopes and Equations of Lines
3.4 Linear Inequalities in Two Variables
3.5 Introduction to Functions |
Math Course Takes 'Real Life' Approach to Algebra
Educational courseware publisher American Education Corp. is taking a new approach to answering the age-old question, "What does algebra have to do with real life?" The company has announced the release of a new course for its A+nyWhere Learning System program. Algebra I: A Function Approach Part 1 is the first semester segment of a full-year algebra course geared to grades 9 and 10, and, in addition to the fundamental concepts and tools of algebra, the course aims to relate the material to "real life."
Taking the fundamentals and applying them to real-world situations using exercises in relevant scenarios allows students to realize the practical uses of linear and quadratic equations, graphs and coordinates, functions, and other algebraic concepts.
The A+nyWhere program is computer based, so students taking courses like Algebra I can use a number of tools incorporated into the software to aid in their assignments and overall comprehension of the material. These tools include onscreen standard and scientific calculators, pictures and diagrams, video tutorials, exercises, practice exams, and, for upper-level courses, interactive feedback |
follows Logic was produced by the WA State Board for Community & Technical Colleges...
see more
This free and open online course in Logic was produced by the WA State Board for Community & Technical Colleges [ as an academic subject, is the systematic study of the standards of correct reasoning. In short, logic is the theory of reasoning. This fully online course is a comprehensive introduction to logic with an emphasis on modern logical theory activities, and is ideal for independent learners, or instructors trying out this course package."״Logic & Proofs is an introduction to modern symbolic logic. It provides a rigorous presentation of the syntax and semantics of sentential and predicate logic. However, the distinctive emphasis is on strategic argumentation. Students learn effective strategies for constructing natural deduction proofs. This learning is supported by the Carnegie Proof Lab: it provides a sophisticated interface, in which students can give arguments by strategically guided forward and backward steps.״
This web page shows links to lectures for a course on Numerical Methods in Engineering taught by the author in the Spring...
see more
This web page shows links to lectures for a course on Numerical Methods in Engineering taught by the author in the Spring Semester of 2009. Click on the lecture links for class notes, Matlab scripts and functions, and assignments. Subjects covered: vectors and matrices in Matlab, graphics in Matlab, programming, numerical linear algebra, solution to equations, numerical integration, data fitting, and ordinary differential equations |
Mathematical World
This accessible series brings the beauty and wonder of mathematics to the advanced high school student, the mathematics teacher, the scientist or engineer, and the lay reader with a strong interest in mathematics. Mathematical World features well-written, challenging expository works that illustrate the fascination and usefulness of mathematics. (ISSN 1055-9426) |
Product Description
Learning about the quadratic equations is really exciting because understanding it helps you understand how all sorts of things work in the real world. Much of the way things move in space can be explained through quadratic equations and this program get you excited about the power of this equation.Grade Level: 8-12. 26 minutes |
Math & Science Accessibility
Fall term 2012, the Distance Learning department and two Math Departments at Portland Community College financed release time for two math faculty, Scot Leavitt and Chris Hughes, to study how to make math content more accessible for online students with disabilities. Since it is blind students that encounter the most serious accessibility problems with online math courses, that is who the study focused on mainly.
The Rule of Four
The rule of four is one of the most useful guiding principles both in teaching, and from the perspective of accessibility. Explicitly, when a concept or idea is introduced and discussed, we try to do so in four different ways:
algebraically
numerically
verbally
graphically
Depending on the student who we are working with, and the particular accommodations that the student has, one or more of these different descriptions may be harder for the student to access than the others. For example, if we are accommodating a student that is hearing impaired then the verbal description will need to be accommodated. This can be achieved in a number of different ways which include: using a sign language interpreter; captioning videos and other audio content. A student who is visually impaired may have more difficulty accessing each of the different descriptions except the verbal.
Specific Best Practices for Math & Science
Graphs
There are many ways to create graphs, including Winplot, Excel, Graph, pgfplots, PSTricks, etc. Any graph or graphic, regardless of how it was created, will always be read as an image by a screen reader. As such, appropriate
alt text must be included for electronic documents and web pages.
Printing a tactile graph on embossed paper is a very time-intensive process. When working with a visually-impaired student it is possible that Disability Services may ask the instructor to choose which are the most important images, as not all images may be printed.
Math/Science in word documents
For MS Word, use the MathType plugin to create math and science equations, formulas and notations. Do not use Microsoft's equation editor.
If you convert to a PDF or export to a webpage, save the original MS Word source document which Disability Services may ask for.
For LibreOffice, just use the native equation editor which easily converts to an accessible format.
Math/Science in PDFs
Math and science equations, formulas and notations are not screen reader accessible in a PDF, so save the source file with the original MathType or LaTeX equations. Disability Services will ask for source files when there is an accommodation need..
Math in PowerPoints
For MS PowerPoint 2013, use the MathType 6.9 plugin to create math and science equations, formulas and notations. Do not use Microsoft's equation editor.
If you convert to a PDF or export to a webpage, save the original MS PowerPoint source document which Disability Services may ask for.
If you are using older versions of PowerPoint or MathType, put the PowerPoint content into a Word document and use MathType to write the equations.
If a student has a time extension which would make taking the test unreasonable to complete in one sitting (imagine taking a six-hour exam in one session!), the instructor should split the test into multiple parts so that the student can take the different parts on different days. |
Precalculus : Real Mathematics, RealIdeal for courses that require the use of a graphing calculator, PRECALCULUS: REAL MATHEMATICS, REAL PEOPLE, 6th Edition, features quality exercises, interesting applications, and innovative resources to help you succeed. Retaining the book's emphasis on student support, selected examples include notations directing students to previous sections where they can review concepts and skills needed to master the material at hand. The book also achieves accessibility through careful writing and design--including examples with detailed solutions that begin and end on the same page, which maximizes readability. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles. Reflecting its new subtitle, this significant revision focuses more than ever on showing readers the relevance of mathematics in their lives and future careers. |
Subject: Mathematics (9 - 12) Title: Interpreting Functions Description: During this lesson, students will be introduced to interpreting functions. A function is a relationship between two sets: the domain (input values) and the range (output values).
Thinkfinity Lesson Plans
Subject: Mathematics Title: Tube Viewer SimulationAdd Bookmark Description: This student interactive, from Illuminations, simulates the effect of viewing an image through a tube. As students move the location of the person or change the length of the tube, the image and measurements also change. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Graph ChartAdd Bookmark Description: This reproducible transparency, from an Illuminations lesson, contains the answers to the similarly named student activity in which students identify the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Graphing What Add BookmarkSubject: Mathematics Title: Gallery WalkAdd Bookmark |
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the company budget). The list goes on.
Algebra is a system of mathematical symbols and rules that are universally understood, no matter what the spoken language. Algebra provides a clear, methodical process that can be followed from beginning to end to solve complex problems. There's no doubt that algebra can be easy to some while extremely challenging to others. For those of you who are challenged by working with numbers, Algebra I For Dummies can provide the help you need.
This easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems. But rest assured, this book is not about memorizing a bunch of meaningless steps; you find out the whys behind algebra to increase your understanding of how algebra works.
In Algebra I For Dummies, you'll discover the following topics and more:
All about numbers – rational and irrational, variables, and positive and negative
Figuring out fractions and decimals
Explaining exponents and radicals
Solving linear and quadratic equations
Understanding formulas andsolving story problems
Having fun with graphs
Top Ten lists on common algebraic errors, factoring tips, and divisibility rules
No matter if you're 16 years old or 60 years old; no matter if you're learning algebra for the first time or need a quick refresher course; no matter if you're cramming for an algebra test, helping your kid with his or her homework, or coming up with next year's company budget, Algebra I For Dummies can give you the tools you need to succeed. The included workbook features over 290 pages with hundreds of practice problems, along with answers and explanations to show you where you went wrong (or right). Add this onto Algebra I For Dummies, and you have an equation for numerical success!
AUTHOR BIO: Mary Jane Sterling has taught mathematics at Bradley University since 1979, and has also taught math and the high school and junior high school would have been lost without it!
Chapter 19 on graphing equations and (slope) intercepts was extremely helpful. I went from a pre-quiz grade of 26 to an actual grade of 93 after studying with the book. Everything is layed out in a simple way and is very clear. I cannot say enough great things about this book!
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Sugar's 'The Silent Crisis Destroying America's Brightest Minds" is a book that is all about educational reality, the facts on the ground, not vacuous educational theories.
Sugar pioneered the SMARTGRADES school notebooks that contain the new learning technology, ACANDY Processing Tools, that empower students for academic success. She transformed my kids into Grade A students, soI speak from experience, not hypothetical theory.
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great for student
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YMDS
Posted January 24, 2010
about the book
this is one of the many books I am working with to help me produce a video
for the New York State regents. I have not shown it to my class because
they may think I am calling them a dummy.It is a very good book.
Yosef M.D. Shulman
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2.2: Reasoning and Proof
Inductive Reasoning
The nth Term – Students enjoy using inductive reasoning to find missing terms in a pattern. They are good at finding the next term, or the tenth term, but have trouble finding a generic term or rule for the number sequence. If the sequence is linear (the difference between terms is constant), they can use methods they learned in Algebra for writing the equation of a line.
Key Exercise: Find a rule for the nth term in the following sequence. Answer: The sequence is linear, each term decreases by . The first term is , so the point can be used. The second term is , so the point can be used. Applying what they know from Algebra I, the slope of the line is , and the intercept is , so the rule is .
True Means Always True – In mathematics a statement is said to be true if it is always true, no exceptions. Sometimes students will think that a statement only has to hold once, or a few times to be considered true. Explain to them that just one counterexample makes a statement false, even if there are a thousand cases where the statement holds. Truth is a hard criterion to meet.
Sequences – A list of numbers is called a sequence. If the students are doing well with the number of vocabulary words in the class, the term sequence can be introduced.
Additional Exercises:
1. What is the next number in the following number pattern?
Answer: This is the famous Fibonacci sequence. The next term in the sequence is the sum of the previous two terms.
2. What is the missing number in the following number pattern?
Answer: Descending consecutive odd integers are being subtracted from each term, so the missing number is .
Conditional Statements
The Advantages and Disadvantages of Non-Math Examples – When first working with conditional statements, using examples outside of mathematics can be very helpful for the students. Statements about the students' daily lives can be easily broken down into parts and evaluated for veracity. This gives the students a chance to work with the logic, without having to use any mathematical knowledge. The problem is that there is almost always some crazy exception or grey area that students will love to point out. This is a good time to remind students of how much more precise math is compared to our daily language. Ask the students to look for the idea of what you are saying in the non-math examples, and use their powerful minds to critically evaluate the math examples that will follow.
Converse and Contrapositive – The most important variations of a conditional statement are the converse and the contrapositive. Unfortunately, these two sound similar, and students often confuse them. Emphasize the converse and contrapositive in this lesson. Ask the students to compare and contrast them.
Converse and Biconditional – The converse of a true statement is not necessarily true! The important concept of implication is prevalent in Geometry and all of mathematics. It takes some time for students to completely understand the direction of the implication. Daily life examples where the converse is obviously not true is a good place to start. The students will spend considerable time deciding what theorems have true converses (are bicondtional) in subsequent lessons.
Key Exercise: What is the converse of the following statement? Is the converse of this true statement also true?
If it is raining, there are clouds in the sky.
Answer: The converse is: If there are clouds in the sky, it is raining. This statement is obviously false.
Practice, Practice, Practice – Students are going to need a lot of practice working with conditional statements. It is fun to have the students write and share conditional statements that meet certain conditions. For example, have them write a statement that is true, but that has an inverse that is false. There will be some creative, funny answers that will help all the members of the class remember the material.
Deductive Reasoning
Inductive or Deductive Reasoning – Students frequently struggle with the uses of inductive and deductive reasoning. With a little work and practice they can memorize the definition and see which form of reasoning is being used in a particular example. It is harder for them to see the strengths and weaknesses of each type of thinking, and understand how inductive and deductive reasoning work together to form conclusions.
Recognizing Reasoning in Action – Use situations that the students are familiar with where either inductive or deductive reasoning is being used to familiarize them with the different types of logic. The side by side comparison of the two types of thinking will cement the students' understanding of the concepts. It would also be beneficial to have the students write their own examples.
Key Exercise: Is inductive or deductive reasoning being used in the following paragraph? Why did you come to this conclusion?
1. The rules of Checkers state that a piece will be crowned when it reaches the last row of the opponent's side of the board. Susan jumped Tony's piece and landed in the last row, so Tony put a crown on her piece.
Answer: This is an example of detachment, a form of deductive reasoning. The conclusion follows from an agreed upon rule.
2. For the last three days a boy has walked by Ana's house at with a cute puppy. Today Ana decides to take her little sister outside at to show her the dog.
Answer: Ana used inductive reasoning. She is assuming that the pattern she observed will continue.
Which is Better? – Students quickly conclude that inductive reasoning is much easier, but often miss that deductive reasoning is more sure and frequently provides some insight into the answer of that important question, "Why?".
Additional Exercises:
1. What went wrong in this example of inductive reasoning?
Teresa learned in class that John Glenn (the first American to orbit Earth) had to eat out of squeeze tubes, and her mom says the food served in airplanes is not very good. She just had a yummy pizza for lunch. She sees a pattern. Food gets better as one approaches the center of the earth. Therefore the food in a submarine must be delicious!
Answer: She carried the pattern too far.
Algebraic Properties
Commutative or Associate – Students sometimes have trouble distinguishing between the commutative and associate properties. It may help to put these properties into words. The associate property is about the order in which multiple operations are done. The commutative is about the first and second operand having different roles in the operation. In subtraction the first operand is the starting amount and the second is the amount of change. Often student will just look for parenthesis; if the statement has parenthesis they will choose associate, and they will usually be correct. Expose them to an exercise like the one below to help break them of this habit.
Key Exercise: What property of addition is demonstrated in the following statement?
Answer: It is the commutative property that ensures these two quantities are equal. On the left-hand side of the equation the first operand is the sum of and , and on the right-hand side of the equation the sum if and is the second operand.
Transitive or Substitution – The transitive property is actually a special case of the substitution property. The transitive property has the additional requirement that the first statement ends with the same number or object with which the second statement begins. Acknowledging this to the students helps avoid confusion, and will help them see how the properties fit together.
Key Exercise: The following statement is true due to the substitution property of equality. How can the statement be changed so that the transitive property of equality would also ensure the statement's validity?
If , and , then .
Answer: The equality can be changed to due to the symmetric property of equality. Then the statement would read:
If , and , then .
This is justified by the transitive property of equality.
Diagrams
Keeping It All Straight – At this point in the class the students have been introduced to an incredible amount of material that they will need to use in proofs. Laying out a logic argument in proof form is, at first, a hard task. Searching their memories for terms at the same time makes it near impossible for many students. A notebook that serves as a "tool cabinet" full of the definitions, properties, postulates, and later theorems that they will need, will free the students' minds to concentrate on the logic of the proof. After the students have gained some experience, they will no longer need to refer to their notebook. The act of making the book itself will help the students collect and organize the material in their heads. It is their collection; every time they learn something new, they can add to it.
All Those Symbols – In the back of many math books there is a page that lists all of the symbols and their meanings. The use of symbols is not always consistent between texts and instructors. Students should know this in case they refer to other materials. It is a good idea for students to keep a page in their notebooks where they list symbols, and their agreed upon meanings, as they learn them in class. Some of the symbols they should know at this point in are the ones for equal, congruent, angle, triangle, perpendicular, and parallel.
Don't Assume Congruence! – When looking at a figure students have a hard time adjusting to the idea that even if two segments or angles look congruent they cannot be assumed to be congruent unless they are marked. A triangle is not isosceles unless at least two of the sides are marked congruent, no matter how much it looks like an isosceles triangle. Maybe one side is a millimeter longer, but the picture is too small to show the difference. Congruent means exactly the same. It is helpful to remind the students that they are learning a new, extremely precise language. In geometry congruence must be communicated with the proper marks if it is known to exist.
Communicate with Figures – A good way to have the students practice communicating by drawing and marking figures is with a small group activity. One person in a group of two or three draws and marks a figure, and then the other members of the group tell the artist what if anything is congruent, perpendicular, parallel, intersecting, and so on. They take turns drawing and interpreting. Have them use as much vocabulary as possible in their descriptions of the figures.
Two-Column Proofs
Diagram and Plan – Students frequently want to skip over the diagramming and planning stage of writing a proof. They think it is a waste of time because it is not part of the end result. Diagramming and marking the given information enables the writer of the proof to think and plan. It is analogous to making an outline before writing an essay. It is possible that the student will be able to muddle through without a diagram, but in the end it will probably have taken longer, and the proof will not be written as clearly or beautifully as it could have been if a diagram and some thinking time had been used. Inform students that as proofs get more complicated, mathematicians pride themselves in writing simple, clear, and elegant proofs. They want to make an argument that undeniably true.
Teacher Encouragement – When talking about proofs and demonstrating the writing of proofs in class, take time to make a well-drawn, well-marked diagram. After the diagram is complete, pause, pretend like you are considering the situation, and ask students for ideas of how they would go about writing this proof.
Assign exercises where students only have to draw and mark a diagram. Use a proof that is beyond their ability at this point in the class and just make the diagram the assignment.
When grading proofs, use a rubric that assigns a certain number of points to the diagram. The diagram should be almost as important as the proof itself.
Start with "Given", but Don't End With "Prove" – After a student divides the statement to be proved into a given and prove statements he or she will enjoy writing the givens into the proof. It is like a free start. Sometimes they get a little carried away with this and when they get to the end of the proof write "prove" for the last reason. Remind them that the last step has to have a definition, postulate, property, or theorem to show why it follows from the previous steps.
Scaffolding – Proofs are challenging for many students. Many students have a hard time reading proofs. They are just not used to this kind of writing; it is very specialized, like a poem. One strategy for making students accustom to the form of the proof is to give them incomplete proofs and have them fill in the missing statements and reason. There should be a progression where each proof has less already written in, and before they know it, they will be writing proofs by themselves.
Segment and Angle Congruence Theorems
Number or Geometric Object – The difference between equality of numbers and congruence of geometric objects was addressed earlier in the class. Before starting this lesson, a short review of this distinction to remind students is worthwhile. If the difference between equality and congruence is not clear in students' heads, the proofs in this section will seem pointless to them.
Follow the Pattern – Congruence proofs are a good place for the new proof writer to begin because they are fairly formulaic. Students who are struggling with proofs can get some practice with this style of writing while already knowing the structure of the proof.
State the "if" side in congruence form.
Change the congruence of segments into equality of numbers.
Apply the analogous property of equality.
Change the equality of numbers back to congruence of segments.
Theorems – The concept of a theorem and how it differs from a postulate has been briefly addressed several times in the course, but this is the first time theorems have been the focus of the section. Now would be a good time for students to start a theorem section in their notebook. As they prove, or read a proof of each theorem it can be added to the notebook to be used in other proofs.
Additional Exercises:
1. Prove the following statement.
If , triangle is isosceles.
Answer:
Statement
Reason
Given
Definition of congruent segments.
Triangle is isosceles.
Definition of isosceles triangle.
Proofs About Angle Pairs
Mark-Up That Picture – Angles are sometimes hard to see in a complex picture because they are not really written on the page; they are the amount of rotation between two rays that are directly written on the page. It is helpful for students to copy diagram onto their papers and mark all the angles of interest. They can use highlighters and different colored pens and pencils. Each pair of vertical angles or linear pairs can be marked in a different color. Using colors is fun, and gives the students the opportunity to really analyze the angle relationships.
Add New Information to the Diagram – It is common in geometry to have multiple questions about the same diagram. The questions build on each other leading the student though a difficult exercise. As new information is found it should be added to the diagram so that it is readily available to use in answering the next question.
Try a Numerical Example – Sometimes students have trouble understanding a theorem because they get lost in all the symbols and abstraction. When this happens, advise the students to assign a plausible number to the measures of the angles in question and work form there to understand the relationships. Make sure the student understands that this does not prove anything. When numbers are assigned, they are looking at an example, using inductive reasoning to get a better understanding of the situation. The abstract reasoning of deductive reasoning must be used to write a proof.
Inductive vs. Deductive Again – The last six sections have given the students a good amount of practice drawing diagrams, using deductive reasoning, and writing proofs, skills which are closely related. Before moving on to Chapter Three, take some time to review the first two sections of this chapter. It is quite possible that students have forgotten all about inductive reasoning. Now that they have had practice with deductive reasoning they can compare it to inductive reasoning and gain a deeper understanding of both. They should understand that inductive reasoning often helps a mathematician decide what should be attempted to be proved, and deductive reasoning proves it.
Review – The second section of chapter two contains information about conditional statements that will be used in the more complex proofs in later chapters. Since the students did not get to use most of it with these first simple proofs, it would be a good idea to draw their attention to it again and talk briefly about the more complex proof that will be coming. |
Algebra Performance Task: Multistep Equations with Tables and Graphs
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PRODUCT DESCRIPTION
10 pages, aligned with common core standards. The cornerstone lesson to what I consider to be easily the best unit I can offer my algebra students. This lesson is part of a series of problems I created that are focused around graphing linear functions and the use of new tools (tables, graphs, equations) to solve problems in context. Students use multiple methods to explore functions and multi-step equations (in one variable) in the course of two application problems: a botany experiment and a car rental conundrum. Underneath the rich content is an important lesson about presentation: a solution to a problem illustrates, generalizes, communicates, and verifies the results. An answer is just a number. Which is better?
This lesson begins with a teacher directed problem that introduces students to interpreting a problem in context and using expressions, functions, tables, and graphs to communicate a solution. Also included is a student self-directed problem that takes these skills and runs with them. The pair of these two problems serve as an excellent introduction to inquiry based learning and lend themselves well to standards based grading systems.
This lesson is designed to be used after students have a basic understanding of how to solve one-variable equations (including those with variables on both sides). They should also have prior knowledge of coordinate graphing and the concept of functions (but not necessarily function notation). For me, this was chapter three in my Algebra 1 course.
Fantastic design and very well scaffolded support for students. I don't see anything on it that I couldn't help walk my 6th graders through at the end of our Algebraic Expressions/Equations unit, yet it's still appropriate for older students to complete more independently.
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Product Questions & Answers
I love your "the plant problem" and "the car Problem" yet as I go over them I wonder if my students have covered this and can work on it independently (as a sponge after exam) in science? 7th grade. I will also check with the 7th grade math teachers to be sure. I am impressed with your products and want to find ways to integrate them into my classes as math extensions. Thank you!
Thank you for your compliments. I think that a typical 7th grader would not have enough algebra to handle either of those problems independently, but I suppose it depends on the child and the curriculum at your school. You should check with the 7th grade math teachers. I'm pretty sure the concepts and the graphing would be entirely appropriate, but the expression writing may be a little challenging without any guidance. Let me know how it goes!
I'm not sure what you mean exactly, but hopefully this will answer your question:
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basic mathematical skills necessary to enter MATH 099. Topics include operations of whole numbers and signed numbers, fractions and decimals, as well as ratio, proportions, and percents. Introduces equations, geometric applications, the laws of exponents, operations with polynomials, and basic factoring. Three class hours weekly. |
Algebra
Extends the application of the familiar algebraic laws for adding and multiplying numbers, matrices and vectors to other contexts. Depending on just which laws are satisfied, the algebraic structures studied are called groups, rings and fields. These concepts underlie much of modern mathematics, and are essential background for research in any area of pure mathematics.
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Elementary Statistics - Updated Solution Manual - 10th edition
Summary: Addison-Wesley is proud to celebrate the Tenth Edition of Elementary Statistics. This text is highly regarded because of its engaging and understandable introduction to statistics. The author's commitment to providing student-friendly guidance through the material and giving students opportunities to apply their newly learned skills in a real-world context has made Elementary Statistics the #1 best-seller in the market. Students learning from Elementary Statisti...show morecs should have completed an elementary algebra course. Although formulas and formal procedures can be found throughout the text, the emphasis is on development of statistical literacy and critical thinking. ...show less
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Mathematical Reasoning for Elementary Teachers:The fourth edition of Mathematical Reasoning has an increased focus on professional development and connecting the material from this class to the elementary and middle school classroom. The authors have provided more meaningful content and pedagogy to arm readers with all the tools that they will need to become excellent elementary or middle school teachers. Thinking Critically. Sets and Whole Numbers. Numeration and Computation. Number Theory. Integers. Fractions and Rational Numbers. Decimals and Real Numbers. Algebraic Reasoning and Representation. Statistics: The Interpretation of Data. Probability. Geometric Figures. Measurement. Transformations, Symmetries, and Tilings. Congruence, Constructions, and Similarities. For all readers interested in mathematical reasoning for elementary teachers.
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Wittmann PrecalculusPrecalculus goes in depth with algebra included with some geometry or even physics. This is the first class in math that really starts applying to real life situations. There is emphasis on trigonometry which is preparing you for calculus |
Department of Mathematics
For Students
There are many resources available for students currently at Towson University, and for prospective students who are considering enrolling at Towson.
For Prospective Students
In addition to the links in the For All Students section below, we have some information that will be of interest primarily to prospective students. If you are undecided about your college major, there is a web page that will describe the advantages and benefits of majoring in mathematics. Also there are links to the Admissions and Financial Aid Offices, and information about the Honors College here at Towson University.
For Current Students
In addition to the links in the For All Students section below, we have some information that will be of interest only to students who are currently enrolled at Towson university. These include links to the textbooks used in mathematics courses, and information about jobs in the Department of Mathematics that are available to current students. Also there is a link to the Mathematics Tutoring Center which is a free resource to all current Towson University students. Towson University's Office of Technology Services (OTS) maintains a software download web page, where students may download and install programs such as Mathematica, Minitab and other useful programs/utilities. Undergraduate research offers the student an an academic experience rarely available in a regular classroom setting, and the opportunity to work closely with a faculty mentor. Undergraduate research is encouraged and financially supported by both the Fisher College of Science and Mathematics and Towson University.
For All Students
The links on this page are to web pages that contain information that is useful to both our current students and to prospective students. Included are pages about the variety of mathematics undergraduate concentrations within the mathematics major program; the two mathematics graduate programs; course descriptions and outlines for all of the undergraduate and graduate mathematics courses we offer; information concerning the Towson University Core Curriculum (GenEd) requirements; links to information about scholarships and award available for mathematics majors; and information on how to transfer courses from other colleges and universities into Towson. The career information web pages are an excellent source of material and further links that provide a wealth of information about careers in mathematics. In addition; there is a link to information about the mathematics placement test, and under the Course Description web page is a link to the Developmental Mathematics Program. |
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Starting at $54Solidly grounded in up-to-date research, theory and technology, Teaching Secondary Mathematicsis a practical, student-friendly, and popular text for secondary mathematics methods courses. It provides clear and useful approaches for mathematics teachers, and shows how concepts typically found in a secondary mathematics curriculum can be taught in a positive and encouraging way. The thoroughly revised fourth edition combines this pragmatic approach with truly innovative and integrated technology content throughout. Synthesized content between the book and comprehensive companion website offers expanded discussion of chapter topics, additional examples and technological tips. An online graphic calculator invites students to interact with the math content of the book, while video clips provide chapter by chapter illustrations of classroom instructional strategies. Each chapter features tried and tested pedagogical techniques, problem solving challenges, discussion points, activities, mathematical challenges, and student-life based applications that will encourage students to think and do. New in this edition: A companion website replete with chapter-by-chapter video lessons, teacher tools, problem solving Q&As, helpful links and resources, and embedded graphing calculators. Updated icons, including NCTM core standards and math history, that connect to key strands that readers will find indispensible as they build their professional knowledge and skills. Problem solving challenges and sticky questions featured in each chapter encourage students to think through everyday issues and possible solutions. Fresh interior design to better highlight pedagogical elements and key features, all to better engage students. |
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