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Linear Algebra by Messer
From Intelligent Perception
The rationale for the second part of the title is that it can serve as an introduction to pure mathematics, proofs etc. That part is tricky and I am not sure this is a good way to start. But the book is good as a first course in linear algebra.
Well written. Good exercises and plenty of them. I wish I saw more connections to calculus.
Contents
You are thrown at the definition of vector space in Section 1.2 without preparation. That's tough. What follows is also tough but inevitable -- using the axioms to prove some "simple" facts about the algebra of vectors. Still, this part is a bit too long.
"Closed" under operations has been a hard topic for the students. You've got to push infinite dimensional spaces / function spaces to make them confront the issue, and the need to use axioms and theorems, less intuition. |
Trigonometry
Retail Price:
$56.95$48.41
Product ID - 13SOSTG | Availability - Now Shipping
Want to get your teen ready for college math? Then, you need Switched-On Schoolhouse Trigonometry for grades 9-12! As a great prep course for advanced math courses, this one-semester, computer-based Alpha Omega curriculum covers topics like right angle trigonometry, trigonometric identities, graphing, the laws of sines and cosines, and polar coordinates. Pre-requisite is Algebra II. Includes quizzes and tests. Order today!Prepare your homeschool high schooler for future math courses and get him the Switched-On Schoolhouse Trigonometry elective for grades 9-12! Practical and informative, this one-semester, computer-based course covers trigonometry in clear, step-by-step lessons that will build your child's confidence in performing advanced math. Made for students who have completed Algebra II, this knowledge-building math course will show your teen how to develop trigonometric formulas and use them in "real world" applications! Plus, to make math lessons more fun, SOS has interactive, exciting multimedia tools like video clips, learning games, and animation to engage your high schooler in learning!
An innovative time-saver, Switched-On Schoolhouse offers homeschool parents a feature no textbook can—automatic grading and lesson planning! Now, you won't have to spend nights pouring over papers trying to check advanced math problems. In addition, this Christian-based SOS course has customizable curriculum, so you can always adjust math lessons to your student's learning pace. A built-in calendar and message center in this Alpha Omega curriculum also make organization a breeze. As an alternative to calculus, this trigonometry course will give your student a clear "big picture" of advanced math, as well as an understanding of how numeric, algebraic, and geometric concepts are used together to build a foundation of higher mathematical thinking. Don't wait to get your teen's mind in shape for college math! Give him a solid head start and order Switched-On Schoolhouse Trigonometry for grades 9-12 from Alpha Omega Publications today! Order |
The master theme of the Tenth Grade will be to continue to develop in students all of the higher cognitive and affective skills, conforming them to the more complex concepts of the secondary school, and to the challenging task of applying principles, concepts, and ideals in an expanding scope of study than they have had in the elementary grades. Now, students will be obliged to thoroughly develop good study habits to help in the intellectual digestion of more difficult subject matter. Students will begin implementing ways in which they will ultimately discover God's plan for each of them, as well as how to prepare themselves for their vocations. They will work toward becoming productive members of their communities, ready and able to defend their Faith and to apply traditional Catholic principles, no matterwhere their vocation takes them. Theology: This year the students will continue the Quest for Happiness Series, with an emphasis on the salvatory role of the Church, guided by the Holy Ghost. This third book of this series, The Ark and the Dove Text, reveals the love which the Holy Ghost has for us. Students will study the Third Person of the Holy Trinity, the Holy Ghost, as well as the nature, origin, structure and history of the Church. Also discussed will be the sacrament of Penance, and the fifth, sixth, and ninth commandments. The Ark & the DoveAnswer Key is a convenient aid for parents or in-home tutors to check over student responses. Also available are The Rosary Novena Booklet and The Way of the Cross Booklet, according to the method of St. Alphonsus Liguori. Algebra II: This year, the student advances to Saxon Algebra II, which includes intermediate algebraic concepts such as: graphical solutions to simultaneous equations, scientific notation, radicals, roots of quadratic equations (including complex roots), properties of real numbers, inequalities and systems of inequalities, logarithms and antilogarithms, exponential equations, basic trigonometric functions, algebra of polynomials, vectors, polar and rectangular coordinate systems, and a wide spectrum of word problems requiring algebra to solve. As with all Saxon math courses, geometry is integrated throughout the program, so that by the end of Saxon Algebra II, all students should be exposed to all of the geometry they would have received in a dedicated geometry course. Saxon Publishers assert that any student who has successfully completed Saxon Algebra II has the necessary tools and skills to score competitively on the SAT and ACT college board tests. Overall, algebra requires a great deal of abstract thought on the part of the student. It will be necessary for the students to work slowly and methodically to learn the rules, theories, and methods of arriving at the abstract answers to these problems. The three-book Saxon set includes: Saxon Algebra II Textbook, Saxon Algebra II Test Forms; Saxon Algebra II Answer Key. SaxonAlgebra II Solutions Manual is also available, and strongly recommended. Literature: This year's literature course is an eclectic look at diverse literary genres. By What Authority, by Robert Hugh Benson, takes place during the Protestant Reformation. Sir Nicholas is the rock-solid head of his household and a devout Catholic who helps "renegade" priests hide from her Majesty's men. The reader will find himself traveling across the English countryside hunting for priests, the next minute witnessing the happenings at the Queen's court.In the midst of all this exists the relationship between a young man and a young lady - one a Catholic, the other aProtestant. Catholic families suffered persecutions of various types. Families were divided; fathers and sons were thrown into jail; and neighbor turned against neighbor. But through it all, the few priests that remained were able to sustain and convert many.Lances of Lynwoodby Charlotte M. Yonge is a story of Eustace, the younger brother of Sir Reginald, the lord of LynwoodCastle and its surrounding lands. One heroic act on the battlefield gains him the favor of the Prince of Wales, who raises young Eustace to the level of Knighthood; but with that honor comes responsibility. The youthful Sir Eustace is hurled into the world of men and must now defend his castle, his orphaned nephew, and if he survives, his honor. In all his encounters, he puts into practice the virtuous qualities of a noble knight. As this exciting story ends, Eustace is hailed a "brave, stouthearted young Knight." The Red Badge of Courage by Stephen Crane is a classic novel of war as seen through the eyes of an untested recruit.This book has been praised for its uncanny recreation of the sights, sounds, and sense of actual combat. Struggling against his fears and feelings of cowardice, young Civil War soldier Henry Fleming endures the nightmare of battle as he wrestles with his conflicting emotions.Experiencing his first trials under fire on a woodland battlefield, he comes to manhood and finds peace of mind in this powerful description of war and haunting interpretation of that bloody symbol of bravery – the red badge of courage. The Ballad of the White Horse is a poem by G. K. Chesterton about the idealized exploits of the Saxon King Alfred the Great, published in 1911. Written in ballad form, the work is usually considered an epic poem. The poem narrates how Alfred was able to defeat the invading Danes at the Battle of Ethandun by God through the agency of the Virgin Mary. In addition to being a narration of Alfred's militaristic and political accomplishments, it is also considered a Catholicallegory. Chesterton incorporates a significant amount of philosophy into the basic structure of the story.
History: This is best described as a hemispheric history, centered on the text, Christ and the America Text, written in the late 1990s by Anne Carroll. Students will learn about the Catholic roots of our country, meet the great Catholic heroes and heroines of North and South America, and learn how the Catholic Church has fared in our hemisphere. All the regions of the Western Hemisphere, including North America, Central America, South America and the Caribbean will be studied. The histories of the United States and other countries in our hemisphere have been deftly intertwined, to show the various relationships and dynamics that entered into the establishment and growth of the diverse nations of the Western Hemisphere. History can be presented from any number of points of view, but, ideally, there is a single interpretation of events, causes, and effects which must be viewed through the lens of a Catholic perspective. Only when we truly understand the past, can we even begin to influence the future. Dr. Carroll does an admirable job of helping the student to understand better what we need to do to help the country we love come closer to God. Other books used: Christ and the Americas Answer Key, Catholics of the Confederacy, Reign of Christ the King. Grammar:The Elegant Essay walks students through the entire essay-building process.From captivating introductions to compelling descriptions and convincing conclusions, students learn to bring their words and their arguments to life.With explanation and practice exercises, it is an excellent extension and refinement of the students writing skills learned to this point. In the Easy Grammar Ultimate Series course, students learn high school English grammar in a highly efficient program which features 180 daily teaching lessons, lasting 10 minutes each.Great for home school!This book teaches important skills in a very direct and logical progression. In Spelling, using the Traditional Catholic Speller 10, students will again learn to spell hundreds of new words and definitions, including Catholic terms. Biology: This year's science course will be centered on Dr. Jay Wile's Biology Text. Using a common-sense Christian approach to the study of Biology, Dr. Wile introduces the student to both major divisions of biology: botany and zoology, and deftly explains the five Kingdoms, and their subordinate classes. In his inimitable writing style, Dr. Wile gives the reader the impression that he is sitting across the table, tutoring the student one-on-one. The text deals insightfully with the difference between macro-evolution and micro-evolution, and shows why Darwinian macro-evolution is simply not possible. Biology Solutions and TestsBook contains: (1) answers to the Study Guide found at the end of each module in the text; (2) tests for each module; and (3) answers to each of the modular tests. |
0321447174
9780321447173 just on the mechanics of how it works. Fully integrated activities are found in the book and in an accompanying Activities Manual. As a result, students engage, explore, discuss, and ultimately reach true understanding of the approach and of mathematics. «Show less... Show more»
Rent Mathematics for Elementary Teachers plus Activities Manual 2nd Edition today, or search our site for other Sybilla Beckmann |
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· Build simulations that illuminate concepts from probability and statistics |
Recent Advances in Numerical Methods features contributions from distinguished researchers, focused on significant aspects of current numerical methods and computational mathematics. The increasing necessity to present new computational methods that can solve complex scientific and engineering problems requires the preparation of this volume with actual new results and innovative methods that provide numerical solutions in effective computing times. Each chapter will present new and advanced methods and modern variations on known techniques that can solve difficult scientific problems efficiently |
Introduction to specific topics in mathematics most useful for
planners.
Topics include: review of the vocabulary of mathematics; analysis
of
linear
and nonlinear functions with economic applications; the
mathematics of
finance; and descriptive statistics. The course will also include
an
introduction to the use of the computer as a tool for data
analysis and
display using spreadsheet
software.
PREREQUISITE:
A passing score on the basic mathematics entrance exam given
during
orientation week. 220A can be waived by a passing score on the
waiver
exam also given during orientation week.
A minimum passing grade on the final exam is required to pass the
course. Problem and computer assignments will count heavily in
determining the course grade. You are encouraged to work jointly
(groups of 2-4 persons) on the homework
and computer projects. Please turn in only one assignment
per
group.
Homework is due by 5:00 p.m. of the date specified. It will be
returned one week from that date. Homework which is submitted
late, but
before the week is over will be marked down a grade. Late homework
must
be submitted before the graded homework is returned. |
Math Spring 2007 Section 1, TR 5:45-7:10pm Science 107
Professor: PHONE: Home Page:
Andrew Diener 3213452
EMAIL ADDRESS: [email protected] OFFICE: OFFICE HOURS: 103F Science 1pm-5pm MTWR.
TEXT: Mathematics for Elementary Teachers, A Contemporary Approach, Seventh Edition, Gary L. Musser, William F. Burger and Blake E. Peterson, Wiley, 2006. COURSE CONTENT: (by catalog) This course includes concepts essential to mathematics for elementary school teaching candidates. Topics include: set theory, numbers and numeration, number theory, rational numbers and problem solving. This course does not meet the general education requirement in mathematics. Prerequisite: MATH 100 or equivalent. CALCULATOR: You must have access to the TI-83+ or TI-84+ graphing calculator on assignments and for part of each test. Tests may be in two parts, one with calculator and one without. HOMEWORK: A list of suggested homework problems will be given for each section covered. I do expect that you will attempt all homework problems. I will grade some, though not all, of these problems. We will spend some class time in groups working on these problems. Any homework must be turned in using the proper format and on time. NO late (or incorrectly formatted) homework will be accepted. Homework will account for 15% of your final grade. (100 pts) QUIZZES: There be will quizzes every week. These quizzes will be very short (approx. 10-15 min.) and will come almost directly from the homework. Each quiz will be worth 10 points and I will count 10 of them. Since I plan to give many of these quizzes (at least 12, hopefully 13) I will drop the extra ones. This does mean that quizzes cannot be made up. Quizzes will count for 15% of your final grade. TESTS: There will be three in-class exams, each exam worth 100 points, and a comprehensive final exam worth 150 points. I will not curve your test grades. If you miss a test for
any reason, you need to notify me no later than the day after the test to set up a time for a make-up. PROJECTS: I strongly encourage students to work together when studying mathematics (except on exams!) and to further this goal there will be several 3 Math 105 December 12, 2006 Name You must show all your work. Partial credit will be given. 1. The following table shows the amounts spent on reducing sizes of rst-grade through thirdgrade public school classes in a certain state. Year 1988 199
EXAM 3 Math 105 April 11, 2008 Name You must show all your work. Partial credit will be given. 1. A laptop computer currently costs $787. The price of the laptop is expected to decrease by 2.9% per year. Find a mathematical model for the price of the
EXAM 1 Math 105 November 2, 2006 Name You must show all your work. Partial credit will be given. 1. Calculate the value of H(t) = -16t 2 + 120 at t = 2, where H(t) is the height of a cliff diver above the water t seconds after he jumped from a 120 fo
QUIZ 3 Name 1. Find the slope and the x-intercept of the line 2x + 3y = 7.2. Do the row operations needed to put the following matrix into nal form and then write down the solutions (if any exist) to the system represented by the matrix. 1 0 0 2
iContents iChapter 1Keeping It In The Ballpark1.1 Studying PhysicsHow do you study for physics? Do you read your physics book the same way your read a book for literature class? Although physics and English literature are both intellectual di
Heinrich Rudolf Hertz(Redirected from Heinrich Hertz) Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894), was the German physicist for whom the hertz, the SI unit of frequency, is named. In 1888, he was the first to demonstrate the existenc
MIDTERM EXAM IISolutions Math 21D Temple-F06 Write solutions on the paper provided. Put your name on this exam sheet, and staple it to the front of your finished exam. Do Not Write On This Exam Sheet. Problem 1. (20pts) (a)Calculate the gradient f (x
MIDTERM EXAM IMath 21D Temple-F06 Write solutions on the paper provided. Put your name on this exam sheet, and staple it to the front of your finished exam. Do Not Write On This Exam Sheet.Problem 1. (15pts) Evaluate R x2y dA where R is the region |
Loci: Resources
Images of F
by Steve Phelps (Madeira High School)
Applet Description
This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a closed figure that happens to be in the shape of the letter F initially. The Geogebra interface allows dragging of points and vectors to make for versatile explorations of basic linear algebra ideas. Suggested activities and exercises using the tool are included on page 2 of this posting and as a separate pdf file for easy printing.
Steve Phelps
Madeira High School
& GeoGebra Institute of OhioClick here or on the screen shot above to open the applet in a separate window.
Investigations
In the Images of F applet on page 1, the columns of the matrix are the elementary vectors e1 and e2. The blue figure is a pre-image initially in the shape of an F. The green figure is the image of the blue F under the transformation given by the matrix.
To answer the questions below, you can drag the tips of the elementary vector to set up the appropriate matrices. You may also need to drag the vertices of the blue F as well.
Warm Up: Set up the following matrices one at a time. Pay particular attention to the lattice points of F and to the lattice points of the image of F.
1.
\left[ \begin{array}{cc} 2 & 3 \\ 0 & 1 \end{array} \right]
2.
\left[ \begin{array}{cc} 1 & 0 \\ 3 & -1 \end{array} \right]
3.
\left[ \begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right]
4.
\left[ \begin{array}{cc} 2 & -1 \\ 2 & 1 \end{array} \right]
5.
\left[ \begin{array}{cc} -2 & 1 \\ 2 & -1 \end{array} \right]
6.
\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]
Investigation 1: Drag the tips of the elementary vectors to set up the following matrices. Discuss the transformations and the resulting image of F under these matrix transformations.
transformations with matrices of the form
\left[ \begin{array}{cc} k & 0 \\ 0 & 1 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & 0 \\ 0 & k \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} k & 0 \\ 0 & k \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 0 & k \\ k & 0 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & 0 \\ k & 1 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & k \\ 0 & 1 \end{array} \right]
Investigation 2: Drag the tips of the elementary vectors to set up matrices that will perform the following transformations. Pay attention to the orientation of the vectors.
Reflection over the x – axis
Reflection over the y – axis
90-degree clockwise rotation around the origin
Half-turn around the origin
90-degree counterclockwise rotation around the origin
Reflection over the line y = x
Reflection over the line y = -x
Copyright 2013. All rights reserved. The Mathematical Association of America. |
More Information
QA-1 Natural Arithmetic
$15.00
Ben Iverson. Volume 1 in set of 3. The basic introduction to Quantum Arithmetic. This should be understandable to a student which is 10 years of age, and is grounded in addition, subtraction and multiplication. A little basic algebra would be helpful. These books are designed for school text books. Although they can be started at a young age, it does not mean that more highly educated persons will find it easy. They may find it quite difficult because they will have certain learnings to forget. 8.5" X 5" |
This course enables students to broaden their understanding of real-world applications of mathematics. Students will analyze data using statistical methods; solve problems involving applications of geometry and trigonometry
MCT4C - Mathematics for College Technology (College):
MEL4E - Mathematics for Work & Everyday Life (Workplace):
This course enables students to broaden their understanding of mathematics as it is applied in the workplace and daily life. Students will use statistics in investigating questions; apply the concept of probability to solve problems in familiar situations; investigate accommodation costs and create household budgets; use proportional reasoning; estimate and measure; and apply geometric concepts to create designs. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
MDM4U - Mathematics: Data Management (University):
This course broadens students understanding of mathematics as it relates to managing data. Students will apply methods for organizing large amounts of information; solve problems involving probability and statistics; and carry out a culminating project that integrates statistical concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. Students planning to enter university programs in business, the social sciences, and the humanities will find this course of particular interest.
MHF4U - Mathematics: Advanced Functions (University):
This course extends students experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students who plan to study mathematics in university and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.
MCV4U - Mathematics: Calculus & Vectors (University):
This course builds on students previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors, and representations of lines and planes in three-dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, rational, exponential, and sinusoidal functions; and apply these concepts and skills to the modeling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who plan to study mathematics in university and who may choose to pursue careers in fields such as physics and engineering. Note: The new Advanced Functions must be taken concurrently with or must precede Calculus and Vectors. |
Purposes
The purpose of this class is to solve a variety of
diverse discrete and counting problems.
Overview
Mathematics is about doing - not about
watching. In this class you will spend most of your time solving
problems. I will help in any way you want, but the mainstay of
the work will be done by you. You will present. You will
work problems. Combinatorics is much more about solving problems
than learning theory. I expect this will appeal to all of
you - as it's matched well with your earliest experiences in
mathematics and probably what draws you the subject in general.
Reading
I have carefully two sources for this class.
Both books are designed as guides and problem sources, more than
explanation sources. By answering a series of leading questions
you can teach yourself. There is also some background text in the
book. It is your responsibility to read that on your own.
Our class time will be spent presenting and working on problems.
The class presentations will be graded roughly as follows:
4 excellent
3.5 very good - at most minor errors
3 some problems, but the main idea of the solution is
clear
2 some correct things
1 attempted
0 no presentation
Priority for presenting problems will be determined
based upon prior performance in the class - lower performance leads to
higher priority. Each student who has not yet presented will have
priority over students who have presented. A third (or more)
problem may be presented in order to replace a prior presentation.
Written Solutions
Each part will have a problem set. Students
are responsible from choosing their own problems from the associated
reading materials. Class presentation problems are not acceptable choices for problem sets. Each problem set is graded
based entirely on the number of problems clearly completed. There
is no partial credit on any particular problem. It either counts
or it does not. In order to count it must be clearly written in
both mathematics and English. Each student will receive credit
for no more than 11 problems per part. Writing a problem that no
one else submits counts for two problems. Remember not to count
on this, though, because others may submit the same problem at some
point. Please regularly check with me regarding what problems you
are choosing. I reserve the right to deem a selection of problems
inadequate and to score them with that limitation. Please be sure
to include a variety of challenges and topics in your problems.
Selecting two problems that are mathematically identical will most
likely result in credit for only one of the two problems.
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.
Academic Dishonesty
While working on homework with one another is
encouraged, all write-ups of solutions must be your own. You are
expected to be able to explain any solution you give me if asked students who miss class
because
of observance of religious holidays the opportunity to make up missed
work.
You are responsible for notifying me by September 11 of plans to
observe
a holiday.
Schedule
August 31
Introduction
September 2 - September 16 Part 1: Graph Theory
September 18 - September 30 Part 2: Strings and Combinations
October 2 - October 16 Part 3: Distributions
October 19 - October 30 Part 4: Partitions
November 2 - November 13 Part 5: Inclusion and Exclusion
November 16 - December 2 Part 6: Recurrence Relations
December 4 - December 21 Part 7: Generating Functions |
Introduction to Scientific, Symbolic, and Graphical Computation
9781568810515
ISBN:
1568810512
Publisher: A K Peters, Limited
Summary: This down-to-earth introduction to computation makes use of the broad array of techniques available in the modern computing environment. A self-contained guide for engineers and other users of computational methods, it has been successfully adopted as a text in teaching the next generation of mathematicians and computer graphics majors |
Acworth, GA ACTArithmetic reasoning on the ASVAB involves basic mathematical operations of real numbers and word problems related to our daily lives. Mathematics knowledge section of the ASVAB includes some more basic math, geometry, and algebraic equations. The most effective way to learn math is from studying and practicing it regularly. |
Call for papers: The interdisciplinary (quarterly) journal of practice, theory, and applied research published by Taylor and Francis -- Abstracted/Indexed in EBSCOhost, MathEduc, ProQuest and other Web indexing services
Special Issue: Computers in K-20 Mathematics Education
Guest Editor: Sergei Abramovich, State University of New York at Potsdam
Just like advances in mathematics research often depend on the methods of calculation available, the effectiveness of mathematics education theories and success of mathematics teaching methods nowadays depend on our knowledge and understanding of how computers can support mathematical learning. The aim of this special issue is to collect scholarly reports on the effective use of computers within the wide range of experiences, grade levels, and curricular topics. Of a special interest are submissions that demonstrate the duality of mathematics learning and computer use in the sense that whereas computers do enable an easy path to mathematical knowledge, mathematics itself can be used to improve the efficiency of computations, which, in turn, enable better access to new mathematical ideas and concepts.
At the pre-college level, the special issue seeks to identify successful experiences in using a computer to communicate the presence of big ideas within seemingly mundane curricular topics and, by the same token, in enabling the study of traditionally difficult and conceptually rich topics through the use of computers. At the college level, the special issue is interested in articles that demonstrate how experimental approach to mathematics that draws on the power of computers to perform numerical and symbolic computations as well as graphical and geometric constructions, makes it possible to balance informal and formal learning of mathematical ideas. Recommended topics to be considered may center on the following questions:
* How are computers used in the preparation of K-12 teachers of mathematics?
* How does the use of computers enable the revision of undergraduate mathematics curriculum?
* How does the use of computers allow one to connect higher concepts to lower concepts and vice versa?
* How can the use of computers contribute to the teaching of upper level mathematics courses or facilitate the revision of advanced mathematics courses to address state-of-the-art in mathematics research?
* How does the use of computers enable the discovery of new knowledge?
* How do computers allow one to teach mathematics differently in the grade school?
* How do computers allow one to teach mathematics differently in the middle school?
* How do computers allow one to teach mathematics differently in the high school?
* How do computers allow one to teach undergraduate mathematics differently?
* How do computers allow one to teach graduate level mathematics courses differently?
Articles are expected to include a theoretical discussion of educational, mathematical, and epistemological issues associated with the use of computers in mathematics education. |
Descripción del producto
Descripción del producto
Packed with fully explained examples, LaTeX Beginner's Guide is a hands-on introduction quickly leading a novice user to professional-quality results. If you are about to write mathematical or scientific papers, seminar handouts, or even plan to write a thesis, then this book offers you a fast-paced and practical introduction. Particularly during studying in school and university you will benefit much, as a mathematician or physicist as well as an engineer or a humanist. Everybody with high expectations who plans to write a paper or a book will be delighted by this stable software.
Biografía del autor
Stefan Kottwitz studied mathematics in Jena and Hamburg. Afterwards, he worked as an IT Administrator and Communication Officer onboard cruise ships for AIDA Cruises and for Hapag-Lloyd Cruises. Following 10 years of sailing around the world, he is now employed as a Network and IT Security Engineer for AIDA Cruises, focusing on network infrastructure and security such as managing firewall systems for headquarters and fleet.
In between contracts, he worked as a freelance programmer and typography designer. For many years he has been providing LaTeX support in online forums. He became a moderator of the web forum latex community.org and of the site golatex.de. Recently, he began supporting the newly established Q and A site tex.stackexchange.com as a moderator.
He publishes ideas and news from the TeX world on his blog at texblog.net.
4.0 de un máximo de 5 estrellasLaTeX for beginners and beyond22 de junio de 2011
Por eloAtl - Publicado en Amazon.com
Formato:Tapa blanda
This book covers all the basics of LaTeX and then some in thirteen chapters totaling about three hundred pages. Each chapter contains a Quiz with solutions given in the Appendix. Its starts with the usual Installation chapter with the installation detailed for Windows, not other environments, maths, fonts (but not much on Unicode), long documents, hyperlinks and bookmarks to finish with a nice and useful chapter on troubleshooting, followed by online resources, answers to quizzes and an index.
The Overall, not LaTeX. It would have been nice to see LaTeX in action. Furthermore, the book has very little on Unicode. The book is about LaTeX, it does not deal at all with XeTeX. It's still useful, but you need to know it. The author uses TeXWorks on Windows. As a result, if you use a Mac and/or another editor, some pages will be useless. But then, I guess you can't really expect a book dedicated to your particular environment. On the negative side,As a disclosure, I need to say that Packt asked me if I would review the book on my site (this is an excerpt of the review). I did not get paid and Packt had no say in the review. The only perk I got was a pdf copy of the book for the review.
When I read about the publication of the book on various forums and blogs, my interest was definitely piqued: the author, Stefan Kottwitz, is a frequent and helpful contributor/moderator on TeX.SX. On the other hand I wondered if anyone would actually want to buy an introductory book to LaTeX, considering the many free tutorials and eBooks available on the Web (although there are many out-of-date ones, so beware!)
After a quick flip through the book, I felt the answer was a very firm "YES". First off, this is certainly an up-to-date book with descriptions of recent packages, and warnings about obsolete ones. While the first few chapter headings read like most other beginner's guide to LATEX, Kottwitz's approach of using complete step-by-step examples throughout the book is something seldom seen in other books or tutorials. By that I mean you don't just get the first few handful of "Hello World" examples, but for much more advanced usage scenarios as well. (BTW, The examples are based on TeXLive and TeXworks.)
Your mileage may vary, but I do feel that such a hand-holding approach (that's what my training course had been described as) -- at least in the early days of learning LaTeX -- is very reassuring. Especially so since LaTeX can be rather intimidating for people who have only used WYSIWYG word processors before. There are pop quizzes are interspersed throughout the content (answers in the appendix).
While the early chapter headings are kind of expected of any beginner's guides, they do still contain valuable nuggets. For example, the microtype package is introduced in Chapter 2, as is how to define your own macros with \newcommand. Imagine a beginner's joy at the even more beautiful typesetting afforded by microtype. And the new-found freedom of defining one's own commands for consistent typesetting of certain materials. Personally I think such tips, introduced at an early stage, would boost beginner's confidence in using LaTeX.
While some might consider the installation instructions of TeXLive and TeXworks in Chapter 1 as frivolous, I certainly welcome the instructions on how to install extra packages in Chapter 11.
Chapter 3 on designing pages is particularly useful, as this seems to be one of the most frequently asked beginner's questions these days. (At least, indicated by the fact that the post on setting page sizes and margins being the 5th all-time most favorite post on my blog.)
I also like the mention of getnonfreefonts in the chapter on fonts. Another favorite chapter of mine is that on Troubleshooting, as this is definitely one of the most important skills if one is to use (and learn!) LaTeX. And everyone who's going to write a thesis or a business report will definitely want to read Chapter 10 on large documents.
Overall, the book does cover everything a beginner should learn about LaTeX, IMHO anyway. My only nitpicks are that the LaTeX logo isn't typeset `properly' in the text; and that the LaTeXed output images seem a tad blurry in the PDF eBook version. But these are just petty nitpicks, really.
So do I recommend LaTeX Beginner's Guide for people interested in learning LaTeX? I'd say Yes. This would be a very nice addition to libraries, or as a communal copy in a research lab, so that newly registered graduate students who're not yet quite busy with their research can spend their first month learning up LaTeX with it. (You can, of course, get your very own copy; I only mention a communal copy as I know some Malaysians -- especially poor grad students -- might be reluctant to fork out about RM120 for a book. Everyone really should fork out money to buy a good book sometime, though.)
7 de 7 personas piensan que la opinión es útil
5.0 de un máximo de 5 estrellasGreat book on LaTeX - not only for beginners28 de mayo de 2011
Por Ingo Bürk - Publicado en Amazon.com
Formato:Tapa blanda
Preface: ========
The author, Stefan Kottwitz, can be found in all common (La)TeX forums as the user Stefan_K and if you're not new to LaTeX you probably already met him online. Since that was the case for me, I knew that he knows what he's talking about and so this book was a must-have for me. I read it in two days and I have to say: I am surprised and amazed. Although it's titled "Beginner's Guide", the target group definitely isn't restricted to beginners. I already wrote larger documents with LaTeX, so I wouldn't consider myself a beginner - and yet I learned a lot just by reading this book once.
Structure: ==========
The book itself is divided into 13 chapters, each being divided into smaller sections. It usually begins with explaining the topic and how to do it in general, followed by "time for action" examples, which then are explained and discussed in detail. That way, it is easy to follow his thoughts, but also to skip certain parts if you want to. I recommend reading everything though, because sometimes he gives little hints which can be really useful.
Content: ========
What can I say - amazing! From how to install a TeX distribution on your computer to how to manage even large projects this book covers everything you (or at least I) need. You really learn how to use LaTeX from scratch and, since I wasn't new to it I know that, he tells you about all the small problems you will sooner or later meet. If I would have had this book at least two years ago, I could have saved myself a lot of time using Google and forums. If you're not experienced with LaTeX, I recommend reading it carefully and really doing your own experiments rather than just copying the examples in order to really learn all the information provided and fully understand what you are doing. In any case I recommend putting little post-its on the pages which seem especially important to you - at least that's what I did and for the moment there are nine of them in my book.
I especially like that the author also talks about typography and how to write "clean" documents rather than giving instructions and commands like "here, do this and that", although this might be a little too much for a complete beginner (which just means in that case you need to read it more carefully). Also, for example, he talks about commonly used, but outdated packages or commands.
Summary: ========
I can really recommend this book, not only to beginners but also to more experienced LaTeX users. There's a lot to learn! Thanks to how it's structed you can also easily use this book to have it on your shelf and look something up in case you need to do so. |
Just went out and got a GMAT review guide but I've been away from math for far too long and the concepts in the quantitative section are over my head. Wondering if someone could recommend a good book/text for a crash course in arithmetic, algebra, and geometry. I found the purplemath link from another post which looks like a good resource for algebra. Anything else would be appreciated.
Manhattan GMAT Numbers Property, Geometry, Algebra, Foundations of GMAT MATH, and Foundations of GMAT VERBAL, Manhattan Gmat Sentence Correction, if you just started studying, get familiar with the manhattan gmat books, in my opinion, you should fork out the 200 bucks, buy them new, and buy the whole set, you'll get the OG Guide #13, and mainly all the tools you'll need to accomplish this task. Books are pretty straightforward, and will teach you the fundamentals for the GMAT. If you buy everything new, you'll get a bunch of question banks + the 6 online CAT exams on the MGMAT.com website, and this will help you strengthen what you've learned on the text book, they cover it all. |
Explaining the concepts of quantum mechanics and quantum field theory in a precise mathematical language, this textbook is an ideal introduction for graduate students in mathematics, helping to prepare them for further studies in quantum physics. The textbook covers topics that are central to quantum physics: non-relativistic quantum mechanics, quantum statistical mechanics, relativistic quantum mechanics and quantum field theory. There is also background material on analysis, classical mechanics, relativity and probability. Each topic is explored through a statement of basic principles followed by simple examples. Around 100 problems throughout the textbook help readers develop their understanding. |
An introduction to functions of a complex variable. The topics covered include complex numbers, analytic and harmonic functions, complex integration, Taylor and Laurent series, residue theory, and improper and trigonometric integrals.
This is an introductory course in complex analysis. Upon completion, students should have a working knowledge of the basic definitions and theorems of the differential and integral calculus of functions of a complex variable and know the similarities and differences between real and complex analysis.
Learning Outcomes
Complex arithmetic, algebra and geometry: Develop facility with complex numbers and the geometry of the complex plane culminating in finding the n nth roots of a complex number.
Differentiable Functions and the Cauchy-Riemann equations: Show knowledge of whether a complex function is differentiable and use the use the Cauchy-Riemann equations to calculate the derivative.
Analytic and Harmonic functions: Determine if a function is harmonic and find a harmonic conjugate via the Cauchy-Riemann equations.
Sequences, Series and Power Series: Determine whether a complex series converges. Show understanding of the region of convergence for power series.
Elementary functions – exponential and logarithm: Understand the similarities and differences between the real and complex exponential function. Compute the complex logarithm. |
We start this first chapter with some definitions to refresh your memory of terms you probably already know. We will point out the difference between exact and approximate numbers, a distinction you may not have made in earlier mathematics classes. Then we will perform the ordinary arithmetic operations—addition and subtraction, multiplication and division—but here it may be a bit different from what you are used to. We will use the calculator extensively, which is probably not new to you, but now we will take great care to decide how many digits of the calculator display to keep. Why not keep them all? We will show that when working with approximate numbers keeping too many digits is misleading to anyone who must use the result of your calculation. As a further complication, we will combine both exact and approximate numbers, as well as positive and negative numbers, or signed numbers. As we proceed, we will point out some rules that will help get us ready for our next chapter on algebra, which is a generalization of arithmetic. |
The Curtis Center aims to provide high quality mathematics activity for
students in local schools. We hope to provide students with a view of
mathematics as a creative reasoning and problem solving activity, with
intrinsic beauty and meaningful application.
The Los Angeles Mathematics
Circle
The UCLA Math Circle is free and open to elementary, middle school and high school students interested in mathematics and eager to learn. Students are divided into 3 groups: Junior circle (elementary school students, grades 1-4); Group A ( students in grades 4-7) and Group B ( students in grades 7-12). The Math Circle meets weekly on Sundays, 2-4 p.m. (3-4 p.m. for the Junior circle). Activities include problem-solving sessions, expository talks on various topics, and preparation for the American Mathematical Competitions (AMC8, AMC10, AMC12, AIME, USAMO).
Mathematics Institute for Young Scholars
This four-week summer day program is designed to deepen secondary
students' understanding of the work of professional mathematicians. The
program focuses on mathematics outside the typical school curriculum and
consists of course lectures, problem solving sessions, seminars and
field trips.
We will not be hosting the Mathematics Institute for Young Scholars in
the summer of 2013. Please check back for updates.
Mathematics Diagnostic Testing Project
As part of its statewide services, MDTP provides two online multiple-choice
tests available for student use to help them prepare for Precalculus and
Calculus level mathematics courses, see These tests were designed to help individual students review their readiness for some mathematics courses and may be useful in preparing for some mathematical placement tests used by California colleges and universities. Each test includes a diagnostic scoring report to help students identify strengths and weaknesses in some topic areas. These tests should be taken without a calculator for best results. The recommended time needed for taking each test is approximately one hour, but there is no enforced time limit. The online tests and resulting diagnostic reports are provided at no charge. For teachers, the UCLA MDTP site serves Los Angeles and Ventura county teachers by distributing, scoring, and reporting the results of tests that measure student readiness for secondary mathematics courses, see |
Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad's matrix and graphing tools offer capabilities that can help students' explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots.... (Show more)(Show less)
New to Mathcad Prime? This brief video illustrates how to leverage the resources on Mathcad's Getting Started tab to learn Mathcad by exploring Help and Tutorials to garner the information required for Just-in-Time learning. ... (Show more)(Show less)
Mathcad 15.0's live math capabilities provide students with timely feedback as they plot graphs, solve equations, or model data. This demonstration illustrates some useful techniques for using Mathcad to help your students be more active in directing their own learning and gain deeper understanding of mathematical concepts.... (Show more)(Show less)
Mathcad offers great features for communicating measurements, calculations, and design intent. This demonstration shows how students can use Mathcad to document and illustrate designs or solve problems in math or engineering. ... |
Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition
The modern landscape of technology and industry demands an equally modern approach to differential equations in the classroom. Designed for a first course in differential equations, the second edition of Brannan/Boyce's Differential Equations: An Introduction to Modern Methods and Applications is consistent with the way engineers and scientists use mathematics in their daily work. The focus on fundamental skills, careful application of technology, and practice in modeling complex systems prepares students for the realities of the new millennium, providing the building blocks to be successful problem-solvers in today's workplace.
The text emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. Section exercises throughout the text provide hands-on experience in modeling, analysis, and computer experimentation. Projects at the end of each chapter provide additional opportunities for students to explore the role played by differential equations in the sciences and engineering.
Brannan/Boyce's Differential Equations 2eis available with WileyPLUS, an online teaching and learning environment initially developed for Calculus and Differential Equations courses. WileyPLUS integrates the complete digital textbook, incorporating robust student and instructor resources with online auto-graded homework to create a singular online learning suite so powerful and effective that no course is complete without it.
Clarity of Applications: Based on the advice of first-edition users and others, the authors have reorganized some topics to make key ideas stand out more clearly, and have added applications that will motivate students by catching their interest and will help to build their skills in modeling with differential equations.
User Friendliness: The 2nd edition is designed to be more student-friendly by adjusting the level and strengthening the emphasis on applications, modeling and the use of computers.
Additional Problems and Exercises: New exercises, projects, and problems invite the student to make conjectures or reach conclusions about complex situations based on computer-generated data and graphs, rather than closed-form solutions.
Stressed Topics: The important link between linear second-order equations and linear systems of dimension two is strengthened in the second edition.
Real-World Applications: New introduction to two-dimensional systems of first-order differential equations in Chapter 3. The author demonstrates the usefulness of eigenvalues in the context of a timely application involving solar energy transfer and storage in a greenhouse.
Reorganization: Sections 4.2 through 4.4 reorganized into two sections. The sections are streamlined and simplified, with optional advanced material moved to exercise sets.
Flexible Organization: Organization of chapters, sections, and projects allows for a variety of course configurations depending on desired course goals, topics, and depth of coverage.
Numerous and Varied Problems: Throughout the text, section exercises of varying levels of difficulty give students hands-on experience in modeling, analysis, and computer experimentation.
Emphasis on Systems: Systems of first order equations, a central and unifying theme of the text, are introduced early, in Chapter 3, and are used frequently thereafter.
Linear Algebra and Matrix Methods: Two-dimensional linear algebra sufficient for the study of two first order equations, taken up in Chapter 3, is presented in Section 3.1. Linear algebra and matrix methods required for the study of linear systems of dimension n (Chapter 6) are treated in Appendix A.
Contemporary Project Applications: Optional projects at the end of Chapters 2 through 10 integrate subject matter in the context of exciting, contemporary applications in science and engineering, such as controlling the attitude of a satellite, ray theory of wave propagation, uniformly distributing points on a sphere, and vibration analysis of tall buildings.
Computing Exercises: In most cases, problems requiring computer generated solutions and graphics are indicated by an icon.
Visual Elements: In addition to a large number of illustrations and graphs within the text, physical representations of dynamical systems and interactive animations available in WileyPLUS provide students with a strong visual component to the subject.
Control Theory: Ideas and methods from the important application area of control theory are introduced in some examples and projects, and in the last section on Laplace Transforms, all of which are optional.
Recurring Themes and Applications: Important themes and applications, such as dynamical system formulation, phase portraits, linearization, stability of equilibrium solutions, vibrating systems, and frequency response are revisited and reexamined in different applications and mathematical settings.
Chapter Summaries: A summary at the end of each chapter provides students and instructors with a birds-eye view of the most important ideas in the chapter.
Available Versions
Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition |
The Mathwise test engine was used successfully five years ago to assess mathematical skills in a traditional British first year engineering mathematics course. Mathwise is a comprehensive set of fifty modules covering topics from the school/university interface in the UK to subjects in the final year of an Honours Mathematics course. The use of Mathwise formal end-of-module assessment has grown over the last few years and this paper will describe some of the areas of growth.
Feedback from the educational experiment in 1994 suggested improvements which, in part, cater for partial credit in formal testing. Mathwise already has some of these features built into its test template giving the chance for the teacher to set intermediate steps in any problem. It also provides for linked answers. In addition, Mathwise has a range of devices to enable the question setter to give partial credit for student answers near but not quite in the correct form. Further improvements have followed in the form of answers in ordered or unordered lists. In packages like Interactive PastPapers it is possible to produce quite sophisticated mathematical tests with automatic marking in a number of feedback modes. |
Euclid's Elements
Description
Euclid's Elements is a remarkable geometry course centered around the propositions of Euclid's Elements.
In the first half of each class students will present prepared talks to explain and demonstrate a geometric proof to the group. Students will learn to make presentations, critically think, and build their confidence as they go through this yearlong course. Students become the teachers of other students!
The second half of each class will have students work through QED™ designed activities to
Euclid's Elements was the first logical presentation of mathematics as designed from a set of first principles. Until publication of modern text books in the 20th century, the Elements was the standard geometry text for all advancedmathematics students. Our course reintroduces this tradition striving to foster excellence in geometry, logic, critical thinking, and public speaking.
Sequence
The Fundamentals
An exciting introduction to Euclid's classic "The Elements". Students will learn logical flow, presentation skills, and the fundamentals of geometry from Euclid's first and second book.
science: students will work with a range of physical problems to determine ideas of measurement, accuracy, and precision.
Information
Each course in Euclid's Elements is a 30-hour workshop for a total of 90 hours of in-class material. Academic year sequences are held in 10 week classes each 3 hours long. Summer camps are held in a single week and cover 30 hours of material.
Each student will receive a full copy of Euclid's Elements. Students are expected to be prepared each week to make a presentation to their classmates.
Classes consist of hands-on activities, lecture, and practice problems. Academic year students are required to complete a minimal amount of practice problems.
Prerequisites
Students should have completed Algebra and should be reading at an 8th grade level. |
More About
This Textbook
Editorial Reviews
Booknews
A text for a junior-level college geometry course for prospective secondary school teachers and other math majors with only calculus as a prerequisite. A diversity of topics are considered in the areas of division ratios and triangles, transformational geometry, projective geometry, conic sections, and axiomatic geometry. Emphasis is placed on the connections between Euclidean and non-Euclidean geometry. Exercises are provided |
Maple
Maple is a powerful mathematical problem-solving and visualization system used worldwide in education, research, and industry. Whether you need to do quick calculations, develop design sheets, teach fundamental concepts, or produce sophisticated high-fidelity simulation models, Maple's world-leading computation engine offers the breadth and depth to handle every type of mathematics.
Maple is available on Windows XP/Vista/7, Mac OS X 10.5+, Sun Solaris, and Linux. Please see Maple System Requirements for more information.
Resources:
Welcome Center The Maplesoft Welcome Center has been designed to provide both new and experienced users of Maplesoft products with a one-stop resource guide that will help you get the most out of your software.
Teacher Resource Center Maple has revolutionized the way teachers and students teach and learn mathematics. Its Smart Document interface allows for an unprecedented level of point-and-click problem solving and visualization. This approach allows teachers and students to focus on concepts, and eliminates the need to memorize command names and syntax.
Student Help Center The Student Help Center provides online forums to get answers to your math questions, online calculators for quick answers to math problems, plus all the resources you need to get up to speed and fully command the power of Maple.
Application Center The Application Center features over 2,000 applicaiotns contributed by the Maplesoft user community.
MaplePrimes MaplePrimes is a web community dedicated to sharing experieces, techniques, and opinions about Maple |
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Books authored by and videos starring E.B. Burger. 1. "The Heart of Mathematics: An invitation to effective thinking" , with Michael Starbird, published by Key College Press and Springer-Verlag in December 1999; Second Edition 2004. 2. "Exploring the Number Jungle: A Journey into Diophantine Anal...
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Kelly's big accomplishments for April included a couple of piano competitions and her CLEP US History II test. After a month or so of work on the Thinkwell Precalculus program we have adjusted her schedule for completion a little to mid-year next year. She likes the program a lot because it is ...
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We all lost a great, lovable companion last week: Scout Tyson. She will be greatly missed by her friends, especially those who worked with Scout at Thinkwell, where she served as the Chief Canine Officer for the past, almost 12 years. (You can read about her job in her guest post at Thinkwell, he...
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Ph.D., Warsaw University This site is a resourse for Margaret Wojcicka's Communtity College of Philadelphia Mathamatics classes. Select your course to find the syllabus, homework and links, as well as previous exams for practice. My office hours for Fall 2006 are Monday, Wednesday, Friday from 11...
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In this assignment you will be expected to write a journal entry that serves to analyze the interrelationships that exist within the Real Number System. You will apply your understanding of the vocabulary words you defined. You will utilize the knowledge you acquired through your research. You may even want to refer to the images in the Photo Gallery.
Journal Prompt: Now that you have become more familiar with the Real Number System, explain the interrelationships that exist between the different subsets. In your writing be sure to address the following: |
This course is the second half of a two-quarter precalculus sequence. The focus of this courseis the development of trigonometry from the unit circle point of view. Exponentialand logarithmic functions and their applications to constant percentage rate growthproblems will be a second focus. Other topics include polar graphs, conic section equations(for the parabola, ellipse, and hyperbola), using augmented matrices to solve linearsystems, arithmetic and geometric sequences, mathematical induction, and the binomialtheorem.
This course will make more sense if you can read or at least browse through the relevant sectionsof the book before each class. Generally we'll cover a section of the book in one to twodays.
Keeping up with the new concepts through homework exercises is essential to success in thecourse. I will frequently collect homework exercises handed out in class, but will not usually collecthomework exercises from the book. We will spend class time discussing the the book exercises, andyou need to do the book exercises to learn the material well. There will be occasional quizzes aswell.
I'm planning to give three tests and a final exam. Some of the tests may have a take-home part.There will be an opportunity to make-up one test by the way I score the final exam. I look at eachsection of the comprehensive final (a test one part , a test two part , etc. ) and look to see onwhich section you have improved the most. If you have, for example, improved the most on thetest two part of the final exam, then the score on the test two portion of the final replaces youroriginal test two score. Of course, if the final exam scores are all lower, your original test scores areleft unchanged. |
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Challenging core courses. A growing number of career cluster classes. And electives to spark all sorts of new interests. iForward's extensive course catalog is always changing to meet students' needs. So check back often. Who knows what you'll discover?
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Math
Core
Algebra I
HONORS|1.0 Credits
Description: Algebra I is the foundation! The skills you'll acquire in this course contain the basic knowledge you'll need for all your high school math courses. Relax! This stuff is important, but everyone can do it. Everyone can have a good time solving the hundreds of real-world problems that are answered with algebra. Each module in this course is presented in a step-by-step way right on your computer screen. You won't have to stare at the board from the back of a classroom. There are even hands-on labs to make the numbers, graphs and equations more real. It's all tied to real-world applications like sports, travel, business and health. This course is designed to give you the skills and strategies for solving all kinds of mathematical problems. It will also give you the confidence that you can handle everything that high school math has in store for you.
Algebra II
HONORS|1.0 Credits
Description: This course connects algebra to the real world. It also demystifies algebra, making it easier to understand and master. The goal is to create a foundation in math that will stay with you throughout high school.
Prerequisites: Geometry
Calculus
1.0 Credits
Description: Students in this course will walk in the footsteps of Newton and Leibnitz.
An interactive text and graphing software combine with the exciting on-line course delivery to make calculus an adventure. The course includes a study of limits, continuity, differentiation, and integration of algebraic, trigonometric, and transcendental functions, and the applications of derivatives and integrals.
Description: For those students needing a slower approach at learning Algebra. This course will be held in Acellus, a different website than Brain Honey. A recommendation from the teacher or guidance counselor is needed.
Geometry
HONORS|1.0 Credits
Description: One day in 2580 B.C., a very serious architect stood on a dusty desert with a set of plans. His plans called for creating a structure 480 feet, with a square base and triangular sides, using stone blocks weighing two tons each. The Pharaoh wanted the job done right. The better our architect understood geometry, the better were his chances for staying alive. Geometry is everywhere, not just in pyramids. Engineers use geometry to bank highways and build bridges. Artists use geometry to create perspective in their paintings, and mapmakers help travelers find things using the points located on a geometric grid. Throughout this course, we'll take you on a mathematical highway illuminated by spatial relationships, reasoning, connections, and problem solving. This course is all about points, lines and planes. Just as importantly, this course is about acquiring a basic tool for understanding and manipulating the real world around you.
Prerequisites: Algebra I
Integrated Math
0.5 Credits
Description: This course will review some of the fundamental math skills you learned in middle school, and then get you up to speed on the basic concepts of algebra. This course will be taught as a "blended" course. Students from the Grantsburg High School will be attending simultaneously as the teacher works with both traditional and online learners. Pre-approval is necessary to take this course. Call Mr. Beesley, Mr. Bettendorf, or Mr. Mark Johnson for more information.
Liberal Arts Mathematics
1.0 Credits
Description: The total weight of two beluga whales and three orca whales is 36,000 pounds. As you'll see in this course, if given one additional fact, you can determine the weight of each whale. To answer this weighty question, we'll give you all the math tools you'll need. The setting for this course is an amusement park with animals, rides, and games. Your job will be to apply what you learn to dozens of real-world scenarios. Equations, geometric relationships, and statistical probabilities can sometimes be dull, but not in this class! Your park guide (teacher) will take you on a grand tour of problems and puzzles that show how things work and how mathematics provides valuable tools for everyday living. Come reinforce your existing algebra and geometry skills to learn solid skills with the algebraic and geometric concepts you'll need for further study of mathematics. We have an admission ticket with your name on it and we promise an exciting ride with no waiting!
Pre-Calculus
1.0 Credits
Description: Students, as mathematic analysts, will investigate how advanced mathematics concepts can solve problems encountered in operating national parks. The purpose of this course is to study functions and develop skills necessary for the study of calculus. The pre-calculus course includes analytical geometry and trigonometry.
Prerequisites: Advanced Algebra or Algebra 2
Elective
AP Calculus AB AB exam given each year in May. With continuous enrollment, students can start the course and begin working on Calculus as early as spring of the previous year.
AP Calculus BC BC exam given each year in May. With continuous enrollment, students can start the course and begin working on Calculus as early as spring of the previous year.
AP Statistics
ADVANCED PLACEMENT|1.0 Credits
Description: Statistics are used everywhere from fast food businesses ordering hamburger patties to insurance companies setting rates to predicting a student's future success by the results of a test. Students will become familiar with the vocabulary, method, and meaning in the statistics which exist in the world around them. This is an applied course in which students actively construct their own understanding of the methods, interpretation, communication, and application of statistics. Each unit is framed by enduring understandings and essential questions designed to allow students a deep understanding of the concepts at hand rather than memorization and emulation. Students will also complete several performance tasks throughout the year consisting of relevant, open-ended tasks requiring students to connect multiple statistical topics together.
iForward is Wisconsin's leading nonprofit online charter school, administered by the award-winning Grantsburg School District. With career-focused academics tailored to meet each student's unique learning style and personal goals, we give middle and high school students the individualized instruction they need to reach their own potential. |
Integers
This math book teaches Integers, a topic that is completely neglected in school except some mention of "there are positive and negative numbers" and then discussion of the rules in operations with integers.
Fact 1: Many kids see their grades drop when they start studying Algebra.
Fact 2: Many kids see their grades drop, or drop more for some, when they start studying Calculus.
Fact 3: Those are not bad students but many are 'A' students.
The common explanation: Algebra and Calculus are conceptually hard (nonsense), and most students are just not naturally inclined to math. Sometimes the explanation is that "the student is not working hard enough." Not true in many cases.
The real explanation: Algebra and Calculus require understanding of numbers, operations, and our place value system. Not in a superficial and operational, can only use but doesn't understand, way. They require a deeper level of understanding. They require students to really understand numbers, what they represent, and how they work.
That's where Integers come into the picture.
Integers are a very important concept. They are not just "negative and positive numbers". An integer is a new idea compared to whole (positive) number (it's an idea about direction).
Fact 4: Schools and math books DO NOT TEACH Integers! None does. Students are told (or read) that "integers are just both positive and negative numbers". They they learn the Rules (e.g. "negative times negative is positive"), which don't really make sense to the poor students who were not taught what integers really are. Naturally, they forget those rules quickly.
Then comes Algebra, with its extensive need for understanding Integers. Without understanding the idea of Integers, how they work, and how to use them, students have no chance in understanding Algebra well, and of solving Word Problems (see the book on those) that require those ideas. Later, when students study calculus, which introduces real numbers, it is even more critical to understand Integers fully and deeply.
This book teaches EVERYTHING to know about Integers. It does so simply. It explains what Integers are; the meaning of operations with integers; and how to use integers. Those "negative times negative is positive" rules become logical and easy to remember because students understand why they are so. Then, when they learn Algebra and Calculus, they are not confused, and they often do better (you do want them to use the Word Problems book before Algebra). |
>My point at the beginning of all this was that when complex numbers are introduced to high school students they don't (and can't) fathom that C is a field, let alone an extension of the R field.
They don't need to be introduced to all the AA lingo - just the operations and identities and work some problems. And I certainly wasn't think "all algebra students". I was think more like "Honors Trig." or "Precalc" or whatever the local instantiation may be. My son attends a decent H.S. where the kids are sometimes well equipped by 10th grade for something like real complex numbers. |
Book description
Enables readers to apply the fundamentals of differential calculus to
solve real-life problems in engineering and the physical sciences
Introduction to Differential Calculus fully engages readers by
presenting the fundamental theories and methods of differential
calculus and then showcasing how the discussed concepts can be applied
to real-world problems in engineering and the physical sciences. With
its easy-to-follow style and accessible explanations, the book sets a
solid foundation before advancing to specific calculus methods,
demonstrating the connections between differential calculus theory and
its applications.
The first five chapters introduce underlying concepts such as
algebra, geometry, coordinate geometry, and trigonometry. Subsequent
chapters present a broad range of theories, methods, and applications
in differential calculus, including:
Concepts of function, continuity, and derivative
Properties of exponential and logarithmic function
Inverse trigonometric functions and their properties
Derivatives of higher order
Methods to find maximum and minimum values of a function
Hyperbolic functions and their properties
Readers are equipped with the necessary tools to quickly learn how to
understand a broad range of current problems throughout the physical
sciences and engineering that can only be solved with calculus.
Examples throughout provide practical guidance, and practice problems
and exercises allow for further development and fine-tuning of various
calculus skills. Introduction to Differential Calculus is an excellent
book for upper-undergraduate calculus courses and is also an ideal
reference for students and professionals alike who would like to gain
a further understanding of the use of calculus to solve problems in a
simplified manner.
Ulrich L. Rohde
, PhD, ScD, Dr-Ing, is Chairman of Synergy Microwave Corporation,
President of Communications Consulting Corporation, and a Partner of
Rohde & Schwarz. A Fellow of the IEEE, Professor Rohde holds several
patents and has published more than 200 scientific papers.
G. C. Jain, BSc, is a retired scientist from the Defense
Research and Development Organization in India.
Ajay K. Poddar, PhD, is Chief Scientist at Synergy Microwave
Corporation. A Senior Member of the IEEE, Dr. Poddar holds several
dozen patents and has published more than 180 scientific papers.
A. K. Ghosh, PhD, is Professor in the Department of Aerospace
Engineering at IIT Kanpur, India. He has published more than 120
scientific papers. |
Purchasing Options
Features
Contains over 30 complete demonstrations that can be used directly in the classroom, including suggestions on how to use them
Provides the tools you need to implement your own creative ideas
Includes a CD-ROM convenient for both PC and Macintosh users that contains all of the Maple code used in the book
Presents all demonstrations in forms that will work in Maple 7, 8, or 9
Assumes no previous experience with Maple
Summary
There is nothing quite like that feeling you get when you see that look of recognition and enjoyment on your students' faces. Not just the strong ones, but everyone is nodding in agreement during your first explanation of the geometry of directional derivatives.
If you have incorporated animated demonstrations into your teaching, you know how effective they can be in eliciting this kind of response. You know the value of giving students vivid moving images to tie to concepts. But learning to make animations generally requires extensive searching through a vast computer algebra system for the pertinent functions. Maple Animation brings together virtually all of the functions and procedures useful in creating sophisticated animations using Maple 7, 8, or 9 and it presents them in a logical, accessible way. The accompanying CD-ROM provides all of the Maple code used in the book, including the code for more than 30 ready-to-use demonstrations.
From Newton's method to linear transformations, the complete animations included in this book allow you to use them straight out of the box. Careful explanations of the methods teach you how to implement your own creative ideas. Whether you are a novice or an experienced Maple user, Maple Animation provides the tools and skills to enhance your teaching and your students' enjoyment of the subject through animation.
Table of Contents
Getting Started The basic command line A few words about Maple arithmetic Comments Assigning names to results Built-in functions Defining functions Getting help and taking the tour Saving, quitting, and returning to a saved worksheet
Simple Animations Animating a function of a single variable Outline of an animation worksheet Demonstrations: Secant lines and tangent lines Using animated demonstrations in the classroom Watching a curve being drawn Demonstration: The squeeze theorem Animating a function of two variables Demonstrations: Hyperboloids Demonstrations: Paraboloids Demonstration: Level curves and contour plots
Editorial Reviews
"Putz designed this book for teachers of precalculus and first and second year calculus to provide a large number of animations to be used to illustrate various concepts of calculus. The accompanying CD-ROM contains all the described examples coded for use in class directly, along with suggestions for how to use these examples... This book will be very useful to faculty. Recommended." -CHOICE, 2004 |
Demystified is your solution for tricky subjects like trigonometry. If you think a Cartesian coordinate is something from science fiction or a hyperbolic tangent is an extremeexaggeration, you need Trigonometry DeMYSTiFieD , Second Edition, to unravel this topic's fundamental concepts and theories at your own pace. This practical guide eases you... more...
Most math and science study guides are a reflection of the college professors who write them-dry, difficult, and pretentious.
The Humongous Book of Trigonometry Problems is the exception. Author Mike Kelley has taken what appears to be a typical t more...
500 Ways to Achieve Your Best Grades. We want you to succeed on your college algebra and trigonometry midterm and final exams. That's why we've selected these 500 questions to help you study more effectively, use your preparation time wisely, and getyour best grades. These questions and answers are similar to the ones you'll find on a typical... more...
From angles to functions to identities - solve trig equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step manner.... more...
Don't be tripped up by trigonometry. Master this math with practice, practice, practice!
Practice Makes Perfect: Trigonometry is a comprehensive guide and workbook that covers all the basics of trigonometry that you need to understand this subject. Each chapter focuses on one major topic, with thorough explanations and many illustrative examples,...
Boost Your grades with this illustrated quick-study guide. You will use it from high school all the way to graduate school and beyond. Clear and concise explanations. Difficult concepts are explained in simple terms. Illustrated with graphs and diagrams. Search for the words or phrases. Access the guide anytime, anywhere - at home, on the train, in... more... |
Should College Classes Ditch the Calculator?
Should College Classes Ditch the Calculator?
According to Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research and Development Center, using calculators in college math classes may be doing more harm than good. In a limited study conducted with undergraduate engineering students and published in the British Journal of Educational Technology, King has determined that our use of calculators may be serving as an alternative to an actual, deep understanding of mathematical material.
"We really can't assume that calculators are helping students," says King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
King, along with co-author and director of the Mathematics Education Centre at Loughborough University, Carol Robinson, conducted the study by interviewing 10 second-year undergraduate students who were enrolled in a competitive engineering program. The students were given a number of mathematical questions dealing with sine waves, which are mathematical curves that describe a smooth repetitive oscillation. To help solve the problems, the students were given the option of using a calculator instead of completing the work entirely by hand. Over half of the students questioned opted to utilize their calculators in order to solve the problems and plot the sine waves.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," says King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the work, King and Robinson interviewed the students about how they approached the material. One student who used the calculator stated that she had trouble remembering the rules for how sine waves operate, and found it generally easier to use a calculator instead. In contrast, however, a student who opted to complete the work without a calculator stated that they couldn't see why anyone would have trouble completing the question, but did admit that it would likely be easier with a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," says King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
Given the small sample size used in the study, it is entirely possible that King's findings are largely anecdotal in how our usage of calculators and understanding of mathematical concepts may positively or negatively correlate. However, King does stress that while all the evidence may not be in, his study does raise important questions regarding how, when and why students choose to use calculators, and in doing so, we may develop a more holistic approach to math instruction |
HSC Maths Course Selection
How to make the right choice at High School to prevent issues later
While we encourage everyone to study the level of maths at which they feel sufficiently challenged, below are some guidelines for the minimum level of maths required for some of our degrees. Please also see Assumed Knowledge for more information.
Type of Degree
Assumed Knowledge
Science with Mathematics Major
Mathematics and Mathematics Extension 1, but study as much maths as you can!
Science or Engineering
Mathematics and Mathematics Extension 1
Commerce with Actuaries, Finance or Accounting Major
Mathematics and Mathematics Extension 1
Health Sciences
Mathematics
Studying at least HSC Mathematics Extension 1 will leave your options open when you choose your university degree. Knowledge of mathematics is a useful skill in any profession, even if it is not required for a particular course. No doubt, you will have to use mathematical tools such as graphs and statistics for the rest of your life.
Learning mathematics also teaches you to approach problems in a systematic and logical way. This is a transferable skill that is particularly important in project management and it is an important skill for life.
HSC Plus
UNSW has introduced a program of rewarding, by means of a bonus points system, performance in Australian Year 12 courses relevant to UNSW undergraduate programs.
This is in recognition of the strong correlation between Year 12 course performance and preparation for, and success in, first year university studies. Students who have undertaken, and done well in, relevant Year 12 courses are generally well prepared for the demands of university study. By employing strategies that enhance the academic achievement of UNSW students, we also improve their employability and increase the range of career options for our graduates. For more information, please see UNSW's HSC Plus page.
Pathways into UNSW Mathematics & Statistics
Below are some examples of mathematics subjects undertaken at high school, and the pathways into first year UNSW courses that assume a certain competency in mathematics.
If you don't have the assumed knowledge, don't panic. The School of Mathematics and Statistics runs a bridging course to build HSC Mathematics knowledge to a level close to HSC Mathematics Extension 1 knowledge.Bear in mind that catching up on knowledge involves hard work and it is always best, if you can, to gain the right amount of knowledge at school.
Essential Mathematics for Higher Education
The Program will be delivered in partnership with TAFE NSW, Randwick College. The program will cover the contents from the HSC two unit mathematics course and will especially be suitable for students who wish to improve their knowledge in mathematics. The course will also be beneficial to students who are having difficulty in their first year of their chosen University Program and the 1st year mathematics courses. The program will be tailored to meet students' needs with a flexible timetable. The program will be run twice during Semester 1 2013. |
Introduction to Proofs This four minute video uses a smart board and a narrator to show students how to create a reasons column and the steps to follow in order to write a 2-column proof in Geometry. It is a good introduction to proofs and the underlying concepts, but no actual examples are done. Author(s): No creator set
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Shock Matrix: -st- Watch Shock master the matrix as he manipulates sounds! The goal for this segment is consonant blend -st-. (1min)Learn Japanese Grammar Video - Absolute Beginner #16 - Suggestions and Invitations |
I think you mentioned that the books used for math and science at Highlands for 7th grade and up are either out of print or hard to find, and I can't recall what you said about what you were going to put into your curriculum packages for homeschoolers. So can you tell me both:
A)What books you actually use at Highlands (or are planning to use going into the future) even if out of print
B)What you currently recommend for homeschoolers or will be putting into your curriculum packages
for both math and science, and for not only 7th and 8th grade, but high school as well.
Thank you so much for all the time you spend answering the questions on this board!
I can tell you easily what we use now for 7th/8th grade math and science, but I don't know yet what we are going to put in our packages for homeschoolers. We'll have to make that decision this summer.
Right now, at Highlands Latin, we use the College of the Redwoods pre-algebra book in 7th grade, and our teachers like it. That's an easy one for homeschoolers because you can buy it on lulu.com for $20! It's self-published by the college and very affordable. We use an out of print Lial/Miller Algebra I book (ISBN 0673188086) in 8th grade.
For science, we don't actually do science in 7th grade. We do world geography, using a course written by one of our teachers, and we add Greek, so there's no room for science. In 8th grade, we do physical science with an emphasis on pre-chemistry using another out of print book - Silver Burdett & Ginn Physical Science ISBN 0382-13472-9.
The problem with these out of print books is that there are no tests or quizzes, and solution manuals are hard to come by. I'm planning to meet with our 7th-8th grade teachers at the end of the year and see what they have to offer me, if anything!
Sorry not to have more information yet, but it's hard to nail the teachers down right now as they work to close out their year.
Tanya,
Do you think you'll have something available for 8th grade science before Fall? I'm trying to make my final choices soonish so I can work on lesson plans in June.
Right now I'm looking at either Holt Physical Science or Prentice Hall Physical Science. I'm also going to look at the book you use at Highlands- but I'm not crazy about making up my own tests. Do you think you would offer some from your school?
My goal is to have something by fall. If we go with the science book we use at school, we will definitely have the tests you need and a syllabus. I won't really know until the middle of June when the teachers are free to work with me. Check with me then, and I'll have a better idea of whether it's going to happen or not.
[QUOTE]In 8th grade, we do physical science with an emphasis on pre-chemistry using another out of print book - Silver Burdett & Ginn Physical Science ISBN 0382-13472-9.[/QUOTE]
This course is on my considering list for fall. You wouldn't happen to know how much lab equipment and any uncommon supplies are needed for this course? Even a rough estimate would be helpful. Thus far I've only used science curricula written specifically for homeschoolers, so the supplies are generally easy to find. |
Qualification through placement, or a grade of C or better in
Math 111 or 115.
Text:
Applied Calculus by Hughes-Hallett, Gleason, Lock, Flath, et al.
Calculator:
Each student is required to have a graphing calculator. My instructions
will predominantly be for TI-83 which is the preferred calculator for this
course.
Overview:
One of the main objectives of this course is for you to understand the basic
concepts of calculus well enough to know when, how, and why to apply them in
real-world situations and to be able to interpret and communicate the results.
To achieve this goal will require practice at a variety of numerical,
graphical, and algebraic methods.
The preface of the book provides additional detail and insight into the
methods you will encounter in this course. Page xii is particularly
well-written but the entire preface is worth reading.
We will work through most of chapters 1 -- 5, and parts of chapter 6.
Attendance:
You are expected to come to every class on time and stay until the end of
class. If you miss one day of class during the summer session, it's
almost as bad as missing an entire week during the fall or spring.
Work Load:
A standard rule of thumb for math classes is that you should
study 2-3 hours outside of class for each hour spent in class. We are
scheduled to meet 9 hours per week in class. This leaves an additional
18-27 hours per week to study outside of class.
Grading:
There will be daily reading and homework, but your grade will only
be based upon your score on the 4 tests. No make-ups will be given
for any of these tests. If you miss a test for any reason, you will
get a 0 on that test. There will be an optional cumulative
test on the last day of class which can replace the lowest of your
4 test scores. There will be no final exam. |
Illustrated Dictionary of Math
Retail Price:
$12.99$11.04
Product ID - RB8055 | Availability - Now Shipping
Looking for a helpful resource that explains math concepts with easy-to-follow examples and colorful, brightly illustrated diagrams? With the Illustrated Dictionary of Math, your child will have over 500 definitions of key math terms and their uses in one convenient 130-page, softbound book. Includes Internet links.
More Details
Does your child know the difference between a rational number and a real number? Is he familiar with a Fibonacci sequence, arithmetic with vectors, or algebraic expressions? With the Illustrated Dictionary of Math from Alpha Omega Publications, your child will solidify his understanding of these math concepts and more with in-depth explanations, easy-to-follow examples, full-color illustrations, and eye-catching diagrams. Divided into four sections—numbers; shapes, space, and measures; algebra; and handling data—this supplemental math resource perfectly complements your homeschool math curriculum. Simply use the comprehensive cross-reference guide and detailed index to locate the current math topic you're studying, and then read the clearly-outlined information. How easy is that? No more confusion, no more frustration. Math will make sense and your child will have a reference tool to remind him of key math facts whenever studying for tests or completing daily lessons.
But there's more! The Illustrated Dictionary of Math also includes an alphabetical list of common money terms as well as a list of commonly used math symbols. Plus for each topic in this math resource book, you'll find internet links to interesting and exciting supplemental websites. Your child will be able to test his math skills with puzzles, games, and quizzes; take virtual tours of the universe from outer space to the innermost parts of atoms; learn how to use mental math tricks to perform difficult calculations in his head; and more! Sound exciting? It is! Don't wait to order the Illustrated Dictionary of Math for your child—add |
Higher Order Derivatives
In this lesson, Professor John Zhu gives an introduction to the higher order derivatives. He explains how the 1st, 2nd, and 3rd derivative relate to one another and goes on to show you example problems.
This content requires Javascript to be available and enabled in your browser.
Higher Order Derivatives
To avoid confusion: treat each
level of derivative as brand new derivative
Higher order derivatives are
easier to solve because of eliminated terms
Higher Order Derivatives
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
This course is an introduction to discrete mathematics, with an emphasis on material used in computer science. Topics include logic, Boolean algebra, coding theory, set theory, combinatorics, and graph theory. (UC, CSU) |
easy-to-understand primer on advanced calculus topics Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams. It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, with introductions to multivariable calculus, differential equations, and numerical analysis. Best of all, it includes practical exercises designed to simplify and enhance understanding of this complex subject. -- |
Calculus and Analysis
In mathematics , catastrophe theory is a branch of bifurcation theory in the study of dynamical systems ; it is also a particular special case of more general singularity theory in geometry . Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation.
In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos . It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter
A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold ). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system.
Disclaimer: CosmoLearning is promoting these educational resources as a courtesy of Gaussian Department of Mathematics (GMath).
Visualizing a function can give a mathematician enormous insight into the function's algebraic and geometrical properties. The easiest way to see what a function looks like is to use a computer as a graphing tool. |
The focused ion beam (FIB) system is an important tool for understanding and manipulating the structure of materials at the nanoscale. Combining this system with an electron beam creates a DualBeam - a single system that can function as an imaging, analytical and sample modification tool. Presenting the principles, capabilities, challenges and applications of the FIB technique, this edited volume comprehensively covers the ion beam technology including the DualBeam. The basic principles of ion beam and two-beam systems, their interaction with materials, etching and deposition are all...
For many everyday transmissions, it is essential to protect digital information from noise or eavesdropping. This undergraduate introduction to error correction and cryptography is unique in devoting several chapters to quantum cryptography and quantum computing, thus providing a context in which ideas from mathematics and physics meet. By covering such topics as Shor's quantum factoring algorithm, this text informs the reader about current thinking in quantum information theory and encourages an appreciation of the connections between mathematics and science.Of particular interest are the new edition continues to serve as a comprehensive guide to modern and classical methods of statistical computing. The book is comprised of four main parts spanning the field:
Optimization
Integration and Simulation
Bootstrapping
Density Estimation and Smoothing
Within these sections, each chapter includes a comprehensive introduction and step-by-step implementation summaries to accompany the explanations of key methods. The new edition includes updated coverage and existing topics as well as new topics such as adaptive MCMC and bootstrapping for correlated data. ...
Business Mathematics focuses on transforming learning and teaching math into its simplest form by adopting learning by application approach. The book is refreshingly different in its approach, and endeavors to motivate student to learn the concept and apply them in real-life situations. It is purposely designed for the undergraduate students of management and commerce and covers wide range of syllabuses of different universities offering this course.
Complex Analysis presents a comprehensive and student-friendly introduction to the important concepts of the subject. Its clear, concise writing style and numerous applications make the basics easily accessible to students, and serves as an excellent resource for self-study. Its comprehensive coverage includes Cauchy-Goursat theorem, along with the description of connected domains and its extensions and a separate chapter on analytic functions explaining the concepts of limits, continuity and differentiability.
...
Discrete Mathematics will be of use to any undergraduate as well as post graduate courses in Computer Science and Mathematics. The syllabi of all these courses have been studied in depth and utmost care has been taken to ensure that all the essential topics in discrete structures are adequately emphasized. The book will enable the students to develop the requisite computational skills needed in software engineering.
...
The second edition of Business Statistics, continues to retain the clear, crisp pedagogy of the first edition. It now adds new features and an even stronger emphasis on practical, applied statistics that will enhance the text's ability in developing decision-making ability of the reader. In this edition, efforts have been made to assist readers in converting data into useful information that can be used by decision-makers in making more thoughtful, information-based decisionsMathematics lays the basic foundation for engineering students to pursue their core subjects. In Engineering Mathematics-III, the topics have been dealt with in a style that is lucid and easy to understand, supported by illustrations that enable the student to assimilate the concepts effortlessly. Each chapter is replete with exercises to help the student gain a deep insight into the subject. The nuances of the subject have been brought out through more than 300 well-chosen, worked-out examples interspersed across the book.
...
Linear Algebra is designed as a text for postgraduate and undergraduate students of Mathematics. This book explains the basics comprehensively and with clarity. The flowing narrative of the book provides a refreshing approach to the subject. Drawing on decades of experience from teaching and based on extensive discussions with teachers and students, the book simplifies proofs while doing away with needless burdensome textual details.
...
The new edition of this introductory graduate textbook provides a concise but accessible introduction to the Standard Model. It has been updated to account for the successes of the theory of strong interactions, and the observations on matter-antimatter asymmetry. It has become clear that neutrinos are not mass-less, and this book gives a coherent presentation of the phenomena and the theory that describes them. It includes an account of progress in the theory of strong interactions and of advances in neutrino physics. The book clearly develops the theoretical concepts from the...
Our knowledge of biological macromolecules and their interactions is based on the application of physical methods, ranging from classical thermodynamics to recently developed techniques for the detection and manipulation of single molecules. These methods, which include mass spectrometry, hydrodynamics, microscopy, diffraction and crystallography, electron microscopy, molecular dynamics simulations, and nuclear magnetic resonance, are complementary; each has its specific advantages and limitations. Organised by method, this textbook provides descriptions and examples of applications for...
This highly-regarded text provides an up-to-date and comprehensive introduction to modern particle physics. Extensively rewritten and updated, this 4th edition includes all the recent developments in elementary particle physics, as well as its connections with cosmology and astrophysics. As in previous editions, the balance between experiment and theory is continually emphasised. The stress is on the phenomenological approach and basic theoretical concepts rather than rigorous mathematical detail. Short descriptions are given of some of the key experiments in the field, and how they...
A basic problem in computer vision is to understand the structure of a real world scene given several images of it. Techniques for solving this problem are taken from projective geometry and photogrammetry. Here, the authors cover the geometric principles and their algebraic representation in terms of camera projection matrices, the fundamental matrix and the trifocal tensor. The theory and methods of computation of these entities are discussed with real examples, as is their use in the reconstruction of scenes from multiple images. The new edition features an extended introduction...
This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy and theology offer a rich intellectual exchange among various current viewpoints, rather than displaying a static picture of accepted views on infinity. The book starts with a historical examination of the transformation...
The Standard Model is the most comprehensive physical theory ever developed. This textbook conveys the basic elements of the Standard Model using elementary concepts, without the theoretical rigor found in most other texts on this subject. It contains examples of basic experiments, allowing readers to see how measurements and theory interplay in the development of physics. The author examines leptons, hadrons and quarks, before presenting the dynamics and the surprising properties of the charges of the different forces. The textbook concludes with a brief discussion on the recent... |
Course in Mathematical Modeling - 99 edition
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This course is the second preparatory algebra course and readies students for the first college-level mathematics course. Concepts studied in the course include rational expressions and equations, functions, radical expressions and equations, complex numbers, and solving equations using factoring, completing the square, and the quadratic formula.
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Numberical Solution of Algebraic Systems Notes
Course: MATH 5485, Fall 2009 School: UCF Rating:
Word Count: 5304
Document Preview, including sound waves, water waves, elastic waves, electromagnetic waves, and so on. For simplicity, we restrict our attention to the case of waves in a one-dimensional medium, e.g., a string, bar, or column of air. We begin with a general discussion of finite difference formulae for numerically approximating derivatives of functions. The basic finite difference scheme is obtained by replacing the derivatives in the equation by the appropriate numerical differentiation formulae. However, there is no guarantee that the resulting numerical scheme will accurately approximate the true solution, and further analysis is required to elicit bona fide, convergent numerical algorithms. In dynamical problems, the finite difference schemes replace the partial differential equation by an iterative linear matrix system, and the analysis of convergence relies on the methods covered in Section 7.1. We will only introduce the most basic algorithms, leaving more sophisticated variations and extensions to a more thorough treatment, which can be found in numerical analysis texts, e.g., [5, 7, 28].
11.1. Finite Differences.
In general, to approximate the derivative of a function at a point, say f (x) or f (x), one constructs a suitable combination of sampled function values at nearby points. The underlying formalism used to construct these approximation formulae is known as the calculus of finite differences. Its development has a long and influential history, dating back to Newton. The resulting finite difference numerical methods for solving differential equations have extremely broad applicability, and can, with proper care, be adapted to most problems that arise in mathematics and its many applications. The simplest finite difference approximation is the ordinary difference quotient u(x + h) - u(x) u (x), h 4/20/07 186
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(11.1)
Peter J. Olver
One-Sided Difference Figure 11.1.
Central Difference Finite Difference Approximations.
used to approximate the first derivative of the function u(x). Indeed, if u is differentiable at x, then u (x) is, by definition, the limit, as h 0 of the finite difference quotients. Geometrically, the difference quotient equals the slope of the secant line through the two points x, u(x) and x + h, u(x + h) on the graph of the function. For small h, this should be a reasonably good approximation to the slope of the tangent line, u (x), as illustrated in the first picture in Figure 11.1. How close an approximation is the difference quotient? To answer this question, we assume that u(x) is at least twice continuously differentiable, and examine the first order Taylor expansion (11.2) u(x + h) = u(x) + u (x) h + 1 u () h2 . 2 We have used the Cauchy formula for the remainder term, in which represents some point lying between x and x + h. The error or difference between the finite difference formula and the derivative being approximated is given by u(x + h) - u(x) - u (x) = h
1 2
u () h.
(11.3)
Since the error is proportional to h, we say that the finite difference quotient (11.3) is a first order approximation. When the precise formula for the error is not so important, we will write u(x + h) - u(x) + O(h). (11.4) u (x) = h The "big Oh" notation O(h) refers to a term that is proportional to h, or, more rigorously, bounded by a constant multiple of h as h 0. Example 11.1. Let u(x) = sin x. Let us try to approximate u (1) = cos 1 = 0.5403023 . . . by computing finite difference quotients cos 1 sin(1 + h) - sin 1 . h
The result for different values of h is listed in the following table. 4/20/07 187
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h approximation error
1 0.067826 -0.472476
.1 0.497364 -0.042939
.01 0.536086 -0.004216
.001 0.539881 -0.000421
.0001 0.540260 -0.000042
1 We observe that reducing the step size by a factor of 10 reduces the size of the error by approximately the same factor. Thus, to obtain 10 decimal digits of accuracy, we anticipate needing a step size of about h = 10-11 . The fact that the error is more of less proportional to the step size confirms that we are dealing with a first order numerical approximation.
To approximate higher order derivatives, we need to evaluate the function at more than two points. In general, an approximation to the nth order derivative u(n) (x) requires at least n+1 distinct sample points. For simplicity, we shall only use equally spaced points, leaving the general case to the exercises. For example, let us try to approximate u (x) by sampling u at the particular points x, x + h and x - h. Which combination of the function values u(x - h), u(x), u(x + h) should be used? The answer to such a question can be found by consideration of the relevant Taylor expansions u(x + h) = u(x) + u (x) h + u (x) h2 h3 + u (x) + O(h4 ), 2 6 h2 h3 u(x - h) = u(x) - u (x) h + u (x) - u (x) + O(h4 ), 2 6 u(x + h) + u(x - h) = 2 u(x) + u (x) h2 + O(h4 ). Rearranging terms, we conclude that u(x + h) - 2 u(x) + u(x - h) + O(h2 ), (11.6) h2 The result is known as the centered finite difference approximation to the second derivative of a function. Since the error is proportional to h2 , this is a second order approximation. u (x) = Example 11.2. Let u(x) = ex , with u (x) = (4 x2 + 2) ex . Let us approximate u (1) = 6 e = 16.30969097 . . . by using the finite difference quotient (11.6): e(1+h) - 2 e + e(1-h) . 6e h2 The results are listed in the following table.
h approximation error 1 50.16158638 33.85189541 .1 16.48289823 0.17320726 .01 16.31141265 0.00172168 .001 16.30970819 0.00001722 .0001 16.30969115 0.00000018
2 2 2 2
(11.5)
where the error terms are proportional to h4 . Adding the two formulae together gives
1 Each reduction in step size by a factor of 10 reduces the size of the error by a factor of 1 and results in a gain of two new decimal digits of accuracy, confirming that the finite 100 difference approximation is of second order.
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However, this prediction is not completely borne out in practice. If we take h = .00001 then the formula produces the approximation 16.3097002570, with an error of 0.0000092863 -- which is less accurate that the approximation with h = .0001. The problem is that round-off errors have now begun to affect the computation, and underscores the difficulty with numerical differentiation. Finite difference formulae involve dividing very small quantities, which can induce large numerical errors due to round-off. As a result, while they typically produce reasonably good approximations to the derivatives for moderately small step sizes, to achieve high accuracy, one must switch to a higher precision. In fact, a similar comment applied to the previous Example 11.1, and our expectations about the error were not, in fact, fully justified as you may have discovered if you tried an extremely small step size. Another way to improve the order of accuracy of finite difference approximations is to employ more sample points. For instance, if the first order approximation (11.4) to the first derivative based on the two points x and x + h is not sufficiently accurate, one can try combining the function values at three points x, x + h and x - h. To find the appropriate combination of u(x -h), u(x), u(x +h), we return to the Taylor expansions (11.5). To solve for u (x), we subtract the two formulae, and so u(x + h) - u(x - h) = 2 u (x) h + u (x) h3 + O(h4 ). 3
Rearranging the terms, we are led to the well-known centered difference formula u (x) = u(x + h) - u(x - h) + O(h2 ), 2h (11.7)
which is a second order approximation to the first derivative. Geometrically, the centered difference quotient represents the slope of the secant line through the two points x - h, u(x - h) and x + h, u(x + h) on the graph of u centered symmetrically about the point x. Figure 11.1 illustrates the two approximations; the advantages in accuracy in the centered difference version are graphically evident. Higher order approximations can be found by evaluating the function at yet more sample points, including, say, x + 2 h, x - 2 h, etc. Example 11.3. Return to the function u(x) = sin x considered in Example 11.1. The centered difference approximation to its derivative u (1) = cos 1 = 0.5403023 . . . is cos 1 The results are tabulated as follows: sin(1 + h) - sin(1 - h) . 2h
This next computation depends upon the computer's precision; here we used single precision in Matlab.
The terms O(h4 ) do not cancel, since they represent potentially different multiples of h4 .
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h approximation error
.1 0.53940225217 -0.00090005370
.01 0.54029330087 -0.00000900499
.001 0.54030221582 -0.00000009005
.0001 0.54030230497 -0.00000000090
As advertised, the results are much more accurate than the one-sided finite difference approximation used in Example 11.1 at the same step size. Since it is a second order 1 approximation, each reduction in the step size by a factor of 10 results in two more decimal places of accuracy. Many additional finite difference approximations can be constructed by similar manipulations of Taylor expansions, but these few very basic ones will suffice for our subsequent purposes. In the following subsection, we apply the finite difference formulae to develop numerical solution schemes for the heat and wave equations.
11.2. Numerical Algorithms for the Heat Equation.
Consider the heat equation 2u u = , t x2 0<x< , t 0, (11.8)
representing a homogeneous diffusion process of, sqy, heat in bar of length and constant thermal diffusivity > 0. The solution u(t, x) represents the temperature in the bar at time t 0 and position 0 x . To be concrete, we will impose time-dependent Dirichlet boundary conditions u(t, 0) = (t), u(t, ) = (t), t 0, (11.9)
specifying the temperature at the ends of the bar, along with the initial conditions u(0, x) = f (x), 0x , (11.10)
specifying the bar's initial temperature distribution. In order to effect a numerical approximation to the solution to this initial-boundary value problem, we begin by introducing a rectangular mesh consisting of points (ti , xj ) with 0 = x0 < x1 < < xn = and 0 = t 0 < t1 < t2 < .
For simplicity, we maintain a uniform mesh spacing in both directions, with h = xj+1 - xj = n , k = ti+1 - ti ,
representing, respectively, the spatial mesh size and the time step size. It will be essential that we do not a priori require the two to be the same. We shall use the notation ui,j u(ti , xj ) where ti = i k, xj = j h, (11.11)
to denote the numerical approximation to the solution value at the indicated mesh point. 4/20/07 190
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As a first attempt at designing a numerical method, we shall use the simplest finite difference approximations to the derivatives. The second order space derivative is approximated by (11.6), and hence u(ti , xj+1 ) - 2 u(ti , xj ) + u(ti , xj-1 ) 2u (ti , xj ) + O(h2 ) x2 h2 ui,j+1 - 2 ui,j + ui,j-1 + O(h2 ), h2
(11.12)
where the error in the approximation is proportional to h2 . Similarly, the one-sided finite difference approximation (11.4) is used for the time derivative, and so ui+1,j - ui,j u(ti+1 , xj ) - u(ti , xj ) u (11.13) (ti , xj ) + O(k) + O(k), t k k where the error is proportion to k. In practice, one should try to ensure that the approximations have similar orders of accuracy, which leads us to choose k h2 . Assuming h < 1, this requirement has the important consequence that the time steps must be much smaller than the space mesh size. Remark : At this stage, the reader might be tempted to replace (11.13) by the second order central difference approximation (11.7). However, this produces significant complications, and the resulting numerical scheme is not practical. Replacing the derivatives in the heat equation (11.14) by their finite difference approximations (11.12), (11.13), and rearranging terms, we end up with the linear system ui+1,j = ui,j+1 + (1 - 2 )ui,j + ui,j-1 , in which = i = 0, 1, 2, . . . , j = 1, . . . , n - 1, (11.14)
k . (11.15) h2 The resulting numerical scheme takes the form of an iterative linear system for the solution values ui,j u(ti , xj ), j = 1, . . . , n - 1, at each time step ti . The initial condition (11.10) means that we should initialize our numerical data by sampling the initial temperature at the mesh points: u0,j = fj = f (xj ), ui,0 = i = (ti ), j = 1, . . . , n - 1. (11.16)
Similarly, the boundary conditions (11.9) require that ui,n = i = (ti ), i = 0, 1, 2, . . . . (11.17)
For consistency, we should assume that the initial and boundary conditions agree at the corners of the domain: f0 = f (0) = u(0, 0) = (0) = 0 , 4/20/07 191 fn = f ( ) = u(0, ) = (0) = 0 .
c 2006 Peter J. Olver2.
A Solution to the Heat Equation.
The three equations (11.1417) completely prescribe the numerical approximation algorithm for solving the initial-boundary value problem (11.810). Let us rewrite the scheme in a more transparent matrix form. First, let u(i) = ui,1 , ui,2 , . . . , ui,n-1
T
u(ti , x1 ), u(ti , x2 ), . . . , u(ti , xn-1 )
T
(11.18)
be the vector whose entries are the numerical approximations to the solution values at time ti at the interior nodes. We omit the boundary nodes x0 = 0, xn = , since those values are fixed by the boundary conditions (11.9). Then (11.14) assumes the compact vectorial form (11.19) u(i+1) = A u(i) + b(i) , where 1 - 2 A= 1 - 2 1 - 2 .. .. . . .. .. . 1 - 2 , i 0 0 . . = . . 0 i
b(i)
(11.20)
.
The coefficient matrix A is symmetric and tridiagonal. The contributions (11.17) of the boundary nodes appear in the vector b(i) . This numerical method is known as an explicit scheme since each iterate is computed directly without relying on solving an auxiliary equation -- unlike the implicit schemes to be discussed below. Example 11.4. Let us fix the diffusivity = 1 and the bar length the initial temperature profile 0 x 1, - x, 5 1 7 x - 2, x 10 , u(0, x) = f (x) = 5 5 7 1 - x, 10 x 1, 4/20/07 192
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= 1. Consider
(11.21)
Peter J. Olver
1
1
1
0.5
0.5
0.5
0.2 -0.5
0.4
0.6
0.8
1 -0.5
0.2
0.4
0.6
0.8
1 -0.5
0.2
0.4
0.6
0.8
1
-1
-1
-3.
Numerical Solutions for the Heat Equation Based on the Explicit Scheme.
on a bar of length 1, plotted in the first graph in Figure 11.2. The solution is plotted at the successive times t = ., .02, .04, . . . , .1. Observe that the corners in the initial data are immediately smoothed out. As time progresses, the solution decays, at an exponential rate of 2 9.87, to a uniform, zero temperature, which is the equilibrium temperature distribution for the homogeneous Dirichlet boundary conditions. As the solution decays to thermal equilibrium, it also assumes the progressively more symmetric shape of a single sine arc, of exponentially decreasing amplitude. In our numerical solution, we take the spatial step size h = .1. In Figure 11.3 we compare two (slightly) different time step sizes on the same initial data as used in (11.21). The first sequence uses the time step k = h2 = .01 and plots the solution at times t = 0., .02, .04. The solution is already starting to show signs of instability, and indeed soon thereafter becomes completely wild. The second sequence takes k = .005 and plots the solution at times t = 0., .025, .05. (Note that the two sequences of plots have different vertical scales.) Even though we are employing a rather coarse mesh, the numerical solution is not too far away from the true solution to the initial value problem, which can be found in Figure 11.2. In light of this calculation, we need to understand why our scheme sometimes gives reasonable answers but at other times utterly fails. To this end, let us specialize to homogeneous boundary conditions u(t, 0) = 0 = u(t, ), whereby i = i = 0 for all i = 0, 1, 2, 3, . . . , (11.22) (11.23)
and so (11.19) reduces to a homogeneous, linear iterative system u(i+1) = A u(i) .
According to Proposition 7.8, all solutions will converge to zero, u(i) 0 -- as they are supposed to (why?) -- if and only if A is a convergent matrix. But convergence depends upon the step sizes. 11.4 Example is indicating that for mesh size h = .1, the time step 4/20/07 193
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k = .01 yields a non-convergent matrix, while k = .005 leads to a convergent matrix and a valid numerical scheme. As we learned in Theorem 7.11, the convergence property of a matrix is fixed by its spectral radius, i.e., its largest eigenvalue in magnitude. There is, in fact, an explicit formula for the eigenvalues of the particular tridiagonal matrix in our numerical scheme, which follows from the following general result. Lemma 11.5. The eigenvalues of an (n - 1) (n - 1) tridiagonal matrix all of whose diagonal entries are equal to a and all of whose sub- and super-diagonal entries are equal to b are k (11.24) , k = 1, . . . , n - 1. k = a + 2 b cos n Proof : The corresponding eigenvectors are vk = k 2k sin , sin , n n ... nk sin n
T
.
Indeed, the j th entry of the eigenvalue equation A vk = k vk reads a sin jk + b n sin (j - 1) k (j + 1) k + sin n n = a + 2 b cos k n sin jk , n
which follows from the trigonometric identity sin + sin = 2 cos + - sin . 2 2 Q.E .D.
In our particular case, a = 1 - 2 and b = , and hence the eigenvalues of the matrix A given by (11.20) are k , k = 1, . . . , n - 1. n Since the cosine term ranges between -1 and +1, the eigenvalues satisfy k = 1 - 2 + 2 cos 1 - 4 < k < 1. Thus, assuming that 0 < 1 guarantees that all | k | < 1, and hence A is a convergent 2 matrix. In this way, we have deduced the basic stability criterion = 1 k , 2 h 2 or k h2 . 2 (11.25)
With some additional analytical work, [28], it can be shown that this is sufficient to conclude that the numerical scheme (11.1417) converges to the true solution to the initialboundary value problem for the heat equation. Since not all choices of space and time steps lead to a convergent scheme, the numerical method is called conditionally stable. The convergence criterion (11.25) places a severe restriction on the time step size. For instance, if we have h = .01, and = 1, then we can only use a time step size k .00005, which is minuscule. It would take an inordinately 4/20/07 194
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large number of time steps to compute the value of the solution at even a moderate times, e.g., t = 1. Moreover, owing to the limited accuracy of computers, the propagation of round-off errors might then cause a significant reduction in the overall accuracy of the final solution values. An unconditionally stable method -- one that does not restrict the time step -- can be constructed by using the backwards difference formula u(ti , xj ) - u(ti-1 , xj ) u + O(hk ) (ti , xj ) t k (11.26)
to approximate the temporal derivative. Substituting (11.26) and the same approximation (11.12) for uxx into the heat equation, and then replacing i by i + 1, leads to the iterative system i = 0, 1, 2, . . . , ui+1,j - ui+1,j+1 - 2 ui+1,j + ui+1,j-1 = ui,j , (11.27) j = 1, . . . , n - 1, where the parameter = k/h2 is as above. The initial and boundary conditions also have the same form (11.16), (11.17). The system can be written in the matrix form A u(i+1) = u(i) + b(i+1) , (11.28)
where A is obtained from the matrix A in (11.20) by replacing by - . This defines an implicit method since we have to solve a tridiagonal linear system at each step in order to compute the next iterate u(i+1) . However, as we learned in Section 4.5, tridiagonal systems can be solved very rapidly, and so speed does not become a significant issue in the practical implementation of this implicit scheme. Let us look at the convergence properties of the implicit scheme. For homogeneous Dirichlet boundary conditions (11.22), the system takes the form u(i+1) = A-1 u(i) , and the convergence is now governed by the eigenvalues of A-1 . Lemma 11.5 tells us that the eigenvalues of A are k = 1 + 2 - 2 cos k , n k = 1, . . . , n - 1.
As a result, its inverse A-1 has eigenvalues 1 = k 1 1 + 2 k 1 - cos n , k = 1, . . . , n - 1.
Since > 0, the latter are always less than 1 in absolute value, and so A is always a convergent matrix. The implicit scheme (11.28) is convergent for any choice of step sizes h, k, and hence unconditionally stable. 4/20/07 195
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Numerical Solutions for the Heat Equation Based on the Implicit Scheme.
Example 11.6. Consider the same initial-boundary value problem considered in Example 11.4. In Figure 11.4, we plot the numerical solutions obtained using the implicit scheme. The initial data is not displayed, but we graph the numerical solutions at times t = .2, .4, .6 with a mesh size of h = .1. On the top line, we use a time step of k = .01, while on the bottom k = .005. Unlike the explicit scheme, there is very little difference between the two -- both come much closer to the actual solution than the explicit scheme. Indeed, even significantly larger time steps give reasonable numerical approximations to the solution. Another popular numerical scheme is the CrankNicolson method ui+1,j - ui,j = ui+1,j+1 - 2 ui+1,j + ui+1,j-1 + ui,j+1 - 2 ui,j + ui,j-1 . 2 (11.29)
which can be obtained by averaging the explicit and implicit schemes (11.14, 27). We can write the iterative system in matrix form B u(i+1) = C u(i) + where -1 2 1+ -1 2 .. 1 . -2 .. . . . , . .. .
1 2
b(i) + b(i+1) ,
1 2 1 2
1+ -1 2 B=
1- 1 2 C=
1-
1 2
. .
.. ..
. . . . .. .
(11.30)
Convergence is governed by the generalized eigenvalues of the tridiagonal matrix pair B, C, or, equivalently, the eigenvalues of the product B -1 C, which are 1- k = 1+ 1 - cos k n k 1 - cos n 196
,
k = 1, . . . , n - 1.
(11.31)
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Peter5.
Numerical Solutions for the Heat Equation Based on the CrankNicolson Scheme.
Since > 0, all of the eigenvalues are strictly less than 1 in absolute value, and so the CrankNicolson scheme is also unconditionally stable. A detailed analysis will show that the errors are of the order of k 2 and h2 , and so it is reasonable to choose the time step to have the same order of magnitude as the space step, k h. This gives the CrankNicolson scheme one advantage over the previous two methods. However, applying it to the initial value problem considered earlier points out a significant weakness. Figure 11.5 shows the result of running the scheme on the initial data (11.21). The top row has space and time step sizes h = k = .1, and does a rather poor job replicating the solution. The second row uses h = k = .01, and performs better except near the corners where an annoying and incorrect local time oscillation persists as the solution decays. Indeed, since most of its eigenvalues are near -1, the CrankNicolson scheme does not do a good job of damping out the high frequency modes that arise from small scale features, including discontinuities and corners in the initial data. On the other hand, most of the eigenvalues of the fully implicit scheme are near zero, and it tends to handle the high frequency modes better, losing out to CrankNicolson when the data is smooth. Thus, a good strategy is to first evolve using the implicit scheme until the small scale noise is dissipated away, and then switch to CrankNicolson to use a much larger time step for final the large scale changes.
11.3. Numerical Solution Methods for the Wave Equation.
Let us now look at some numerical solution techniques for the wave equation. Although this is in a sense unnecessary, owing to the explicit d'Alembert solution formula, the experience we gain in designing workable schemes will serve us well in more complicated situations, including inhomogeneous media, and higher dimensional problems, when analytic solution formulas are no longer available. Consider the wave equation 2u 2u = c2 , t2 x2 0<x< , t 0, (11.32)
modeling vibrations of a homogeneous bar of length with constant wave speed c > 0. We 4/20/07 197
c 2006 Peter J. Olver
impose Dirichlet boundary conditions u(t, 0) = (t), and initial conditions u (0, x) = g(x), t We adopt the same uniformly spaced mesh u(0, x) = f (x), ti = i k, xj = j h, where 0x . (11.34) u(t, ) = (t), t 0. (11.33)
. n In order to discretize the wave equation, we replace the second order derivatives by their standard finite difference approximations (11.6), namely u(ti+1 , xj ) - 2 u(ti , xj ) + u(ti-1 , xj ) 2u (ti , xj ) + O(h2 ), 2 2 t k (11.35) u(ti , xj+1 ) - 2 u(ti , xj ) + u(ti , xj-1 ) 2u 2 (t , x ) + O(k ), x2 i j h2 Since the errors are of orders of k 2 and h2 , we anticipate to be able to choose the space and time step sizes of comparable magnitude: k h. Substituting the finite difference formulae (11.35) into the partial differential equation (11.32), and rearranging terms, we are led to the iterative system ui+1,j = 2 ui,j+1 + 2 (1 - 2 ) ui,j + 2 ui,j-1 - ui-1,j , i = 1, 2, . . . , j = 1, . . . , n - 1, (11.36)
h=
for the numerical approximations ui,j u(ti , xj ) to the solution values at the mesh points. The positive parameter ck > 0 (11.37) = h depends upon the wave speed and the ratio of space and time step sizes. The boundary conditions (11.33) require that ui,0 = i = (ti ), ui,n = i = (ti ), u(i+1) = B u(i) - u(i-1) + b(i) , where 2 2 (1 - 2 ) 2 2 (1 - 2 ) 2 .. .. B= . . 2 .. .. . .
2
i = 0, 1, 2, . . . .
(11.38)
This allows us to rewrite the system in matrix form (11.39)
, u(j)
2 2 (1 - 2 ) 198
2 u1,j j u2,j 0 . . = . , b(j) = . . . . 0 u n-2,j 2 j un-1,j (11.40)
c 2006 Peter J. Olver6.
Numerically Stable Waves.
The entries of u(i) are, as in (11.18), the numerical approximations to the solution values at the interior nodes. Note that the system (11.39) is a second order iterative scheme, since computing the next iterate u(i+1) requires the value of the preceding two, u(i) and u(i-1) . The one difficulty is getting the method started. We know u(0) since u0,j = fj = f (xj ) is determined by the initial position. However, we also need to find u(1) with entries u1,j u(k, xj ) at time t1 = k in order launch the iteration, but the initial velocity ut (0, x) = g(x) prescribes the derivatives ut (0, xj ) = gj = g(xj ) at time t0 = 0 instead. One way to resolve this difficult would be to utilize the finite difference approximation gj = u1,j - gj u(k, xj ) - u(0, xj ) u (0, xj ) t k k u1,j = fj + k gj . However, the approximation (11.41) is only accurate to order k, whereas the rest of the scheme has error proportional to k 2 . Therefore, we would introduce an unacceptably large error at the initial step. To construct an initial approximation to u(1) with error on the order of k 2 , we need to analyze the local error in more detail. Note that, by Taylor's theorem, u(k, xj ) - u(0, xj ) k 2u u c2 k 2 u u (0, xj ) = (0, xj ) , (0, xj ) + (0, xj ) + k t 2 t2 t 2 x2 where the error is now of order k 2 , and, in the final equality, we have used the fact that u is a solution to the wave equation. Therefore, we find u(k, xj ) u(0, xj ) + k u c2 k 2 2 u (0, xj ) + (0, xj ) t 2 x2 c2 k 2 c2 k 2 f (xj ) fj + k gj + = f (xj ) + k g(xj ) + (f - 2 fj + fj-1 ) , 2 2 h2 j+1 199
c 2006 Peter J. Olver
(11.41)
to compute the required values7.
Numerically Unstable Waves.
where we can use the finite difference approximation (11.6) for the second derivative of f (x) if no explicit formula is known. Therefore, when we initiate the scheme by setting u1,j = or, in matrix form, u(0) = f , u(1) =
1 2 1 2
2 fj+1 + (1 - 2 )fj +
1 2
2 fj-1 + k gj ,
(11.42)
B u(0) + k g + 1 b(0) , 2
(11.43)
we will have maintained the desired order k 2 (and h2 ) accuracy. Example 11.7. Consider the particular initial value problem utt = uxx , u(0, x) = e- 400 (x-.3) ,
2
ut (0, x) = 0,
0 x 1, t 0,
u(t, 0) = u(1, 0) = 0,
subject to homogeneous Dirichlet boundary conditions on the interval [ 0, 1 ]. The initial data is a fairly concentrated single hump centered at x = .3, and we expect it to split into two half sized humps, which then collide with the ends. Let us choose a space discretization 1 consisting of 90 equally spaced points, and so h = 90 = .0111 . . . . If we choose a time step of k = .01, whereby = .9, then we get reasonably accurate solution over a fairly long time range, as plotted in Figure 11.6 at times t = 0, .1, .2, . . . , .5. On the other hand, if we double the time step, setting k = .02, so = 1.8, then, as plotted in Figure 11.7 at times t = 0, .05, .1, .14, .16, .18, we observe an instability eventually creeping into the picture that eventually overwhelms the numerical solution. Thus, the numerical scheme appears to only be conditionally stable. The stability analysis of this numerical scheme proceeds as follows. We first need to recast the second order iterative system (11.39) into a first order system. In analogy with u(i) R 2n-2 . Example 7.4, this is accomplished by introducing the vector z(i) = u(i-1) Then B -I z(i+1) = C z(i) + c(i) , where C= . (11.44) I O 4/20/07 200
c 2006 Peter J. Olver
Therefore, the stability of the method will be determined by the eigenvalues of the coeffiu cient matrix C. The eigenvector equation C z = z, where z = , can be written out v in its individual components: B u - v = u, ( B - 2 - 1) v = 0, u = v.
Substituting the second equation into the first, we find 1 v. The latter equation implies that v is an eigenvector of B with + -1 the corresponding eigenvalue. The eigenvalues of the tridiagonal matrix B are governed by Lemma 11.5, in which a = 2(1 - 2 ) and b = 2 , and hence are or Bv = + + 1 =2 1 - 2 + 2 cos k n , k = 1, . . . , n - 1.
Multiplying both sides by leads to a quadratic equation for the eigenvalues, 2 - 2 ak + 1 = 0, where 1 - 2 2 < ak = 1 - 2 + 2 cos k < 1. n (11.45)
Each pair of solutions to these n - 1 quadratic equations, namely = ak k a2 - 1 , k (11.46)
yields two eigenvalues of the matrix C. If ak > 1, then one of the two eigenvalues will be larger than one in magnitude, which means that the linear iterative system has an exponentially growing mode, and so u(i) as i for almost all choices of initial data. This is clearly incompatible with the wave equation solution that we are trying to approximate, which is periodic and hence remains bounded. On the other hand, if | ak | < 1, then the eigenvalues (11.46) are complex numbers of modulus 1, indicated stability (but not convergence) of the matrix C. Therefore, in view of (11.45), we should require that h ck < 1, or k< , (11.47) = h c which places a restriction on the relative sizes of the time and space steps. We conclude that the numerical scheme is conditionally stable. The stability criterion (11.47) is known as the Courant condition, and can be assigned a simple geometric interpretation. Recall that the wave speed c is the slope of the characteristic lines for the wave equation. The Courant condition requires that the mesh slope, which is defined to be the ratio of the space step size to the time step size, namely h/k, must be strictly greater than the characteristic slope c. A signal starting at a mesh point (ti , xj ) will reach positions xj k/c at the next time ti+1 = ti + k, which are still between the mesh points xj-1 and xj+1 . Thus, characteristic lines that start at a mesh point are not allowed to reach beyond the neighboring mesh points at the next time step. For instance, in Figure 11.8, the wave speed is c = 1.25. The first figure has equal mesh spacing k = h, and does not satisfy the Courant condition (11.47), whereas the 4/20/07 201
c 2006 Peter J. Olver
Figure 11.8.
The Courant Condition.
second figure has k = 1 h, which does. Note how the characteristic lines starting at a 2 given mesh point have progressed beyond the neighboring mesh points after one time step in the first case, but not in the second.
4/20/07
202
c 2006 "complex analysis" integer |
Thinking Mathematicallyterm course in Liberal Arts Mathematics, Survey of Mathematics, Finite Mathematics, and Mathematics for education majors. Known for their well-conceived, relevant applications and meticulously annotated examples, Bob Blitzer's books balance the needs of appropriate coverage and student motivation while helping students develop strong critical thinking skills. This text gives students the skill-building and practice essential at this level, along with the applications and technology needed to foster an appreciation of mathemati... MOREcs through their college careers and beyond. Blitzer's goal is to show students how mathematics can be applied to their lives in interesting, enjoyable, and meaningful ways. His accessible writing style helps students relate to and grasp the material. This general survey of mathematical topics helps a diverse audience, with different backgrounds and career plans, to understand mathematics. Blitzer provides the applications and technology readers need to gain an appreciation of mathematics in everyday life. Demonstrates how mathematics can be applied to readers'lives in interesting, enjoyable, and meaningful ways. Features abundant, step-by-step, annotated Examplesthat provide a problem-solving approach to reach the solution; annotations are conversational in tone, explaining key steps and ideas as the problem is solved. Begins each section with a compelling vignette highlighting an everyday scenario, posing a question about it, and exploring how the chapter section subject can be applied to answer the question. A highly readable reference for anyone who needs to brush up their mathematics skills. |
0387946144
9780387946146
Mathematical Analysis: Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level. «Show less
Mathematical Analysis: Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical... Show more»
Rent Mathematical Analysis today, or search our site for other Axler |
GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus.
On the one hand, GeoGebra is a dynamic geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards.
On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.
The GeoGebraWiki is a free pool of educational materials for GeoGebra. Everyone can contribute and upload materials there:
International GeoGebraWiki - pool of educational materials for GeoGebra and the German GeoGebraWiki
The Dynamic Worksheets
GeoGebra can also be used to create dynamic worksheets:
Pythagoras
visualisation of Pythagoras' theorem
Ladder against the Wall
application of Pythagoras' theorem
Circle and its Equation
connection between a circle's center, radius and equation
Slope and Derivative of a Function (3 sheets)
relation between slope, derivative and local extrema of a function
Derivative of a Polynomial
interactive exercise to practice finding the derivative of a cubic polynomial
Upper- and Lower Sums of a Function
visualisation of the backgrounds of Riemann's Integral |
Comprehensive Instruction
To prepare students for algebra, the mathematics curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills. The development of these concepts and skills is intertwined, each supporting the other and reinforcing learning.
Teachers can help by providing students with sufficient practice distributed over time and including a conceptually rich and varied mix of problems to support their learning. In addition, teachers should encourage and support students in their efforts to master difficult mathematics content. Students who believe that effort, not just inherent talent, counts in learning mathematics can improve their performance.
Multimedia Overview
Developing Conceptual Understanding, Fluency, and Problem Solving
Use this multimedia overview to learn about the value of simultaneously teaching concepts, procedures, and problem solving; the importance of practice distributed over time in developing automaticity and improving fluency, including the use of technology-based tools; and the relationship between student beliefs about learning and mathematics performance.
(8:37 min) |
Math 7
The regular sequence course for Grade 7 students, this course focuses on mastery of math skills needed to apply concepts encountered in higher-level courses. Students review basic arithmetic skills, then apply these skills to operations with whole numbers, integers, decimals, fractions, and percents. Further topics include ratio, proportion and geometry. Students experience more abstract thinking through the introduction of variables and learn to solve simple equations. Additional math time is allotted for a math lab, which provides opportunities for students to reinforce math concepts through a hands-on approach. Students who successfully complete this course move on to Pre-Algebra in Grade 8. |
is the 6th edtion of barron's math2c good enough for math level 2? Everyone seems to say that barron's is the best for math 2 but do they mean the new versions or is the 6th edition(old version) good enough? Is it hard like the other ones? |
Buy now
Detailed description Math Essentials, Middle School Level gives middle school math teachers the tools they need to help prepare all types of students (including gifted and learning disabled) for mathematics testing and the National Council of Teachers of Mathematics (NCTM) standards. Math Essentials highlights Dr. Thompson's proven approach by incorporating manipulatives, diagrams, and independent practice. This dynamic book covers thirty key objectives arranged in four sections. Each objective includes three activities (two developmental lessons and one independent practice) and a list of commonly made errors related to the objective. The book's activities are designed to be flexible and can be used as a connected set or taught separately, depending on the learning needs of your students. Most activities and problems also include a worksheet and an answer key and each of the four sections contains a practice test with an answer key.
From the contents The Author.
Notes to the Teacher.
Section 1: Number, Operation, and Quantitative Reasoning.
Objectives.
1. Compare and order fractions, decimals (including tenths and hundredths), and percents, and find their approximate locations on a number line.
5. Generate the formulas for the circumference and the area of a circle; apply the formulas to solve word problems.
6. Generate and apply the area formula for a parallelogram (including rectangles); extend to the area of a triangle.
7. Generate and apply the area formula for a trapezoid.
8. Apply nets and concrete models to find total or partial surface areas of prisms and cylinders.
9. Find the volume of a right rectangular prism, or find a missing dimension of the prism; find the new volume when the dimensions of a prism are changed proportionally.
Practice Test.
Section 4: Graphing, Statistics, and Probability.
Objectives.
1. Locate and name points using ordered pairs of rational numbers or integers on a Cartesian coordinate plane.
2. Construct and interpret circle graphs.
3. Compare different numerical or graphical models for the same data, including histograms, circle graphs, stem-and-leaf plots, box plots, and scatter plots; compare two sets of data by comparing their graphs of similar type.
4. Find the mean of a given set of data, using different representations such as tables or bar graphs.
5. Find the probability of a simple event and its complement.
6. Find the probability of a compound event (dependent or independent). |
The Math Lab
The Math Lab offers free tutoring and Internet access to currently enrolled full-time and part-time HCC students. As a service organization, our purpose is to serve our students in ways that advance our mission.
Mission of the Math Lab:
1.
To provide the greatest possible assistance to every student who comes to us without regard to race, color, religion, national origin, sex, age or disability.
2.
To help students improve their academic performance.
3.
To motivate the process of learning Mathematics.
4.
To help students gain confidence and to acquire the belief that it is possible for them to succeed in their study of Mathematics.
Some Rules and How to use the lab:
Here are some tips for getting the most out of the lab:
*
Students utilizing the Math lab must have a valid HCC Student ID or a Copy of their current fee statement and a photo ID.
*
For students needing tutoring, please sit in the designated Tutoring Area. Students seated at the computers or in the back of the lab will not be approached by tutors to be given tutoring assistance.
*
Try to have specific homework problems or questions ready for the tutors to look at.
*
It is usually better to make frequent shorter visits than come in for one or two long sessions just before a test.
*
Find several tutors with whom you are comfortable. Usually you learn more this way because no two tutors are going to explain things exactly the same way.
*
Use a computer tutorial keyed to your textbook on a regular basis. The lab staff will help you get started with a tutorial program and they will continue to help you with software and math questions.
*
Ask other students to work with you. Often they are quite willing to help.
*
No smoking, eating or drinking is allowed in the lab.
*
The lab's phone is not for student use. Pay phones can be found on the first floor, for your convenience.
*
Children are NOT permitted in the lab.
*
Cell phones are to be put on vibrate or silent mode. If you have your cell phone on vibrate, please do not leave it sitting on the desk. The vibration can be loud and very distracting for your fellow students. Please answer and/or complete any calls in the hall or stairwell, so as to minimize distracting noise in the lab and hallways.
*
Students utilizing the computers for general Internet use are limited to 15 minutes. If the lab staff asks you to give up your seat for a student needing to work on class work, you are expected to comply as quickly as possible. |
KSU Math Emporium Is About More Than Just Numbers
The second floor of the Kent State University Library is being transformed into a 250-computer, state-of-the-art learning center dubbed the Math Emporium, pictured in this rendering.
Kent State University
Kent State University is making a big investment in student success, as well as increasing freshman retention rates, by completely revamping the way basic math courses are taught.
The second floor of the university library is in the midst of a $1 million renovation project that is turning former storage space into a 250-computer, state-of-the-art learning center dubbed the Math Emporium.
Andrew Tonge, chair of the Department of Mathematical Sciences at Kent State, said students whose individual assessments show a need for basic math instruction will be introduced this fall to ALEKS – an artificially intelligent computer learning system.
Tonge said the adaptive program allows students to choose their own pathways through various math topics at their own pace while constantly assessing their progress.
"Our students have not been succeeding as well as we'd like in elementary math courses," he said. "(With ALEKS), students will be learning all the materials they need to learn – not what the instructors think they need to learn."
The ALEKS software was chosen over other models, Tonge said, because it has proven to be the most effective mathematics software on the market that adapts to individual student needs.
"Evidence from other universities shows that when teaching in this type of format, students do much better … You get at least 25 percent more students moving forward rather than getting stuck in elementary (math) courses, then dropping out, which is what we don't want them to do," he explained.
The 11,000-square-foot math emporium will be the new campus "classroom" for students whose placement assessments show a need for Basic Algebra I-IV courses, formerly called core math.
"These are the courses that students should have learned sufficiently in high school to succeed in college courses," Tonge said. "We need to get them up to speed. There's a strong correlation between success in first math courses and retention at Kent State."
Instead of reporting to the math building for classes this fall, students enrolled in any of the four basic algebra courses will attend "class" in the Math Emporium, which will be staffed at all times with teaching faculty, graduate students and peer coaches.
Tonge said instructors will be able to continually monitor student progress through the ALEKS software, as it provides statistics on everything from what a student is currently working on to how much time they've devoted to various topics. Faculty will be proactive in providing individualized help to those who need it.
Students in the basic algebra classes benefit from interacting with both the software and the instructional team in the Math Emporium. "They're classes that provide one-on-one interaction," he said.
The emporium will be staffed from 7:30 a.m. to 9 p.m. Mondays through Thursdays; 7:30 a.m. to 6 p.m. Fridays; 10 a.m. to 6 p.m. Saturdays and noon to 8 p.m. Sundays.
Tonge said another advantage of Kent State using the ALEKS model is that the software is Web-based. "Students will be able to actually do work 24 hours a day, seven days a week, wherever they are," he said.
The Math Emporium renovation project should be finished by Aug. 15, then computers and furniture will be moved in and set up for the start of classes Aug. 29. A dedication event is planned for 4 p.m. Sept. 13.
Great! Otherwise, nowadays, students are very likely to avoid STEM, especially math. Students should take math from school level so that they don't have to face any problem in future. Schools also should take the matter seriously. Unfortunately most of the public schools do not do so. Students can try online math tutoring services like tutorteddy.com for free math help / solutions (video / text) if schools cannot offer.
"(With ALEKS) students will be learning all the materials they need to learn - not what the instructors think they need to learn" is not accurate - the appropriate message was somehow lost in translation.
The curriculum is unchanged: courses still contain the same material instructors have agreed students need to learn to give them a good chance of success in college level courses. The article suggests there is a disconnect here, but this is not the case. What will change is not what students need to learn (what we think they need to learn), but how and when they learn. The instructors and the software will work together to help the students come to grips with basic material they need for success in college and beyond. The instructor's role will be different from before, but the overall objectives for student learning will not change. |
Math for Elementary Teachers II
Welcome to Mth126: "Continued study of the mathematical
concepts and techniques that are fundamental to, and form the
basis for, elementary school mathematics. Topics include:
use of probability and statistics to explore real-world
problems; representation and analysis of discrete mathematical
problems using counting techniques, sequences, graph theory,
arrays and networks; use of functions, algebra and the basic
concepts underlying the calculus in real-world-applications."
News and Updates - Spring 2010
See the Homework and Handouts table
below to download or print the syllabus, calendar, and all
homework sets and class notes.
Note:
Bolded problems can be done
without "Chebyshev's theorem",
which we didn't get to in class
today. However, even through "Chebyshev's
theorem has an odd name (for
Westerners, at least), it is
actually pretty simple to use in
practice -- see if you can do
the homework on these problems!
1. Go to the
National Library
of Virtual
Manipulatives
(NLVM) and
access
the applet we
viewed today
in class. Have
the computer
generate five
problems, and
solve them using
the online
applet.
2. Still
using the
NLVM Algebra
Balance Scale
applet,
select "Create
Problem" and
enter the
following
problems (one at
a time). Solve
each problem
using efficient
and inefficient
sequences of
steps. Does it
matter which way
you proceed?
Homework:
For sequences (1) thru (4), find
the next term in the sequence.
Then classify the sequence as
(a) arithmetic, (q) quadratic,
or (o) other. Finally, if the
sequence is arithmetic, give an
explicit rule yn for the nth
term. Otherwise, write a
recursive rule for the sequence.
11,
13, 15, 17, ...
18, 15, 12,
9, ...
1, 1, 2, 4,
7, 11, 16, ...
4, 8, 14,
22, ...
For sequences (5) and (6),
find the pattern and state the
next three terms in the
sequence. |
School Search
Mathematics
Our Belief We believe that the success of a student depends on three factors:
1. The amount of interest the child puts into his education. 2. The amount of interest the parent puts into the child's education. 3. The learning environment created by the school.
Philosophy
Our expectation is that every student at August Martin High School will take four years of mathematics. All students are encouraged to take the most challenging mathematics courses at which they can function and succeed.
To meet the demands of the College Preparatory initiative and to help students transition to college and technical careers, our goal is to help students master the Sequential Mathematics curriculum. Those students who require Bridge to Algebra before entering Mathematics A, will be exposed to a variety of teaching strategies. All students will receive supplementary supportive assistance when needed.
Bilingual courses will be offered in Bridge to Algebra and Mathematics A.
Mathematical Placement
1. Entering freshman who score below the 26th percentile on the Queens Math Test, DRP or the CAT, will be programmed for MG1.
2. Those scoring within the 25th and 50th percentile on the Queens Math Test, DRP or the CAT, will be programmed in M$A M$A is an extended period program running 7or 8 periods per week.
3. Entering freshman scoring above the 50th percentile on the Queens Math Test, DRP or the CAT, will be programmed for M$1.
4. Students who pass Bridge to Algebra MG1 and MG2 will be programmed for M$A.
Visit Math Links at Library Website
Students are welcome to visit Math Assistant Principal regarding any inquiry about mathematics class. Students should come only during their lunch period unless there is an emergency. Students are discouraged from cutting their classes in order to see the Math Assistant Principal. You must get a pass from your class teacher.
Math Sequence
Mathematics A (M#A, M#B, M#C, M#D) – Prentice Hall A four term sequence Math A Regents Course. Each course is a double period class. M#A begins the sequence in the first term and so on. The Math A course includes logic,algebra, coordinate & analytical geometry, probability, systems of equation, trigonometry of right triangle, statistics, locus and construction.
Mathematics A (M&1, M&2, M&3 or M#1, M#2, M#3)
A three term sequence Math A Regents Course. The sequence M&1, M&2 and M&3 is an accelerated program where each course is a single period class. The sequence M#1, M#2 and M#3 is an accelerated program with extra support. Each course is a double period class. Mathematics A (M$A, M$B, M$C & M$D)
A four term sequence Math A Regents Course. Each course is a single period class.M$A begins the sequence in the first term and so on. M$D is given in theLast term and culminates in the Regents Math A Examination. Each class is the prerequisite for the next class. The Math A course includes logic, algebra, coordinate & analytical geometry, probability, systems of equation, trigonometry of right triangle, statistics, locus & construction. Mathematics B (*M$E, M$F, M$G & M$H- four terms) (**M$4, M$5, M$6- three terms)
Prerequisite: Passed the Math A Regents Exam *A four term sequence Math B Regents Course. M$E is given in the first term and so on. M$H is given in the last term and culminates in the Regents Math B Examination.
**A three term sequence Math B Regents Course. M$4 is given in the first term and so on. M$6 is given in the last term and culminates in the Regents Math B Examination. Each class is the prerequisite for the next class. The Math B course includes relations and functions, exponential functions, logarithmic functions, trigonometry, complex numbers, transformational geometry, probability, sequences, binomial theorem and statistics.
Intermediate Algebra (MG3 & MG4 or MGS)
An enriched course which contains algebra and geometry including matrices, determinants, sequences, trigonometry and complex fractions.
Students must have attained a passing score in the Course 1 Regents Exam or Math A Regents Exam along with the approval of the Assistant Principal-Mathematics.
Pre-Calculus (MEP1 & MEP2)
A course for those students planning to take Advanced Placement Calculus. The courseincludes topics such as sequences, series, methods of convergence for series, analytic geometry of parametric equations, polar graphs, Theory of Equations, matrices & limits. |
Could you suggest me a good TI calculator?
Could you suggest me a good TI calculator?
i have decided to buy a good TI calculator. The cost isn't really a factor.. since almost all of their calculators come under 349 USD.
My area of work is mostly calculus-based physics and analytical geometry [coordinate geometry]. In mathematics, i'd also be needing features for basic basic statistics and probability computation.
Advanced algebraic support and data visualization is a must in addition to vector algebra and vector calculus. From what I saw at their site, one of the Graphing calculators would be a good option with me, because I deal with a lot of function plots and data visualization. And since programming is one of my hobbies, I'd also need a programmable calculator which can be programmed with the TIGCC library.
And since i already have Mathematica with me and do most of my stuff with that.. the only reason i want a calculator is for portability [a laptop is gonna cost a lot more :P].. so the smaller the better.
Also.. is there any calculator which has some specific set of functions for chemistry based applications?
I would strongly suggest the TI-89. It can simplify algebraic expressions, do basic (and a few not-so-basic) integrals, and some other useful stuff. It won't substitute for your physics knowledge. But my good ol' 89 saved me on my PhD qualifier (no seriously, I would have failed if not for its diff eq solver), so I'd recommend it to practically anyone in the physical sciences.
In regards to vector algebra, it can do dot and cross products. It can allegedly also do coordinate transformations, though I've never dared to actually try this. It can also do vector calculus...sort of. You'll need to turn double, triple, line, and surface integrals into iterated integrals by hand. But once you have it in this form, you can plug it into the 89 and get an expression. Not sure what you mean by data visualization, but it can graph continuous functions and sequences. As for programming, I don't know what the TIGCC library is, but the 89 has some sort of programming language. I've seen people write some pretty impressive programs for the 89. It can even play Super Mario. Not that you want to use your calculator for games, but if it can do this, then I'm sure it can do some pretty useful mathematical applications.
thanks a lot for the reply.. even i was thinking of the TI-89 since it's quite a popular model. But.. what do you think about the Voyage 200 or the TI-92 series? The only reason i didn't want to go for it was that it might be bulky and as i said.. portability was a major concern in my case. It's photographs hardly give me an idea of what it's real life size is..
I do know that the Voyage 200 qualifies as a computer and is not allowed in SAT exams and stuff.. but the exams that i have to give in india don't allow any calculator at all.. so it doesn't really matter to me. |
Assessment Rules
CMod description
Several important topics are considered in this module,
including series and their summation, the binomial theories, infinite series,
Taylor expansion. The use of matrices to describe physical phenomena and solve
mathematical problems is described. Basic ideas in probability are introduced,
permutations and combinations, distributions expectation and variance, and the
relevance to data analysis.
Curriculum Design: Outline Syllabus
Series and their formal representation, summation sign. Convergence of infinite sums. Geometric series and summation of infinite geometric series. Binomial expansion and binomial coefficients Taylor expansion and Taylor polynomials. Series representations of trigonometric and exponential functions.
Differential equations and their role in physics. Separable first order differential equations. Second order differential equations for conservative systems and the method of integrating multiplier. Example of a harmonic oscillator problem.
Linear ordinary differential equations. General and particular solutions of ODE. Properties of the function exp(Dx). The method of auxiliary equation for solving homogeneous ODE's. ODE's describing driven systems, harmonic force and the phenomenon of resonance. |
Posts Tagged 'time'
High school math is the next step in a child's life where they are taken out of the basic principles of mathematics and ushered into a world of greater depth, learning skills that they will carry with them into adulthood and hopefully exercise their critical thinking skills in a way that will help them in their career field of choice. Starting with the first year of high school, 9th grade math builds upon the basics and also introduces new components of practical math application, such as consumer math and finances.
9th Grade Math
An example of a typical 9th grade math course will include the following :
Consumer Mathematics
Applied Business Math
Pre-Algebra
Algebra 1
Geometry
Some students may excel beyond the required courses in their first year and go on to study Algebra 2, Advanced Algebra, and perhaps even venture into calculus. Every student is different and now is the time for them to truly understand and put into practice all of the equations they were taught up to transitioning from elementary education, to high school.
An important element of approaching 9th grade math, is to consider the student and what interests they have, and the goals they have for themselves in the future. If they have plans to attend a college, and what degree they wish to achieve can affect the course material they pursue while working toward their high school diploma. When looking at 9th grade math the entire four years of high school should also come into focus and contribute to what and how the student studies.
Along with learning new math expressions students will begin to learn how to use a variety of problem solving strategies. They will learn how to determine what information is needed while problem solving, and how to use multiple representation to explain situations ( numerically, algebraically, graphically, etc.) They will begin to understand how to reason, prove their reasoning mathematically, and then communicate their reasoning to others.
It is important for teachers to take the time to connect with their students on this subject. Young people who are struggling in grasping basic concepts, or did not fully learn foundational principles in elementary and middle school will have a much more difficult time transitioning into higher math. Algebra in particular can become a source of frustration without basic understanding. Parents should also be aware of where their child is at, and what they should be working on. Having teachers and parents who are willing to be involved can help a student feel more comfortable with asking questions and less inhibited in the math world.
9th grade truly is a transition from child to young adult. Diving into the world of higher math is one step closer to equipping the next generation with the tools they need to succeed. With proper support from teachers, parents, and a school that facilitates an atmosphere of learning through asking questions and hands on application, 9th grade math is one more brick in the foundation of a child's education that they can build upon for the rest of their lives. |
Description of Saxon Algebra 1/2: Teacher CD by Saxon
Based on Saxon's proven methods of incremental development and continual review strategies, the Algebra 1/2 program combines pre-algebra mathematics with a full pre-algebra course and an introduction to geometry and discrete mathematics.
This helped my first two children to be ready for calculus in college--they were able to test out of college algebra. My youngest child needed to go at a slower rate and had to take college Math in college. So it is great for those who have a bent for math and science. |
PhD training
If you are applying to take a PhD degree in pure mathematics you can
give a relatively wide area, e.g. "Algebra", "Geometry", or "Topology"
as your proposed Field of Study on the on-line application form. If you do have more specific interests you are encouraged to
make these known under "Further Information" or on a separate sheet.
You are also encouraged to contact directly a prospective supervisor.
We offer postgraduate projects in the following broad fields of
mathematics: |
Absolutely wonderful book Get this for your 6th grade student. It reviews math they have had and teaches how to figure out word problems. It has great explanations and examples. It has practice problems and then quizzes at the end of a unit. Answers are either right after the problems or at the back of the book. Parents can use this to help kids, and math teachers can certainly use it for probably 5th through 8th or ...
Worked really well for me I bought this book (and a bunch of others) because I wanted to prepare myself for a college math assessment test and score high enough that I could jump straight into statistics (I know, pretty ambitious!).
My current situation?
1) I haven't studied algebra (or any math at all for that matter!) in about 20 years
2) I didn't learn math in English
So, after opening the first algebra ...
very good start to teach yourself the bases of the probability theory.. I find this book one of the simplest books that help to explain the elementary of the probablity theory. The auther used very easy examples along with solved problems to explain the basics of probability. Applications of probability theory like Simulation, Game theory, and actuarial science have been addressed in a very simple way. If you are searching for the simplest book to understand ...Math I had a probability and statistics class and this book was somewhat helpful. I actually found that the teacher's instruction was more understandable than the book. But it wasn't completely unuseful. Itis good to have for class.
I used this book for an online stats with ease. I did not have the luxary of an instructor guide me through problems and have my questions clarified in class. I took the class online and I had no problem inderstanding how to wrok out the problems. It goes into detail on how to work out problems and gives examples step by step. There are also a lot of practice problems in the text with odd numbers answered for review. The CD in also a great ...
Student Solutions Manual Elementary Statistics: A Step By Step Approach Purchased this solution manual to assist with obtaining correct answers to the textbook problems. However, this book was not worth the $60 plus dollars it cost, including shipping to obtain it. This manual is extremely thin in size and contain; and should have been added as an appendix inclusion inside the textbook, vice making it a separate purchase. I recommend spending more time with course ...
not good experience Contacted seller twice before receiving the book and seller did not respond to either inquiry. When I received the book, it was in WAY worse condition then stated. The binder was in awful shape and the pages were filled with writing....in ink. This was not stated in description. I contacted seller after receiving the book to discuss this, again, no response. I would not buy again from this ...
Lucky 13 The book goes into details about the real world of using math. Recommend this book for any math class. It consist of all math problems from simple math to trig. and geo. algebra, etc...great to have for future classes
Ths is the best book! Three cheers for Bluman! I owned the 5th edition for school and accidentally returned it. Wow that was a big mistake. I got the 6th edition and it just was a relief that I had found it. Probably more expensive than I really wanted to spend, but it was worth it. So, now I'm doing a self-study and I'm enjoying the book tremendously!
Decent Introdution To Business/Consumer math Before reading this book, I already understood many of the concepts presented, but I had never learned them "formally", so I gave it a shot. I liked the format, the concepts were presented in small portions with quizzes after each portion (then a final quiz for the whole chapter). If I had to do it over again, I would definitely choose this book.
Hope this helps... |
Number Theory & Abstract Algebra
Number Theory & Abstract Algebra
I'm currently taking a course, "Abstract Algebra I & Number Theory" and i'm wondering:
what is the difference between abstract algebra and number theory? the two topics seem meshed together.
i tried googling both of them and it doesn't really help. it's hard to tell the differences between the two.
can anyone give me a solid answer?
edit: i'm mostly wondering because we also have a course "Abstract Algebra II," and a course "Topics in Number Theory," both of which require "Abstract Algebra I & Number Theory" as a prerequisite. i'm only required to take one and i'd rather take the one i'm more interested in and better at.
Gosh, I have trouble seeing the similarity. Abstract algebra is the study of abstract groups, rings, fields, and such; it studies properties and how they generalize. They also classify groups and other such creatures.
Number theory is about whole numbers, divisibility, modular relations, and the like. It then uses other fields like algebra (to gain insight into integers considered as a group under addition, or a ring under addition/multiplication mod p, etc.), real analysis (generating functions, sequences, analytic approximations of the discrete), complex analysis (continuations of number-theoretical functions, special functions like zeta), combinatorics, graph theory, etc.
yes, 2 very similar topics my little half-wit friend.
I afraid I can't help you further today, I now take my grandmother to doctors.
sorry for my bad english.
love you all, love to your mothers
C to the T to the remBath
xxx
Number Theory & Abstract Algebra
Number Theory and Algebra are related.
Number Theory uses many areas of mathematics to solve problems, as CRGreathouse pointed out. Algebra seems to be one of the most popular. You'll get to learn about Fermat's Little Theorem (Euler's Theorem), which is one of the most common applications from Group Theory to Number Theory.
Jimmypoopens: I'm currently taking a course, "Abstract Algebra I & Number Theory" and i'm wondering: what is the difference between abstract algebra and number theory?
I understand his question. He has a book on Abstract Algebra that starts with the development of the integers, much of which he might also find in a Number Theory book. I suggest he look beyond the beginning chapters for his answer.
I see where he's coming from too. Remainders mod p form a field... remainders mod n in general form a group. Modular arithmetic and Abstract Algebra are essentially the same thing. How about the whole concept of the "algebraic number" too? Abstract Algebra has applications in number theory.
There's plenty of Number Theory that has nothing do do with Abstract Algebra, though. Just wiki Analytic Number Theory and you'll find plenty. |
The 56 activities in this collection give students the opportunity to directly experience, through dynamic visualization and manipulation, the topics covered in precalculus. It finishes with a dynamiHow does a scale change in an equation effect the points on the graph? Given an equation, students often confuse what happens to an equation in function form with what happens to the points in a list...Using this virtual manipulative you may: graph a function; trace a point along the graph; dynamically vary function parameters; change the range of values displayed in the graph; graph multiple functi... More: lessons, discussions, ratings, reviews,...
Enter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each co... More: lessons, discussions, ratings, reviews,...
Explore how the parameters in a quadratic equaiton in standard form affect the graph of the equation. Dynamically change the parameters a, b, and c and immediately see the effect on the graph. Try tPlay this customizable game by entering functions that "hit" certain coordinates while avoiding others. Players (or teachers) can add as many of the coordinates to target or avoid, as well as set colo... More: lessons, discussions, ratings, reviews,...
Explore how the parameters in a quadratic equation in vertex form affect the graph of the equation. Dynamically change the parameters and immediately see how the graph changes. Try to change the par... More: lessons, discussions, ratings, reviews,...
This is a Java graphing applet that can be used online or downloaded. The purpose it to construct dynamic graphs with parameters controlled by user defined sliders that can be saved as web pages or emThis App provides a way for students to study and learn how to identify the coefficients of a function from a graph. Students can choose linear functions, quadratic functions, and absolute value funct |
MATH 210: Calculus III
Navigation:
Labs
The goal of the lab projects is to develop geometric intuition by using computer graphics to help visualize and understand the mathematical objects discussed in the course.
The Labs are part of the course. They meet once a week in room 1200 SEO on Tuesday or Thursday. Your Lab may meet at a different time from your lecture---see the schedule on the "Sections" page.
Lab Schedule
Below is the schedule for the weekly computer lab. Note that in addition to the potential quizzes scheduled below, your instructor may decide to give additional quizzes during the lectures.
More information about the quizzes will be given by your instructor. |
0495389617
9780495389613
1111803870
9781111803872 you expand your reasoning abilities as it teaches you how to read, write, and think mathematically. Packed with real-life applications of math, it blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. The authors' five-step problem-solving approach makes learning easy. More student-friendly than ever, the text offers a rich collection of student learning tools. Enhanced WebAssign online learning system is available for an additional charge. With ELEMENTARY AND INTERMEDIATE ALGEBRA, 4e, algebra makes sense! «Show less |
, step-by-step explanations make the material crystal clear. E... MOREstablished the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues. Elementary Linear Algebra 10th edition gives an elementary treatment of linear algebra that is suitable for a first course for undergraduate students. The aim is to present the fundamentals of linear algebra in the clearest possible way; pedagogy is the main consideration. Calculus is not a prerequisite, but there are clearly labeled exercises and examples (which can be omitted without loss of continuity) for students who have studied calculus. Technology also is not required, but for those who would like to use MATLAB, Maple, or Mathematica, or calculators with linear algebra capabilities, exercises are included at the ends of chapters that allow for further exploration using those tools.
A concluding chapter covers twenty applications of linear algebra drawn from business, economics, physics, computer science, ecology, genetics, and other disciplines. The applications are independent and each includes a list of mathematical prerequisites.
This text comes with WileyPLUS.
This online teaching and learning environment integrates the entire digital textbook with the most effective instructor and student resources to fit every learning style. With WileyPLUS:
Students achieve concept mastery in a rich, structured environment thatís available 24/7 Instructors personalize and manage their course more effectively with assessment, assignments, grade tracking, and more. WileyPLUS can complement the textbook or replace the printed text altogether. |
Pre-Algebra Math
Pre-Algebra Math
The following options are provided for TMCC students whose math test scores have placed them below the Math 095 (beginning algebra) level. Each option covers the same pre-algebra topics preparing the student to test into and succeed in Math 095.
Currently Available Options
Option 1. Quick Review of Pre-Algebra Math
This option is for students who need only a little brushing up on their math skills. It consists of a series of self-tests with answers covering pre-algebra topics. No lessons are provided. For details, see Pre-Algebra Quick Review.
Option 2. ALEKS Online Course
This online self-paced course uses the ALEKS learning system. ALEKS provides an initial assessment of the student's math skills, an individualized curriculum based upon that initial assessment, guided practice problems, worked examples, video tutorials, and regular assessments during the course. The 18-week ALEKS access code to take this class currently costs $67.00 (if purchased online). For details, see ALEKS Pre-Algebra Course.
Option 3. Free Online Course
This course uses a free online Pre-Algebra textbook that provides lessons, worked examples, and practice problems with answers. Video lessons covering many of the course's topics and additional practice problems are also provided. For details, see Free Online Pre-Algebra Course.
Options Available Soon
Option 4. Pre-Algebra Workbook
This pencil-and-paper option is for students who do not want to work on a computer (or simply want additional practice). A workbook is provided at minimal cost to the student. The student works through assigned chapters and sections in the workbook.
Option 5. No-Credit Skills Center Class
This option serves students who want an instructor to provide guidance through the course, feedback on tests, tutoring help, etc. There are no formal lectures: students in the class select one of the four math review options above. There is a fee for the class. |
Algebra II
Algebra II is typicality an upper math class taught to juniors. Out of any other class offered to highschool students it is a leading indicator of college and work success according to a study reported by the Washington Post. Most students take algebra I and geometry before algebra II, however some advanced students may be encouraged to take it their freshmen year after taking algebra I in 8th grade. The class has significant focus on higher level operations in algebra including radical, logarithmic, and exponential functions. Modeling and application is particularly important.
Looking to compare unit and pacing guides? We have a small collection of planning resources from unit plans to full semester course plans. Take what you need, all we ask is that you upload any great planning guides you have. |
Browse Bestsellers
The Eighth Edition of this highly dependable book retains its best features–accuracy, precision, depth, and abundant exercise sets–while substantially updating its content and pedagogy.Now in its third edition, Mathematical Concepts in the Physical Sciences provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts intoDesigned for students in various disciplines of engineering, science, mathematics, management and business, this effective study tool includes hundreds of problems with step-by-step solutions and ... > read more
Eminently suited to classroom use as well as individual study, Roger Myerson's introductory text provides a clear and thorough examination of the models, solution concepts, results, and ... > read more
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Focusing on helping students to develop both the conceptual understanding and the analytical skills necessary to experience success in mathematics, we present each mathematical topic in this text ... > read more |
Maths The Basic Skills is part of a suite of resources accompanied by 3 workbooks and 3 worksheet packs - available to buy separately. These resources have been designed specifically for the Adult Numeracy Curriculum, covering Entry Levels 1, 2 and 3 and Levels 1 and 2.
Synopsis:
The textbook targets the higher levels of the Adult Numeracy Curriculum, Entry Level 3, Level 1 and Level 2. Covering all of the three subject areas of the curriculum in one book, it offers revision of Entry Level 1 and 2 topics where appropriate. Exercises progress from simple numerical questions gradually increasing in difficulty to incorporate numbers into language. Suggestions for alternative methods of learning are provided for students who are struggling to comprehend a particular |
MATH 180 and 184 Workshops, Fall 2012
This website gives information about the weekly workshops associated
to MATH 180 and 184. These workshops are a key component of these
courses. Surveys of students done in the past indicate that the
vast majority think the workshops have helped them do better in
the course, and analysis of student performance on common final
exams supports this perception.
Different problem sets are used in the two courses, but the
structure of the workshops is identical. Problem sets and solutions
will be posted after all workshops for a given week are
complete; see the links immediately below.
Workshop Overview
The weekly workshops are 80 minutes long and start the second
week of classes. There are 12 workshops. The workshops are an important learning
element in the course. In MATH 180, workshops are worth 7.5% of the course grade; in MATH 184 they are worth 10%.
The primary activity in the workshops is working on weekly
Practice Problems, using prepared problem sets. The activity is
to be done in groups, and is to be facilitated rather than
tutored. In addition to this group work, in most workshops
students will individually work on a Problem to Hand In at the
end of the workshop.
Two TAs, one a graduate student and one an undergraduate, lead
the workshops. They do not tutor, but rather they facilitate.
Workshop Learning Goals
Actively think about and solve problems involving calculus
Interact with peers to discuss mathematics and problems involving
mathematics
Acquire and reinforce basic problem-solving skills including
reading problems carefully and writing down information contained
in the problem as a prelude to solving the problem
Workshop Details
Groups have three or four students each. They are formed at
the start of term and then generally remain intact through the
remainder of the term. Groups work on blackboards. Students in
each group take turns writing at the blackboard, and are not
sitting down and working individually.
The workshop grade component is based on group participation and
performance on the Quizzes. Effort rather than a correct answer
is the main criteria for earning thse marks. Click here
for details on how workshop grades are computed.
The 80 minutes for each workshop are usually broken down as
follows:
65 minutes for groups working on the Practice Problems
15 minutes for students working individually on the Quizzes
There will be no Quiz at the first or last workshop or at one
midterm workshop in which a survey will instead be administered.
Workshops Tips
Your instructor will cover the relevant facts in class before
you come to your weekly workshop. To take full advantage of the
workshop, review your notes from class and/or the textbook before
coming to your workshop. The workshops are your chance to practice
working on new problems, not to learn the basic material or work
on homework problems. But do try some homework problems on the
relevant material before your workshop. The better prepared you
are, the more you will get out of the workshops.
The workshops are not a replacement for homework. In addition
to attending the workshops, you must conscientiously work on
the homework your instructor assigns in order to do well in the
course. Participating fully with your group in the workshops
should make it easier for you to solve problems yourself on homework
assignments and exams.
Remember: the point of the workshops isn't getting
the final answer correct. It is the process of activelyengaging in problem solving with your peers. This process
will help you more than simply being told how to solve
problems by your instructor or TA. Mathematical problem solving
is a skill that is best learned by doing rather than watching.
Relax and make the most of this experience. It's probably
different from anything you've done in the past and may
seem daunting at first. Once you get the "hang" of
it, you'll find the workshops a rewarding, fun activity,
just like the thousands of students who've done these workshops
in past years. Who knows, maybe you'll even want to volunteer
to become a TA next year!
This website is maintained by Rajiv Gupta,
the workshop program coordinator. |
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Pre Algebra worksheets Need to learn about Pre Algebra? Shmoop makes learning more fun and relevant for students in the digital age. Find all the information you need to gain a better understanding of Pre Algebra, from study guides, practice tests, and more. Why go anywhere else? Shmoop has you covered. |
Resumen del libro
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Derived from extensive teaching experience in Paris, this second edition now includes over 100 exercises in probability. New exercises have been added to reflect important areas of current research in probability theory, including infinite divisibility of stochastic processes, past-future martingales and fluctuation theory. For each exercise the authors provide detailed solutions as well as references for preliminary and further reading. There are also many insightful notes to motivate the student and set the exercises in context. Students will find these exercises extremely useful for easing the transition between simple and complex probabilistic frameworks. Indeed, many of the exercises here will lead the student on to frontier research topics in probability. Along the way, attention is drawn to a number of traps into which students of probability often fall. This book is ideal for independent study or as the companion to a course in advanced probability theory
indice: Preface to the Second Edition; Preface to the First Edition; 1. Measure theory and probability; 2. Independence and conditioning; 3. Gaussian variables; 4. Distributional computations; 5. Convergence of random variables; 6. Random processes; Where is the notion N discussed?; Final suggestions: how to go further?; References; Index |
Computer Usage: Students are assumed to be versed in the use MathCAD or MATLAB to perform scientific computing such as numerical calculations, plotting of functions and performing integrations. Students will develop and visualize solutions to moderately complicated field problems using these tools |
Spread of Numeracy in Early America
An entertaining and informative history of the birth of the American passion for numbers, tracing the history of numeracy from its origins in the Enlightenment to its flowering in mid-nineteenth century America.
A Mathematical Study of Human Thought
This monograph reports a thought experiment with a mathematical structure intended to illustrate the workings of a mind. It presents a mathematical theory of human thought based on pattern theory with a graph-based appro ...
Beyond the Numbers
A valuable guide to a successful career as a statistician A Career in Statistics: Beyond the Numbers prepares readers for careers in statistics by emphasizing essential concepts and practices beyond the technical tools p ...
Laboratories for Decision Making
Emphasizes applied Inter-disciplinary approach in business context- Provides increased motivation in classroom and linkage to other courses outside the classroom. Includes guided analysis and open-ended discussion questi ...
Enjoy a wide range of dissertations and theses published from graduate schools and universities from around the world. Covering a wide range of academic topics, we are happy to increase overall global access to these wor ...
This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to hav ...
This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. Thi ...
This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. Thi ...
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1820 Excerpt: ... any two ha ...
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1820 edition. Excerpt: ...to ...
Significant research activity has occurred in the area of global optimization in recent years. Many new theoretical, algorithmic, and computational contributions have resulted. Despite the major importance of test proble ...
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague. In A Combination of Geometry Th ...
Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on ...
Combinatorial chemistry represents a revolution in the way the pharmaceutical industry identifies and optimizes leads for drugs. This text attempts to provide a basic knowledge and a practical guide to perform combinator ...
The aim of this work is the definition of the polyhedral compactification of the Bruhat-Tits building of a reductive group over a local field. In addition, an explicit description of the boundary is given. In order to ma ...
Improve your algebra and problem-solving skills with A COMPANION TO CALCULUS! Every chapter in this companion provides the conceptual background and any specific algebra techniques you need to understand and solve calcul ...
Enjoy a wide range of dissertations and theses published from graduate schools and universities from around the world. Covering a wide range of academic topics, we are happy to increase overall global access to these wor ... |
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For a newly developing "first course" in mathematics which may replace Liberal Arts Math or College Algebra.This inexpensive introduction to mathematical models explores mathematics in the context of real applications that provide meaning and motivation. Technology is used throughout as a tool to solve problems and investigate solutions. |
Platonic Realms: Coping with Math Anxiety This page is a "minitext" within a larger math site authored by B. Sidney Smith, a doctoral candidate at University of Colorado, Boulder. His essay explores the social and educational roots of current attitudes toward and myths about math. He also includes some practical strategies for coping with math anxiety.
Help for Math Anxiety Authored by the mathematics staff at Middle Tennessee State University, this web page offers practical advice for tackling math anxiety. Although written for students in the traditional classroom, the six brief readings are helpful for anyone. They cover topics such as taking a math study skills inventory, how math is different from other subjects, how to study and take tests, and offers web links for additional information. |
Synopsis
Whether you're new to geometry or just looking for a refresher, this completely revised and updated third edition of Geometry Success in 20 Minutes a Day offers a 20-step lesson plan that provides quick and thorough instruction in practical, critical skills. Stripped of unnecessary math jargon but bursting with geometry essentials, Geometry Success in 20 Minutes a Day is an invaluable resource for both students and adults.
Found In
eBook Information
ISBN: 9781576858 |
Commercial site with one free access a day. Students use mapping diagram to create a relation, then they can check if it is a function from the mapping diagram, ordered pairs and graph. After studen... More: lessons, discussions, ratings, reviews,...
This applet demonstrates an exponential growth model which plots population P_i for i=1 to i=600 given user input for the initial population P_0 and growth rate G. The difference equation used is P_(i... More: lessons, discussions, ratings, reviews,...
This applet demonstrates a logistic growth model which plots population P_i for i = 1 to i = 600 given user input for the initial population P_0, growth rate G and carrying capacity CC. The difference |
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Prealgebra and Introductory Algebra Book Description
Lial/Hestwood's "Prealgebra and Introductory Algebra, "2e, gives students the tools necessary to succeed in developmental math courses and prepares them for future math courses and the rest of their lives. The Lial developmental team creates a pattern for success by emphasizing problem-solvingPopular Searches
The book Prealgebra and Introductory Algebra by Margaret L Lial, Diana L Hestwood, John Hornsby
(author) is published or distributed by Addison Wesley Publishing Company [0321433467, 9780321433466].
Prealgebra and Introductory Algebra has Paperback binding and this format has 1,254 number of pages of content for use.
The printed edition number of this book is 2.
This book by Margaret L Lial, Diana L Hestwood, John Hornsby |
Most problems in mathematics do not involve linear relationships but things that are curved. For example, the distance needed for a car to stop when the brakes are applied is related to the square of the speed of the car and the intensity of a light beam varies inversely with the square of the distance from the light source. In this chapter you will learn about the building blocks used to create these other kinds of problems, monomials and polynomials. |
Standards-Based Resources
Math
recommended books
Explore this month's posting of book reviews. You and your students may enjoy reading these noteworthy releases:
Codes and Ciphers
Authored By: Sean Callery
Published: January 2008
Callery covers the history of codes and ciphers from hieroglyphics to cyberspace. He looks at code-breaking devices, Native American smoke signals, flags and semaphore, Braille, Morse code, and mono- and polyalphabetic letter substitution. The book features sidebars, many illustrations, a glossary, and recommendations for further reading.
Resource Type: Recommended Non-PBS Book
Subject: Math
Game Theory: A Very Short Introduction
Authored By: Ken Binmore
Published: November 2007
Game theory, a branch of applied mathematics, is the study of games when they are played rationally. To reach a larger audience, this brief introduction explains game theory without mathematical equations. An expert in the field, the author examines game theory in the sciences, particularly evolutionary biology, psychology, ethics, politics, and economics. The book contains many illustrations. |
The purpose of this course is to allow students who are already using ALEKS as part of one of their math courses at DHS an extra opportunity to progress.
Excerpts from .com
Assessment and LEarning in Knowledge Spaces is a Web-based, artificially intelligent assessment and learning system. ALEKS uses adaptive questioning to quickly and accurately determine exactly what a student knows and doesn't know in a course. ALEKS then instructs the student on the topics she is most ready to learn. As a student works through a course, ALEKS periodically reassesses the student to ensure that topics learned are also retained. ALEKS courses are very complete in their topic coverage and ALEKS avoids multiple-choice questions. A student who shows a high level of mastery of an ALEKS course will be successful in the actual course she is taking. … ALEKS also provides the advantages of one-on-one instruction, 24/7, from virtually any Web-based computer for a fraction of the cost of a human tutor.
Educational Objectives (College / Career Readiness Focus):
1.Students will increase mastery of mathematics skills and concepts using the ALEKS online program. |
The SimCalc Project aims to democratize access to the Mathematics of Change for mainstream students by combining advanced simulation technology with innovative curriculum that begins in the early g... More: lessons, discussions, ratings, reviews,...
Students are asked to predict what an absolute value graph will look like with given parameters. Then they can use a graphing utility that is on the same page to test their predictions. Feedback is g... More: lessons, discussions, ratings, reviews,...
With this one-variable function grapher applet and function evaluator, users can rotate axis/axes, change scale, and translate by using mouse or by entering data. The web site also contains informatio... More: lessons, discussions, ratings, reviews,...
This tool lets you plot functions, polar plots, and 3D with just a suitable web browser (within the IE, FireFox, or Opera web browsers), and find the roots and intersections of graphs. In addition, yo... More: lessons, discussions, ratings, reviews,...
The "Scrambler" is an amusement park ride with a central rotating hub with three central arms, with spokes at the end of each arm that rotate in a different direction. This applet, approved by the ElGuided activities with the Graph Explorer applet, designed to let students learning about quadratic functions explore: the parabolic shape of the graphs of quadratic functions; how coefficients affect... More: lessons, discussions, ratings, reviews,...
Commercial site with one free access per day. Students are given a set of points and are asked to "zap" as many points as possible. They can use either polynomial form or vertex form. Students can ge |
Welcome to the "World of Math"
Mrs. Ann Funk
Courses Offered: Algebra I,
Geometry, Algebra II,
Trigonometry, Statistics
Course Title: Algebra I
Course Description:This course is intended for, but not limited to, freshmen and sophomores. Emphasis is on basic algebraic concepts and skills, highlighted by some key area in introductory geometry, basic logic and introductory trigonometry.
Outline of Topics:
Working with Real Number
Solving Equations and Problems
Polynomials
Fractions
Applying Fractions
Introduction to Fractions
System of Linear Equations
Inequalities
Rational and Irrational Numbers
Quadratic Functions
Course Title: Geometry
Course Description:This course is intended for, but not limited to, freshmen and sophomores. Emphasis is on geometry which encompasses proofs, properties, constructions, polygonal areas & perimeters and right triangle trigonometry.
Outline of Topics:
Geometric Figures
Proof
Parallelism
Congruent Triangles
Congruence
Similarity
Polygons
Special Quadrilaterals
Right Triangles
Coordinate Geometry
Areas & Volumes of Solids & Plane Figures
Course Title: Algebra II
& Trigonometry
Course Description:This course is intended for, but not limited to, juniors and seniors. Emphasis is on advanced algebra, trigonometry, problem-solving and the use of graphing calculators.
Outline of Topics:
Modeling Problem Situations
Exploration of Polynomials
Rational, Irrational & Complex Expressions
Exploring and Applying Functions
Variation
Sequences and Series
Exploring Logarithmic and Exponential Functions
Angles, Trigonometry & Vectors
Transformations of Graphs & Data
Periodic Models
Course Title: Statistics
Course Description:This one semester course is offered to any student who has successfully completed Algebra II. This course will present concepts, principles and methods of statistics from two perspectives: descriptive and inferential. Statistical topics include organizing data, sampling, measures of central tendency, probability, correlation, random variables, hypothesis testing, confidence intervals, and inference.
Outline of Topics:
Exploring Data
The Normal Distribution
Examining Relationships
Producing Data
Sampling Distributions
Introductions to Inference
Inference for Distributions
Inference for Proportions
Probability
Random Variables
Binomial & Geometric Distributions
Course Title:
Trigonometry
Course Description:This one semester course if offered to any student who has successfully completed Algebra II. This course will further reinforce trigonometric and exponential properties, functions and their graphs. Logarithmic and vector applications, along with problem solving through calculator use, will also be utilized to enhance concepts within trigonometry. |
MAT
106
- Math for Elementary Education I
This is the first course of a two-semester sequence which explores the mathematics content in grades K-6 from an advanced standpoint. Topics include: problem solving; functions and graphs; and numbers and operations. This course is open to elementary education and early childhood students only. |
use graphs to analyze the nature of changes in quantities
in linear relationships.
Students in the middle grades should
learn algebra both as a set of concepts and competencies tied to the representation
of quantitative relationships and as a style of mathematical thinking
for formalizing patterns, functions, and generalizations. In the middle
grades, students should work more frequently with algebraic symbols than
in lower grades. It is essential that they become comfortable in relating
symbolic expressions containing variables to verbal, tabular, and graphical
representations of numerical and quantitative relationships. Students
should develop an initial understanding of several different meanings
and uses of variables through representing quantities in a variety of
problem situations. They should connect their experiences with linear
functions to their developing understandings of proportionality, and they
should learn to distinguish linear relationships from nonlinear ones.
In the middle grades, students should also learn to recognize and generate
equivalent expressions, solve linear equations, and use simple formulas.
Whenever possible, the teaching and learning of algebra can and should
be integrated with other topics in the curriculum.
Understand patterns,
relations, and functions
The study of patterns
and relationships in the middle grades should focus on patterns that relate
to linear functions, which arise when there is a constant rate of change.
Students should solve problems in which they use tables, graphs, words,
and symbolic expressions to represent and examine functions and patterns
of change. For example, consider the following problem:
Charles
saw advertisements for two cellular telephone companies. Keep-in-Touch
offers phone service for a basic fee of $20.00 a month plus $0.10 for
each minute used. ChitChat has no monthly basic fee but charges $0.45
a minute. Both companies use technology that allows them to charge for
the exact amount of time used; they do not "round up" the time to the
nearest minute, as many of their competitors do. Compare these two companies'
charges for the time used each month.
Students might begin
by making a table, picking convenient numbers of minutes, and finding
the corresponding costs for the two companies, as shown in figure 6.8a.
Using a graphing calculator, students might then plot the points as ordered
pairs (minutes, cost) on the coordinate plane, obtaining a graph for each
of the two companies (see fig. 6.8b). Some students might describe the
pattern in each graph verbally: "Keep-in-Touch costs $20.00 and then $0.10
more per minute." Others might write an equation to represent the cost
(y) in dollars in terms of the number of minutes (x),
such as y = 20.00 + 0.10x.
Fig. 6.8. Students can compare the
charges for two telephone companies by making a table
(a) and by representing the charges on a graphing calculator
(b).
p.
223
Before the students solve the
problem, a teacher might ask them to use their table and graph to focus
on important basic issues regarding » the
relationships they represent. By asking, "How much would each company
charge for 25 minutes? For 100 minutes?" the teacher could find out if
students can interpret and extend the patterns. Since the table identifies
only a small number of distinct points, a teacher could ask why it is
legitimate to connect the points on the graph to make a line. Students
might also be asked why one graph (for ChitChat) includes the origin but
the other (for Keep-in-Touch) does not (see fig. 6.8b). Most students
will recognize that the ChitChat graph includes the origin because there
is no charge if no calls are made but the Keep-in-Touch graph includes
(0, 20) because the company charges $20.00 even if the telephone is not
used.
Many students will naturally seek
a formula to express these patterns, but questions such as the following
would be a good catalyst for others: How can you find the cost for any
number of minutes for the Keep-in-Touch plan? For the ChitChat plan? What
aspects of the stated price schedule are indicated in the graph? How?
Students are likely to observe the constant difference between both the
successive entries in the table and the coordinates of the points for
each company along a straight line. They may explain the pattern underlying
the function by saying, "Whenever you talk for one more minute, you pay
$0.10 more (or $0.45 more), so the points go up the same amount each time."
Others might say that a straight line is reasonable because each company
charges a constant amount for each minute. Teachers should encourage students
to explain their observations in their own words. Their explanations will
provide the teacher with important insights into the students' thinking,
particularly how well they recognize and represent linear relationships.
A solution to the stated problem
requires comparing data from the two companies. A teacher might want to
ask additional questions about this comparison: Which company is cheaper
if you use the telephone infrequently? If you use it frequently? If you
cannot spend more than $50.00 in a month but you want to talk for as many
minutes as possible, which company would be the better choice? Considering
questions such as these can lay the groundwork for a pivotal question:
Is there a number of minutes that costs the same for both companies? Such
questions could give rise to many observations. For example, most students
will notice in their table that something important happens between 50
and 60 minutes, namely, using ChitChat becomes more expensive than using
Keep-in-Touch. From the graph, some students may observe that this shift
occurs at about 57 minutes: Keep-in-Touch is the cheaper company when
a customer uses more than 57 minutes in a month. Experiences such as this
can lay a foundation for solving systems of simultaneous equations.
p.
224
The problem could also easily
be extended or adapted in ways that would draw students' attention to
important characteristics of the line graph for each company's charges.
For example, to draw attention to the y-intercept, students
could be asked to use a graphing calculator to examine how the graph would
be affected if Keep-in-Touch increased or decreased its basic fee or if
ChitChat decided to begin charging a basic fee. Students' attention could
be drawn to the slope by asking them to consider the steepness of the
lines using a question such as, What happens to the graph for Keep-in-Touch
if the company increases its cost per minute from $0.10 to $0.15? Through
experiences such as these, students should develop a general understanding
of, and » facility with, slope and y-intercept
and their manifestations in tables, graphs, and equations.
The problem could also easily
be extended to nonlinear relationships if, for instance, the companies
did not charge proportionally for portions of minutes used. If they rounded
to the nearest minute, then the cost for each company would be graphed
as a step function rather than a linear function. In another variation,
a nonlinear pricing scheme for a third company could be introduced.
Another important topic for class
discussion is comparing and contrasting the merits of graphical, tabular,
and symbolic representations in this example. A teacher might ask, "Which
helps us see better the point at which the two companies switch position
and Keep-in-Touch becomes the more economical—a table or a graph?"
"Is it easier to see the rate per minute from the graph or from the equation?"
or "How can you determine the rate per minute from the table?" Through
discussion, students can identify the strengths and the limitations of
different forms of representation. Graphs give a picture of a relationship
and allow the quick recognition of linearity when change is constant.
Algebraic equations typically offer compact, easily interpreted descriptions
of relationships between variables.
Represent and analyze
mathematical situations and structures using algebraic symbols
Working with variables
and equations is an important part of the middle-grades curriculum. Students'
understanding of variable should go far beyond simply recognizing that
letters can be used to stand for unknown numbers in equations (Schoenfeld
and Arcavi 1988). The following equations illustrate several uses of variable
encountered in the middle grades:
27 = 4x + 3
1 = t(1/t)
A = LW
y = 3x
The role of variable as "place
holder" is illustrated in the first equation: x is simply taking
the place of a specific number that can be found by solving the equation.
The use of variable in denoting a generalized arithmetic pattern is shown
in the second equation; it represents an identity when t takes
on any real value except 0. The third equation is a formula, with A,
L, and W representing the area, length, and width, respectively,
of a rectangle. The third and fourth equations offer examples of covariation:
in the fourth equation, as x takes on different values, y
also varies.
p.
225
Most students
will need extensive experience in interpreting relationships among quantities
in a variety of problem contexts before they can work meaningfully with
variables and symbolic expressions. An understanding of the meanings and
uses of variables develops gradually as students create and use symbolic
expressions and relate them to verbal, tabular, and graphical representations.
Relationships among quantities can often be expressed symbolically in more
than one way, providing opportunities for students to examine the equivalence
of various » algebraic expressions. Fairly
simple equivalences can be involved: the cost (in dollars) of using Keep-in-Touch
can be expressed as y = 0.10x + 20, as y =
20 + 0.10x, as 20 + 0.10x = y,
and as 0.10x + 20 = y. Complex symbolic expressions
also can be examined, such as the equivalence of 4 + 2L + 2W
and (L + 2)(W + 2) – LW when
representing the number of unit tiles to be placed along the border of a
rectangular pool with length L units and width W units;
see the "Representation" section of this chapter for a discussion of this
example.
A problem such as the one in figure
6.9 (adapted from Educational Development Center, Inc. 1998, p. 41) could
give students valuable experience in deciding whether two expressions
are equivalent. A teacher might encourage students to begin solving this
problem by drawing several more boxes of various sizes so they can look
for a pattern. Some students will probably note that the caramels are
also arranged in a rectangular pattern, which is narrower and shorter
than the rectangular arrangement of chocolates. Using this observation,
they might report, "To find the length and width of the caramel rectangle,
take 1 off the length and 1 off the width of the chocolate rectangle.
Multiply the length and width of the caramel rectangle to find the number
of caramels." If L and W are the dimensions of the
array of chocolates, and C is the number of caramels, then this
generalization could be expressed symbolically as C = (L – 1)(W – 1).
Other students might find and use the number of chocolates to find the
number of caramels. For example, for a 35 box of chocolates,
they might propose starting with the 15 chocolates, then taking off 3
because "there is one less column of caramels" and then taking off 5 "because
there is one less row of caramels." This could be expressed generally
as C = LW – L – W. Although
both expressions for the number of caramels are likely to seem reasonable
to many students, they do not yield the same answer. For the 35 array, the first
produces the correct answer, 8 caramels. The second gives the answer 7
caramels. Either examining a few more boxes with different dimensions
or reconsidering the process represented by the second equation would
confirm that the second equation needs to be corrected by adding 1 to
obtain C = LW – L – W + 1. The algebraic
equivalence of (L – 1)(W – 1) and LW – L – W
+ 1 can be demonstrated in general using the distributive property of
multiplication over subtraction.
Super
Chocolates are arranged in boxes so that a caramel is placed in
the center of each array of four chocolates, as shown below. The
dimensions of the box tell you how many columns and how many rows
of chocolates come in the box. Develop a method to find the number
of caramels in any box if you know its dimensions. Explain and
justify your method using words, diagrams, or expressions.
Fig. 6.9. The Super Chocolates problem
Through a variety of experiences
such as these, students can learn the strengths and limitations of various
methods for checking the equivalence of expressions. In some instances,
the equivalence of algebraic expressions can be demonstrated geometrically;
see the "Geometry" section of this chapter for a demonstration that (a + b)2 = a2 + 2ab + b2.
p.
226
Most middle-grades students will
need considerable experience with linear equations before they will be
comfortable and fluent in transforming or solving them. Although students
will probably acquire facility with equations at different times during
the middle grades, by the end of grade 8, students should be able to solve
equations like 84 – 2x = 5x + 12 for
the unknown number, to recognize as identities such equations as 1 = t(1/t)
(when t is not 0), to apply formulas such as V = r2h,
and to recognize that equations such as y = –3x
+ 10 represent linear functions that are satisfied by many ordered
pairs (x, y). Students should be able to use equations of the
form y = mx + b to represent linear relationships,
and they should know how the values of the slope (m) and the
y-intercept (b) » affect
the line. For example, in the "cellular telephone" problem discussed earlier,
they should recognize that y = 0.10x + 20 and y
= 0.45x are both linear equations, that the graph of the
latter will be steeper than that of the former, and that the former intersects
the y-axis at (0, 20) rather than at the origin.
Students' facility with symbol
manipulation can be enhanced if it is based on extensive experience with
quantities in contexts through which students develop an initial understanding
of the meanings and uses of variables and an ability to associate symbolic
expressions with problem contexts. Fluency in manipulating symbolic expressions
can be further enhanced if students understand equivalence and are facile
with the order of operations and the distributive, associative, and commutative
properties.
Use mathematical models
to represent and understand quantitative relationships
A major goal in the middle
grades is to develop students' facility with using patterns and functions
to represent, model, and analyze a variety of phenomena and relationships
in mathematics problems or in the real world. With computers and graphing
calculators to produce graphical representations and perform complex calculations,
students can focus on using functions to model patterns of quantitative
change. Students should have frequent experiences in modeling situations
with equations of the form y = kx, such as relating
the side lengths and the perimeters of similar shapes. Opportunities can
be found in many other areas of the curriculum; for example, scatterplots
and approximate lines of fit can model trends in data sets. Students also
need opportunities to model relationships in everyday contexts, such as
the "cellular telephone" problem.
p.
227
Students also should have experience
in modeling situations and relationships with nonlinear functions, such
as compound-interest problems, the relationship between the length of
the radius of a circle and the area of the circle, or situations like
the one in figure 6.10. If students have only a few points to examine,
it can be difficult to see that » the
graph for this problem is not linear. As more points are graphed, however,
the curve becomes more apparent. Students could use graphing calculators
or computer graphing tools to do problems such as this.
Consider
rectangles with a fixed area of 36 square units. The width (W)
of the rectangles varies in relation to the length (L)
according to the formula W = 36/L. Make a table
showing the widths for all the possible whole-number lengths for
these rectangles up to L = 36.
Look
at the table and examine the pattern of the difference between
consecutive entries for the length and the width. As the length
increases by 1, the width decreases, but not at a constant rate.
What do you expect the graph of the relationship between L
and W to look like? Will it be a straight line? Why or
why not?
Solution:
The graph is not a straight line because the rate of change is
not constant. Instead the graph appears to be a curve that bends
sharply downward and then becomes more level.
Fig. 6.10. A problem involving a
nonlinear relationship, with an associated table and graph
p.
228
When doing experiments or dealing
with real data, students may encounter "messy data," for which a line
or a curve may not be an exact fit. They will need experience with such
situations and assistance from the teacher to develop their ability to
find a function that fits the data well enough to be useful as a prediction
tool. In their later study of statistics, students may learn sophisticated
methods to determine lines of best fit for data. In the middle grades
when students encounter a set of points suggesting a linear relationship,
they can simply use a ruler to try several lines until they find one that
appears to be a good fit and then » write
an equation for that line. An example of this sort of activity, related
to a scatterplot of measurements, can be found in the "Data Analysis and
Probability" section of this chapter. With a graphing calculator or computer
graphing software, students can test some conjectures more easily than
with paper-and-pencil methods.
Analyze change in various contexts
In their study of algebra,
middle-grades students should encounter questions that focus on quantities
that change. Recall, for example, that ChitChat charges $0.45 a minute
for phone calls. The cost per minute does not change, but the total cost
changes as the telephone is used. This can be seen quite readily from
the two graphs in figure 6.11. The meaning of the term flat rate
can be seen in the cost-per-minute graph, which shows points along
a horizontal line at y = 0.45, representing a constant rate
of $0.45 a minute. The total-cost graph shows points along a straight
line that includes the origin and has a slope of 0.45.
Fig. 6.11. These two graphs represent
different relationships in ChitChat's pricing scheme.
Students may be confused
when they first encounter two different graphs to represent different
relationships in the same situation. Teachers can assist students in understanding
what relationships are represented in the two graphs by asking such questions
as these: When the number of minutes is 4, what do the values of the corresponding
point on each graph represent? When the number of minutes is 8? Why is
the y-value of the cost-per-minute graph constant at 0.45? How
much does the total-cost graph increase from 5 minutes to 6 minutes? Why?
How would each graph change, if at all, if the cost per minute were changed
to $0.20?
A slight modification of the problem,
such as the addition of a third telephone company with a different pricing
scheme, can allow the analysis of change in nonlinear relationships:
Quik-Talk advertises
monthly cellular phone service for $0.50 a minute for the first 60 minutes
but only $0.10 a minute for each minute thereafter. Quik-Talk also charges
for the exact amount of time used. »
A teacher might ask students
to graph the rates of change in this example. Figure 6.12 shows the cost-per-minute
graph and a total-cost graph for Quik-Talk's pricing scheme. Students
can answer questions about the relationships represented in the graphs:
Why does the cost-per-minute graph consist of two different line segments?
How can we tell from the graph that the pricing scheme changes after 60
minutes? Why is part of the total-cost function steeper than the rest
of the graph?
Fig. 6.12. These two graphs represent
different relationships in Quik-Talk's pricing scheme.
Comparing the cost-per-minute
graph to the total-cost graph for phone calls can help students develop
a clearer understanding of the relationship between change (cost per minute)
and accumulation (total cost of calls). These concepts are precursors
to the later study of change in calculus.
Students' examination of graphs
of change and graphs of accumulation can be facilitated with specially
designed computer software. Such software allows students to change either
the number of minutes used in one month by dragging a horizontal "slider"
(see fig. 6.13) or the cost per minute by dragging a vertical slider.
They can then observe the corresponding changes in the graphs and in the
symbolic expression for the relationship. Technological tools can also
help students examine the nature of change in many other settings. For
example, students could examine distance-time relationships using computer-based
laboratories, as discussed in the "Measurement" section of this chapter.
Such experiences with appropriate technology, supported by careful planning
by teachers and interactions with classmates, can help students develop
a solid understanding of some fundamental notions of change.
Fig. 6.13. Computer software can
help students understand some fundamental notions of change. |
In this part of Flash & Math you will find a large and growing collection of self-contained learning and teaching modules for mathematics - math appplets, i.e. mathlets. Mathlets can be used by instructors for classroom demonstration, and by students for self-paced study and exploration.
They are all programmed in ActionScript and run in the Flash Player. If you are interested in developing you own mathlets, visit the second part
of our site: ActionScript 3 Tutorials for Flash Developers.
Come back to our site often for new mathlets!
A FEW SAMPLE MATHLETS
New: Slope Field Applet
The applet plots the slope field for any ordinary differential equation of the form dy/dx=f(x,y)
entered by the user. The user controls the x and y ranges as well. Approximate solutions
based on the slope field can be drawn with the mouse.
New: Practice Sketching Derivative
The applet illuminates the relationship between graphs
of functions and their derivatives. A graph of
a randomly generated function f(x) is provided. The student attempts to sketch the graph
of the derivative f '(x) by dragging and shaping a curve. The answer is evaluated for correctness
at a click of a button providing instant feedback.
Contour Diagram Plotter and 3D Function Grapher Applets Combined
This applet combines a contour diagram plotter and a 3D function grapher, and allows the user
to toggle between the two. Since both contour maps and 3D graphs are very sensitive
to the choice of x and y ranges, it is often very hard to interpret a contour diagram
without seeing the corresponding 3D graph. This applet gives an opportunity to compare these two
ways to visualize functions of two variables.
Implicit Equations Grapher
This applet graphs user-defined implicit equations of the form
f(x,y)=g(x,y) in a user-defined x,y ranges. The syntax used for input
is the same as graphing calculator syntax and the applet is very easy to use.
The results are comparable with those provided by CAS like Maple.
Matching Formulas to Data Plot points,
enter any parametric family of functions, dynamically change parameters while observing the
changes in the least square error. Quick and easy matching formulas to data points.
Parametric Curves on the Plane Graph
paramertic curves in rectangular as well as polar coordinates. Curves can be traced showing the direction of motion
and the way a curve is traversed. The x and y ranges adjust automatically.
Derivative Plotter
Enter a formula for an arbitrary function, the applet will plot it. Try sketching the derivative by drawing
on screen. Reveal the real derivative dynamically by dragging a slider while the lineal element corresponding to the tanget line
and its slope are displayed.
Sequences and Series of Functions Plotter
Enter a formula for a sequence or a series of functions, the applet will plot consecutive terms or partial sums at a click
of a button. In particular, you can enter Taylor series or Fourier series.
You can also enter a piecewise defined limit function to be plotted. Many demo examples are provided.
3D Function Graphers
Enter a formula for f(x,y) in terms of x and y, the ranges for x and y, and the graphers display the graph of the function
of two variables f(x,y) in 3D. You can rotate the graph in real time and change its transparency to see the surface
clearly.
Parametric Surfaces in Rectangular Coordinates
Enter parametric formulas for the x, y, and z coordinates, and the grapher will display the corresponding surface in 3D.
You can rotate the graph in real time and change its transparency for better understanding of the surface.
Spherical Coordinates Presented Interactively
Move a point in 3D and see its spherical coordinates change. Many multiple choice practice problems will help you gain an insight into spherical coordinates.
Especially that pesky phi coordinate! |
Courses I teach
Courses at Normandale
updated March, 2011
Math 1050 Mathematical Foundations I
This is the first of threeI typically teach one or two sections of this course each year. I am teaching one right now and I am scheduled to teach one section Fall 2011.
Math 1060 Mathematical Foundations II
This is the second of three math content courses for intending elementary teachers; it follows up on Math 1050. The course focuses on geometry and measurement.
I teach at least one section of this per year. I am teaching one right now and am scheduled to teach one in the fall.
I usually teach one section of this course each semester. I am teaching one right now and am scheduled to teach a night section in the fall.
Math 1510 Calculus I
I most recently taught Calculus Fall 2010.
Math 0600/0700 Elementary and Intermediate Algebra (Math Center)
I taught two sections each semester over the last year. I am not scheduled to be in the Math Center in the next couple of semesters.
Courses at other institutions
Math 121 Calculus I (at MSU, Mankato)
I taught this fall semester 2006.
Math 181 Intuitive Calculus (at MSU, Mankato)
This course is required for elementary licensure students who are pursuing a middle-school mathematics specialty. It focuses on the major ideas of calculus-limits, differentiation and integration-with an emphasis on the concept of change over mathematical formalism. I taught this course in fall 2008.
Math 201 Elements of Mathematics I (at MSU, Mankato)
This is the first of two
Math 202 Elements of Mathematics II (at MSU, Mankato)
This is the second of two math content courses for intending elementary teachers; it follows up on Math 201. The course focuses on geometry, data, probability and algebra.
Math 483 Advanced Perspectives on 5-8 Mathematics (at MSU, Mankato)
This course focuses on the math content and pedagogy relevant to the middle grades for secondary licensure students. There is a special emphasis on curriculum and planning, and on learning to think and solve problems like a middle school student.
ME 5821 Teaching Probability and Statistics (University of Minnesota)
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Hi, a group of teachers from Hastings Middle School attended the CMP conference last February and were given your website in order to download files for the interactive whiteboards. We are especially interested in the Wumps and Tupelo Township activities that were shown at the conference. When we tried to get to the activities at your website, it said that the file could not be found. Is there any way you could let us know how we could get your CMP whiteboard/smartboard activites? |
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