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Foundation Mathematics and Statistics, 1st Edition
Foundation Mathematics and Statistics provides the reader with a firm understanding of the maths and stats they will need for a computing degree or diploma. The book will give the reader competency in a range of mathematical tools required for technical subjects, and the confidence they will need in the classroom. Explanations of mathematical tools are supported by real world examples to make this subject accessible. Graded exercises enable the reader to practice and revise each topic.
Starting with the basics of arithmetic and algebraic manipulation, the book covers everything from exponentials to logarithms. Providing a general grounding in proportions, ratios and percentages, this book will also help readers to understand probability and set theory. Finally, coverage includes the summary and presentation of statistical data and the drawing of histograms | 677.169 | 1 |
MGraw-Hill SRA Explorations and Applications
christy7177
Reviewed on Thursday, November 20, 2008
Grades Used: 7th grade
Dates used: 2008
My daughter is in the 7th grade. She was in public school (until recently). She was doing alg 1 and failing miserablly. I was shocked as I worked with her. There was just so much material that she had not yet mastered (that should definately be mastered before alg 1). Some of the things that came up was "What is half of 100%?" and "What does 3/4 plus 1/2 equal?" She couldn't answer. I decided to test her to see where she was at in math. She couldn't do anything from fractions on. The school refused to put her back a grade, tutor her... nothing. I decided to take her out for the year to try and catch her up. I have looked into everything and bought so much stuff for Math. Manipulatives, MUS, TT, Passport to mathematics, Life Of Fred... along with websites, Cosmeo, Netflix's math videos... But, this is one of my favorites. The Scope and Sequences was just perfect. Not too easy and not too advanced. It has real life problems (applied math), regular mechanical problems, math games and is a large colorful book. It pleasing just to look at and doesn't feel threatening like some math books can feel. We are using the 6th garde version because I felt it best suited my daughter. I am just extremely impressed with the book and very much recommend this! It is better then anything I have found so far.
janine
Reviewed on Monday, May 5, 2003
Grades Used: 4th and 5th
Dates used: 2002-2003
This is an excellent curriculum, especially for homeschoolers. It uses a lot of games and life situations while asking the student to think for themselves.
Concepts are revisited in different contexts and word problems are integrated so well that students who have trouble with them soon stop noticing that they're even doing them!
Eventually the homeschooling community will discover this treasure. We've been using it for two years now and plan to continue. Readily available online used (so don't let the price of the Teacher's Edition stop you.)
L
Reviewed on Sunday, May 4, 2003
Grades Used: 2nd and 5th Grades
Dates used: Nov-present day
I love MGraw-Hill kids! My youngest uses the Little Critter version, my oldest's version is colorful, but more mature, (of course). The books are fun, and seem to be thorough. I can buy them, (inexpensively), at any Staples Store. (Be sure to get a Teachers Rewards card for Staples! ) I am very happy with this program.
Shannon in SC
Reviewed on Tuesday, July 18, 2000
Grades Used: 2
Dates used: 2000
This worktext is wonderful if you like colorful full curriculum workbook style math for your child. It is very similar to Scott Foresman Math which we used last year (1st grade level), but is is more challenging. The workbook has 160 daily lessons. With Scott Foresman there were 16 chapters of varying length so I was never sure how much to cover each day. SRA's daily lessons eliminates that problem. Teachers manuals are not necessary in the younger levels. We really like this. | 677.169 | 1 |
Quadratic Functions- Goal-directed Instructional Design Plan
Quadratic Functions- Goal-directed Instructional Design Plan
The goal that will be addressed during this lesson plan is to provide the concepts, methods, and tools for understanding characteristics and transformations of graphs from their parent functions using a goal-directed instructional design plan.
The goal that will be addressed during this lesson plan is to provide the concepts, methods, and tools for understanding characteristics and transformations of graphs from their parent functions using a goal-directed instructional design Quadratic Functions- Goal-directed Instructional Design Plan to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Quadratic Functions- Goal-directed Instructional Design Plan
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Discussion
Kevin Karkkainen
(Teacher (K-12))
Your lesson does a great job of including a wide variety of teaching practices and assesment options. I think this will give you the opportunity to reach a large majority of students in your class no matter what their learning style. I also really like the use of manipulatives in your lesson.
Technical Remarks:
I would have liked to see the handouts to see what they looked like. Other educators might want to use them too!
Time spent reviewing site:
10min
4 years ago
Leslie Sniegowski
(Teacher (K-12))
This lesson plan does a great job of incorporating direct classroom instruction, peer collaboration, and a variety of assessment options. I really liked the inclusion of the "Guess Who" game which allows students to work with other classmates and interact with the material they are learning. The lesson plan accomodates a variety of learning types to make sure that all students in the classroom have the same access to the learning material.
Technical Remarks:
None.
Time spent reviewing site:
15 minutes
4 years ago
Deborah Johnson
(Teacher (K-12))
Your lesson chooses to use a variety of teaching method, maximizing on the Universal Design for Learning. I saw in your lesson that students participate in whole class discussion where I assume you lecture and provide examples for students, they then have an opportunity to have hands –on learning with the manipulatives, followed by eventually using the applet online to cement learning and "Guess Who" collaboration game. Providing students with so many different opportunities for learning will keep students engaged and also maximize on all learning styles. Good job!
Technical Remarks:
I noticed that you wrote that the lesson plan had worksheets and an applet that went along with the lesson on the teacher website. By any chance could this link be provided on MERLOT so that educators could actually use and implement this lesson? It sounds like a great UDL resource and would be nice to be shared fully. | 677.169 | 1 |
9780226317199 / 0226317196
Shipping prices may be approximate. Please verify cost before checkout.
About the book:
In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe adopts a hands-on approach, illustrating key concepts with numerous concrete examples.
The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing way, with an emphasis on finitely generated versus finitely presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group." Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields 0226317196 Publisher: University of Chicago Press | 677.169 | 1 |
A Model for Mathematics Education Reform. A four-year, problem based mathematics curriculum for high schools, to meet the needs of both college-bound and non-college-bound students. IMP is designed to replace the four-course sequence typical of most high school mathematics programs with an integrated course. While the traditional Algebra I-Geometry-Algebra II/Trigonometry-Precalculus structure emphasizes rote learning of mathematical skills and concepts, the IMP curriculum is problem-based, consisting of four-to-eight-week units that are each organized around a central problem or theme. Motivated by this central focus, students solve a variety of smaller problems, both routine and non-routine, that develop the underlying skills and concepts needed to solve the central problems in that unit. | 677.169 | 1 |
Mazen Shahin
Applied mathematics has increasingly come to the forefront in research, conferences, in teaching, and with our students. Mathematical biology in particular is a growing field of interest among many mathematicians, so the timing of Explorations of Mathematical Models in Biology with MATLAB® couldn't have been better. With the use of MATLAB® throughout the book, the reader is introduced to a wide variety of biological data and models, which are then translated into mathematical models and analyzed. The main goal of Shahin's book is to help students create models for problems in biology, ecology, and the environmental sciences. With the increasing interest in mathematics and biology and publications such as Math & Bio 2010: Linking Undergraduate Disciplines, there will be an increasing need to know how to model such data and to be able to get our students ready for related professions.
The approach Shahin takes throughout the book is to look at the applications using difference equations and matrices in MATLAB®. This is a very elementary approach but it works well. Each example is very clear and easy to follow. When reading through the examples, one can tell that Shahin used lots of time and care writing up each solution. When analyzing the models, Shahin's preference is to rely on graphical and numerical , rather than theoretical, techniques.
Using MATLAB® was a good decision, as many of the calculations are long. For example, in "Exploration 4.1: A Population Movement Model," once the MATLAB® functions are written to iterate the general form for the two-state Markov Chain, the example goes on to ask about what will happen after 100 years. This is where MATLAB® really comes in handy along, especially the resulting graphs.
The only negative about MATLAB®, which of course affects any computer program, is to learn how to setup the various programs. Shahin has very clear examples, however, which provide a solid basis for the introductory steps of using MATLAB®. For example, on pages 242–247, "Model 4.3: Plant Population Dynamics" is a great example of how a biology can be modeled using a second-order linear difference equation and how MATLAB® is used to help in the calculations of the eigenvalues and eigenvectors.
Reordering of some of the chapters and sections might have improved the book, as would having some group discovery projects. The chapter on "Modeling with Matrices" might fit better before the introductory chapter on difference equations. Also, sections 3.4–3.6 could be split off from chapter 3 and made into a new chapter before chapter 4. Group projects could ask students to obtain data and analyze it; these might also go into a supplemental guide for instructors to use in the classroom with students.
Another area of concern with the book is getting students access to MATLAB®. There is a 30-day trial version and a student edition costs $99.00, which could be high in price for some students. The publishers might want to look into a way to offer a combined package of MATLAB® with the book.
Overall, the book is a great resource to use across many diverse fields. I see the book being used in a mathematical biology class, as an independent study, or in an honors course. There are many modern day problems presented in the book, such as the SIR Model of Infectious Diseases, which our students need and want to see.
Students have always wanted to know the answer to the question, "How will I use this in the real-world?" As important as this question was years ago, it is even more important today, as technology has consumed us and we seem to be always running to catch up. Students want to know applications more than ever before. This book is a great example of how we, as instructors, can provide our students with more applications to be better prepared for the future.
Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He has written several book reviews for the MAA and his research fields are mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks, along with online homework various software. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes. | 677.169 | 1 |
This algebra program includes several helpful functions for algebra II classes. Functions include: an option to solve a system of equations with any number of variables, the quadratic formula with solutions and graph details, a trinomial factoring option, an option for compounded interest that solves for the balance, principal, rate, or time in years, an option for continuously compounded interest that solves for the balance, principal, rate, or time in years, an option to find the standard deviation and mean of a set of data, and an option to expand a binomial using Pascal's triangle. | 677.169 | 1 |
Product Description
This Saxon Algebra 1/2 kit includes the hardcover student text, softcover answer key & softcover test booklet, and the solutions manual. Containing 123 lessons, this text is the culmination of pre-algebra mathematics, a full pre-algebra course and an introduction to geometry and discrete mathematics. Topics covered include prime and compositeThe Solutions Manual features solutions to all textbook practices and problem sets. Early solutions contain every step, while later solutions omit obvious steps. Final answers are in bold type for accurate, efficient grading.
I love Saxon. Have taught it to my kids since they were in 1st grade. But this set of books wasn't well organized. Part of the answers are in one book, part in another. I still haven't found the explanation for the practice sets. Usually Saxon is great! What happened with this level?
This saxon is confusing, too repetitive in some areas, has insufficient instruction in other areas, and is NOT A MASTERY program.
There are too many problems per lesson. An "average" student cannot complete the lesson in an hours time. Students are left feeling that they are "poor" math students.
BETTER CHOICE would be video text algebra (it also has a pre algebra component) - i would HIGHLY recommend Video Text, which is a MASTERY program that teaches students the "why" of what they are learning | 677.169 | 1 |
Permutation groups are one of the oldest topics in algebra. Their study has recently been revolutionized by new developments, particularly the Classification of Finite Simple Groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This text summarizes these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many examples of applying the Classification of Finite Simple Groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.
Reviews:
"...an up-to-date study about this subject that is mainly aimed at beginning graduate students. Its style is compact, and it is comprehensive, easy to read and capable of motivating its readers | 677.169 | 1 |
Mathematics textbook now available on Scholar Works
All mathematics majors and minors at Grand Valley are required to take MTH 210: Communicating in Mathematics. The course is intended as an introduction to mathematical proofs and serves as a transition from the problem solving calculus courses to the more theoretical upper-level courses.
Because the course is a supplemental writing skills course (SWS), Ted Sundstrom said it was difficult to find a suitable textbook to accompany the course in the 1990s. According to Sundstrom, professor of mathematics, there were very few texts that taught writing in mathematics. Another problem, he said, during that time was that the books were not easy to use in a course that emphasized active learning. In the late 1990s, he decided to write a book to address those issues.
His text, "Mathematical Reasoning, Writing and Proof," contains four features new to most mathematic textbooks: writing and instruction to write mathematical proofs, instruction to construct proofs, active learning in mathematics, and use of elementary number theory and congruence arithmetic throughout the text.
Pearson Education Inc. published the first two editions of the text but the third edition is now available on Grand Valley's Scholar Works as a free open-source text and has been approved by the American Institute of Mathematics. Since the text launched in 2013, it has been downloaded more than 2,200 times. A print on-demand copy of the text is also available for purchase on Amazon.com.
Sundstrom joined Grand Valley in 1973. He has published numerous works, including, "Abstract Algebra: An Inquiry-Based Approach" (coauthored with Jonathan Hodge and Steven Schlicker), "Exploring Calculus with a Graphing Calculator" (coauthored with Charlene Beckmann), and he has been published in "The Mathematics Teacher" and the "Journal of Algebra." | 677.169 | 1 |
Hello, I am just getting restarted with mathematics as an adult. Although it was always easy, I hated math when I was a kid/in school. I have recently decided to give it an honest second chance to see if I have any interest for a possible career change. So I got Algebra II for Dummies. I am already running into the same problem I had in school that teachers would never answer. Man would I be excited if someone could really answer some these questions for me. I just went over the review for factoring. Understanding is no problem, and even figuring out the answer before reading the explanation was kind of fun I have to admit. But I am still left with "OK, but what does this do for me? Why does factoring enable me to accomplish? When/where would this actually be applied?"
I might actually like this afterall if I could figure out an actual application for all of it. Sorry if I sound like a dork!
December 19th 2010, 05:23 PM
dwsmith
You never stop factoring in math. It is useful for all math courses you take.
Thanks for the reply, mate, but that isn't what I am looking for. I mean, what does factoring enable a person to do? If factoring itself has no real purpose or value by itself, but only used to prepare for higher level math, that is fine. But then, why can't that just be explained with an example of some higher level math and what exactly that would be used for? Basically I found the whole concept of math to be a form of mental masturbation. I would really like to change my mind about that.
December 19th 2010, 05:33 PM
dwsmith
Suppose you are putting up a parabolic dome.
You have the equation for the parabola but don't know where it should intersect the ground. If you factor your equation, you can obtain the intercepts.
December 19th 2010, 06:24 PM
pickslides
Also when finding a min/max of a function you will need to know how to factor.
December 19th 2010, 07:51 PM
Djeayzonne
Thanks for that real-world application! Any chance I could see that in action somewhere? | 677.169 | 1 |
Employability Skills: Brush Up Your Electronics!< Less
Employability Skills: Brush Up Your Electrical up your Maths. Now published as a portable, learning, reference and subject revision guide, students, teachers and hobbyists can have their own low-cost portable version... More > Up Your MechanicsPublished eight times a year it provides schools, colleges, universities and other educational buyers worldwide with a single access point for all their Design Technology needs.
Towards the back of this eBook you will find a list of advertisers and their websites where suppliers offer additional discounts for schools regular purchases. Each issue features a particular aspect of the use of technology in education. In just forty pages it is impossible to cover all the latest developments in design technology. We have therefore chosen page titles that will point you in the direction of further information and how it can be applied to teaching technology in education eBook titles with accompanying software can be found from the link above.< Less
GCSE Maths Teachers Pack V10. eBook covers all the topics of this popular software title used in schools and colleges worldwide for over twenty years.
See Additional Notes for instructions to... More > download the highly interactive PC unlimited user educational software site licence. The software is designed to work as an interactive textbook on your PC screen. Comprising hundreds of menu selected colorful topics where the graphic images (from your eBook) are brought to life for every value change along with many additional learning software features.
Chapters (250 topics):- Number Systems, Number Conversion, Number Types, Compound Measures, Roots, Angles and Parallels, Triangle Ratios, Triangle Angles, Percentages, Ratios, Fractions, Vectors, Geometry, Circle Angles, Area, Surface Area and Symmetry, Volume, Laws, Algebra, Mathematical Rules, Powers and Indices, Simplifying, Linear Equations, Graphing, Slope and Translation, Curves and Angle Conversion, Data Analysis, Additional Notes.< Less
GCSE Maths V10 - eBook covers all the topics of this popular software title used in schools and colleges worldwide for over twenty years and now available as a Tablet and Smartphone App.
See... More > Additional Notes at the back of the book for instructions to download the accompanying interactive App which brings the 250+ topics to life by allowing you to insert your own values. Visually on a phone or tablet it looks almost identical to the eBook content, except you can input your own values which update the graphics and calculations to reflect these changes. There is also an optional comprehensive PC version to download with even more features both applications can be unlocked with your eBook purchase receipt for no additional charge.
A combined eBook, App and PC educational package at a tiny fraction of the previously published pricePublished eight times a year it provides schools, colleges, universities and other educational buyers worldwide with a single access point for all their Design Technology needs. education technology eBooks | 677.169 | 1 |
College Algebra with Modeling and Visualization, CourseSmart eTextbook, 4th Edition
Description
Gary Rockswold teaches algebra in context, answering the question, "Why am I learning this?" By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswold's focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book. | 677.169 | 1 |
Search Results
This series of lectures, created by Salman Khan of the Khan Academy, focuses on topics covered in a first year course in differential equations. A basic understanding of differentiation and integration from Calculus...
This collection, created by Salman Khan of the Khan Academy, features videos on geometry. A basic understanding of Algebra I necessary to understand the fundamental elements featured in this collection. Altogether, the...
This series of videos, authored by Salman Khan of the Khan Academy, features non-trigonometry pre-calculus topics. A solid understanding of all of the topics in the "Algebra" playlist should make this playlist pretty...
This series of lectures, created by Salman Khan of the Khan Academy, features topics covered in the first two or three semesters of college calculus; Everything from limits to derivatives to integrals to vector...
This math unit from Illuminations asks students to inscribe and circumscribe regular polygons. They will then analyze the area of the polygons, using either an applet or a graphing calculator. The unit includes two... | 677.169 | 1 |
This volume consists of a collection of survey articles by invited speakers and original articles refereed by world experts that was presented at the fifth China–Japan–Korea International Symposium. The survey articles provide some ideas of the application as well as an excellent overview of the various areas in ring theory. The original... more...
Karoubi 's classic K-Theory, An Introduction is to provide advanced students and mathematicians in other fields with the fundamental material in this subject . K-Theory, An Introduction is a phenomenally attractive book: a fantastic introduction and then some. serve as a fundamental reference and source of instruction for outsiders who would be... more...
Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing... more...
Just the facts (and figures) to understanding algebra.
The Complete Idiot's Guide® to Algebra has been updated to include easier-to-read graphs and additional practice problems. It covers variationsof standard problems that will assist students with their algebra courses, along with all the basic concepts, including linear equations and inequalities,... more... | 677.169 | 1 |
The math skills needed for a successful foodservice career—now in a new edition Culinary Calculations, Second Edition provides the mathematical knowledge and skills that are essential for a successful career in today's competitive foodservice industry. This user-friendly guide starts with basic principles before introducing more specialized... more... | 677.169 | 1 |
Shipping prices may be approximate. Please verify cost before checkout.
About the book:
The goal of Elementary and Intermediate Algebra: Concepts and Applications, 4e is to help todays students learn and retain mathematical concepts by preparing them for the transition from skills-oriented elementary and intermediate algebra courses to more concept-oriented college-level mathematics courses, as well as to make the transition from skill to application. This edition continues to bring your students a best-selling text that incorporates the five-step problem-solving process, real-world applications, proven pedagogy, and an accessible writing style. The Bittinger/Ellenbogen/Johnson series has consistently provided teachers and students with the tools needed to succeed in developmental mathematics. This revision has an even stronger focus on vocabulary and conceptual understanding as well as making the mathematics even more accessible to students. Among the features added are new Concept Reinforcement exercises, Student Notes that help students avoid common mistakes, and Study Summaries that highlight the most important concepts and terminology from each chapter.
Used books: 1 - 25 of 25
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Jt Media & Textbooks via United States
Softcover, ISBN 0321205081 Publisher: Addison Wesley, 2003 Used - Good, Usually ships in 1-2 business days, 3rd edition as listed. *NO MyMathLab access code!* Access code is one-time use, and it has been used and is NOT included. Only NEW books would have this access code. Used books have used access codes Textbook only. tight binding and there could be highlighting and markings. Book shows wear but still great for class. Check out my feedback, and buy with confidence Good. US Edition...
Softcover, ISBN 0321205081 Publisher: Addison Wesley, 2003 Used - Good. 0321205081-Wesley, 2002321205081 Publisher: Addison Wesley Longman, 2003 Used - Good, Usually dispatched within 1-2 business days, This Book is in Good Condition. Clean Copy With Light Amount of Wear. 100% Guaranteed.
Softcover, ISBN 0321205081 Publisher: Addison Wesley, 2003 Used - Very Good, Usually ships in 1-2 business days, Very Good condition! Huge seller with millions of transactions! Satisfaction Guaranteed! | 677.169 | 1 |
Louise Huttner of Burlington County College designed this activity to help students learn and practice their calculus skills. Students will collect data on the monthly temperature of a city. They will then graph and...
Calculus may seem to be quite dismal to some, but it comes alive through the fine work of Gabriela R. Sanchis. Sanchis wrote this excellent piece on teaching calculus by drawing on the historical evolution of some of...
Crafting new instructional aids for math courses can be difficult, but this helpful new article from Markus Hohenwarter and Judith Preiner will bring much joy to the hearts of mathematics teachers everywhere. This... | 677.169 | 1 |
In this course, the second in the calculus sequence, integral calculus is investigated along with an introduction to infinite series. Subjects studied include Riemann sums, the definite integral and the Fundamental Theorem of Calculus, Techniques of Integration, area between curves, disk and washer method for computing volumes, improper integrals, trapezoid rule. Introduction to sequences and series, p-series, ration test for convergence, power series, Taylor series and representation of a functions as a power series. | 677.169 | 1 |
Introduction To Graph Theory
9780073204161
ISBN:
0073204161
Pub Date: 2004 Publisher: McGraw-Hill College
Summary: Written by one of the leading authors in the field, this text provides a student-friendly approach to graph theory for undergraduates. Much care has been given to present the material at the most effective level for students taking a first course in graph theory. Gary Chartrand and Ping Zhang's lively and engaging style, historical emphasis, unique examples and clearly-written proof techniques make it a sound yet acc...essible text that stimulates interest in an evolving subject and exploration in its many applications.This text is part of the Walter Rudin Student Series in Advanced Mathematics.
Chartrand, Gary is the author of Introduction To Graph Theory, published 2004 under ISBN 9780073204161 and 0073204161. One hundred ninety one Introduction To Graph Theory textbooks are available for sale on ValoreBooks.com, and six used from the cheapest price of $18.50 | 677.169 | 1 |
JIRÍ ANDEL is Vice Dean of the Faculty of Mathematics and Physics at Charles University, Prague, Czech Republic. He is the author of eighty-five scientific papers and four books written in Czech, two of which have been translated into German. Mathematics of Chance is the first of his books to be translated into English. more...
Use these hands-on activities to introduce students to a variety of mathematical concepts that are directly tied to the NCTM Standards and to recent research in the field of math instruction. Parent letters, attitude surveys, group rating sheets, and a student response journal are included in a special assessment section. Grades 3-7. (Great for the... more...
A solutions manual to accompany Statistics and Probability with Applications for Engineers and Scientists Unique among books of this kind, Statistics and Probability with Applications for Engineers and Scientists covers descriptive statistics first, then goes on to discuss the fundamentals of probability theory. Along... more...
Exploring Probability in School provides a new perspective into research on the teaching and learning of probability. It creates this perspective by recognizing and analysing the special challenges faced by teachers and learners in contemporary classrooms where probability has recently become a mainstream part of the curriculum from early childhood... more...
Exercises and Solutions in Statistical Theory helps students and scientists obtain an in-depth understanding of statistical theory by working on and reviewing solutions to interesting and challenging exercises of practical importance. Unlike similar books, this text incorporates many exercises that apply to real-world settings and provides much... more...
What are the chances of a game-show contestant finding a chicken in a box? Is the Hanukkah dreidel a fair game? Will you be alive ten years from now? These are just some of the one-of-a-kind probability puzzles that acclaimed popular math writer Paul Nahin offers in this lively and informative book. Nahin brings probability to life with... more...
Mastering Mathematics provides flexible online and print teaching and learning resources. The service focuses on strands within the curriculum to improve progression throughout Secondary Mathematics. Mastering Mathematics Student Books and eBooks are organised into progression strands in line with Mastering Mathematics Teaching and Learning Resources:.... more... | 677.169 | 1 |
have developed techniques and methods that facilitate learning Algebra and try to make it interesting by showing some application in real life. Algebra 2 is a very important subject and has the foundation for all future math. If this subject is mastered, students can learn easier higher levels of math. | 677.169 | 1 |
Algebra 2, Study Guide & Intervention Workbook / Edition 1
Study Guide and Intervention/Practice Workbook provides vocabulary, key concepts, additional worked out examples and exercises to help students who need additional instruction or who have been absent.See more details below | 677.169 | 1 |
Hundreds
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In addition to Maths tuition, I also now offer one-to-one training on
some of the fundamentals of this impressive program, whether you're
just starting to use it, or want ideas for how to get the most from
your spreadsheet.
Teacher Training:
During the course of my PGCE teacher training year, I wrote a number of
short essays, and two rather more substantial ones, on different
aspects of teaching and learning. They are available here.
Twitter: Stay informed about updates and new resources by following me: @the_chalkface | 677.169 | 1 |
Solutions Manual for use with Elementary Algebra
Elementary Algebra Math Zone Student Access Card
Elementary Algebra w/MathZone
Elementary Algebra
Elementary Algebra
Elementary Algebra w/ Mathzone
Summary
Video Lectures (DVD):In the videos, qualified teachers work through selected exercises from the text book, following the solution methodology employed in the text. The video series is available on DVD or online as an assignable element of MathZone. The DVDs are closed-captioned for the hearing impaired, subtitled in Spanish, and meet the Americans with Disabilities Act Standards for Accessible Design. Instructors may use them as resources in a learning center, for online courses, and/or to provide extra help students who require extra practice. | 677.169 | 1 |
Additional Details
This is an outstanding math program for the math-gifted student. It is rigorous and oriented to the independent problem-solver. The texts are based on the premise that students learn math best by solving problems lots of problems and preferably difficult problems that they don't already know how to solve. Most sections, therefore, begin by presenting problems and letting students intuit solutions BEFORE explaining ways to solve them. Even if they find ways to answer the problems, they should read the rest of the section to see if their answer is correct and if theirs is the best or most efficient way to solve that type of problem. Textual instruction, then, is given in the context of these problems, explaining how to best approach and solve them. Throughout the text there are also special, blue-shaded boxes highlighting key concepts, important things to retain (like formulas), warnings for potential problem-solving pitfalls, side notes, and bogus solutions (these demonstrate misapplications). There are exercises at the end of most sections to see if the student can apply what's been learned. Review problems at the end of each chapter test understanding for that chapter. If a student has trouble with these, he should go back and re-read the chapter. Each chapter ends with a set of Challenge Problems that go beyond the learned material. Successful completion of these sets demonstrates a high degree of mastery. A unique feature in this series is the hints section at the back of the book. These are intended to give a little help to selected problems, usually the very difficult ones (marked with stars). In this way, students can get a little push in the right direction, but still have to figure out the solution for themselves. The solution manuals do contain complete solutions and explanations to all the exercises, review problems and challenge problems. It is best for students not to access these until they have made several attempts to solve the problems first. I particularly like one of the motivating boxes in the text that coaches, "If at first you don't know how to solve a problem, don't just stare at it. Experiment!". That pretty much sums up the philosophy of the course, encouraging children to take chances, become aggressive problem solvers, and attack problems with confidence. I wonder how far some children would go if they were encouraged this way instead of being spoon fed?
Though this course is used in classroom settings, the texts are student-directed, making them perfect for the independent learner or homeschooler. Students should start the introductory sequence with the Prealgebra book. Afterwards, begin the Introduction to Algebra. Students will be prepared for both the Introduction to Countingand Probability and Introduction to Number Theory courses after completing the first 11 chapters of Algebra. It won't matter whether they do these along with Algebra, put aside Algebra and complete the other two or finish Algebra first and then do them. All of them should be completed prior to the Introduction to Geometry book. If you are coming into this course from another curriculum, you will probably want to take the Algebra diagnostic test to decide where to enter this program. Even if your student has finished Algebra 2 elsewhere, you will want to make sure that all of the material from this series has been covered before continuing on to the Intermediate series.
. Taken together, these constitute a complete curriculum for outstanding math students in grades 6-10 and one that prepares them for competitions such as MATHCOUNTS and the American Mathematics Competitions. The material is challenging and in-depth; this is not a course for the mathematically faint of heart. If your child loves math, is genuinely math-gifted, or is interested in participating in math competitions, you definitely need to give this one serious consideration.
Customer Reviews
I found this math program last year for my 5th grade math whiz-kid (We don't know where that one child got it from!) After skipping two years of Saxon and still being bored I decided to look at something else for him and this fit the bill perfectly It was challenging and economical and the online support was amazing Using Art of Problem Solving's website we were able to access fabulous videos for each lesson My son loved the instructor because he was both clear and engaging Furthermore they have an online learning system--Alcumus--that is currently free We used this as a supplement to the book and although it does not cover all the material in the text it was a great help for the sections that overlap Finally the answer manual is very well done with detailed solutions explained thoroughly which was a good thing since I needed help remembering things I haven't learned in years Overall we were very very pleased with this text and the supporting website features We can't recommend it highly enough! | 677.169 | 1 |
Description
This non-traditional introduction to the mathematics of scientific computation describes the principles behind the major methods, from statistics, applied mathematics, scientific visualization, and elsewhere, in a way that is accessible to a large part of the scientific community. The authors use many hands-on examples to provide the reader with the tools and insight necessary for the effective and intelligent use of Mathematica or other software packages for this purpose.
Introductory material includes computational basics, a review of coordinate systems, an introduction to facets (planes and triangle meshes), and an introduction to computer graphics. The scientific computing part of the book covers topics in numerical linear algebra (basics, solving linear system, eigen-problems, SVD, and PCA) and numerical calculus (basics, data fitting, dynamic processes, root finding, and multivariate functions). The visualization component of the book is separated into three parts: empirical data, scalar values over 2D data, and volumes. | 677.169 | 1 |
Mathematical translations
This lesson develops knowledge of algebraic expressions and their verbal equivalents. Students will establish a foundation for future Algebra I tasks by identifying mathematical symbols and expressions through group work and individual tasks. This lesson contains modifications for the novice high English Language Learner (ELL).
A lesson plan for grades 6–12 English Language Development and Mathematics
Please excuse my dear Aunt Sally: StudentsModifications
Place an ELL student with a native English speaker who will be beneficial to the ELL student's learning to work together in a group or place an intermediate or an advance ELL student with a novice student to work together in a group.
After completing the mathematical expression chart, read the expressions aloud to the student.
Decrease the number of questions to be answered on the homework assignment but remember to incorporate all the concepts for this lesson.
Alternative assessments
On the chapter test give students multiple choice questions or matching questions instead of short answer questions where spelling can be an issue.
Allow students extended time on the test.
Give the students an oral section of the test so they can verbally convey the answer instead of having to write the answer.
Critical vocabulary
Addition
Subtraction
Division
Multiplication
Exponents
Equal
Translation
Algebraic expression
Comments
This lesson plan was developed during the English Language Development Standard Course of Study lesson planning institutes hosted by the North Carolina Department of Public Instruction and LEARN NC, June and July, 2004. It includes specific strategies, instructional modifications, and alternative assessments, which make this lesson accessible to limited English proficient students. Please note that this lesson has been aligned with the goals and objectives of the North Carolina English Language Development standards.
Common Core State Standards
Mathematics (2010)
Grade 6
Expressions & Equations
6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Grade 7
7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Solve word problems leading to equations of the form px + q = r and p(x...
High School: Algebra
Creating Equations
ALG.CE.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
North Carolina curriculum alignment
English Language Development (2005)
Objective 0.03: Record information on familiar topics and academic areas using graphic organizers.
Mathematics (2004)
Grade 7
Objective 5.02: Translate among different representations of algebraic expressions, equations and inequalities.
Grade 8
Goal 5: Algebra - The learner will understand and use linear relations and functions.
Objective 5Grade 9–12 — Introductory Mathematics
Goal 4: Algebra - The learner will understand and use linear relations and functions.
Objective 4LEARN NC, a program of the UNC School of Education, finds the most innovative and successful practices in K-12 education and makes them available to the teachers and students of North Carolina - and the world. | 677.169 | 1 |
Lecture, discussion, student presentation group work, and
use of a computer algebra system.
Entrance
Skills and Knowledge:
List the required skills
and/or knowledge without which a student would be highly unlikely to receive a grade of A, B,
C, or Credit (or for Health and Safety, would endanger self or others) in the
Target Course. | 677.169 | 1 |
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Summary
This books aims to make mathematics accessible to non-experts and the lay reader. Providing an entertaining overview of the subject, the text includes several fascinating mathematical conundrums. The author's writing ability is second to none and Acheson provides a clear, unpatronising explanation of mathematics. The book contains several cartoons, sketches and photos making it entertaining and compulsive reading.
Classification
Dewey Decimal: 511
Book Details
Physical Description: 208 pages
Edition Info: Paperback; 2004-11 | 677.169 | 1 |
Description
Contains Mathematica-based learning modules on topics traditionally covered in the A-level syllabus of English and Welsh schools. Suitable for freshman level college calculus courses in North America.
The topics covered in the book roughly correspond to those traditionally studied in the English and Welsh mathematics A-level school syllabus. The first year undergraduate with mathematics A-level will find the book most useful for revision. In North American terms, the book is suitable for freshman calculus. An electronic supplement by Phillip Kent is included. Contents
Getting Started with Mathematica | Foundations (Functions & Graphs, Trigonometry, Sequences) | Calculus (Differentiation, Integration, Series) | Vectors and Matrices | Complex Numbers Related TopicsCalculus and Analysis | 677.169 | 1 |
Synopsis
Geometric measure theory provides the framework to understand the structure of a crystal, a soap bubble cluster, or a universe. Measure Theory: A Beginner's Guide is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Morgan emphasizes geometry over proofs and technicalities providing a fast and efficient insight into many aspects of the subject.
New to the 4th edition: * Abundant illustrations, examples, exercises, and solutions. * The latest results on soap bubble clusters, including a new chapter on "Double Bubbles in Spheres, Gauss Space, and Tori." * A new chapter on "Manifolds with Density and Perelman's Proof of the Poincaré Conjecture." * Contributions by undergraduates.
eBook Details
Academic Press, September 2008
ISBN:
9780080922409
Language:
English
Download options:
PDF (Adobe DRM)
You can read this item using any of the following Kobo apps and devices: | 677.169 | 1 |
Written by three gifted-and funny-teachers, How to Ace Calculus provides humorous and readable explanations of the key topics of calculus without the technical details and fine print that would be found in a more formal text. Capturing the tone of students exchanging ideas among themselves, this unique guide also explains how calculus is taught,... more...
A Course in Abstract Harmonic Analysis is an introduction to that part of analysis on locally compact groups that can be done with minimal assumptions on the nature of the group. As a generalization of classical Fourier analysis, this abstract theory creates a foundation for a great deal of modern analysis, and it contains a number of elegant results... more...
Numerical Analysis for Engineers: Methods and Applications demonstrates the power of numerical methods in the context of solving complex engineering and scientific problems. The book helps to prepare future engineers and assists practicing engineers in understanding the fundamentals of numerical methods, especially their applications, limitations,... more...
This book introduces the application of microeconometric methods for modelling various aspects of economic activity for small to large size enterprises, using methods that are based on both time-series and cross-section approaches. The information obtained from using these estimated models can then be used to inform business decisions that improve... more...
... more...
Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A 5 in them. The book surveys known facts about surfaces with an action of A 5 , explores A 5 -equivariant geometry of the quintic del Pezzo threefold V 5 , and gives a proof of its A 5 -birational... more... | 677.169 | 1 |
By Elizabeth G. Bremigan, Ralph J. Bremigan, and John D. Lorch
Mathematics for Secondary School Teachers, which is intended for prospective educators in middle and high school, balances discovery learning with direct instruction.
Written to develop a deeper understanding of the mathematics that will be taught, the book presents topics of central importance in the secondary school mathematics curriculum, notably, functions, polynomials, trigonometry, exponential and logarithmic functions, numbers and operations, and measurements.
Beyond the goals of conceptual understanding and computational fluency, readers of the book can devise their own mathematical explanations and arguments; create examples and visual representations; remediate typical student errors and misconceptions; and analyze students' work.
A broad collection of exercises of varying degrees of difficulty is integrated with the text. Instructors are able to emphasize the following:
Basics: The exploration of key precollege topics from intuitive and rigorous points of view;
Connections: The exploration of relationships among topics, using tools from college-level mathematics;
Extensions: The exploration of college-level mathematical topics that have a compelling relationship to precollege mathematics.
By taking into account the diverse mathematical backgrounds of preservice teachers and the varied goals of teacher preparation programs, the authors have written a textbook whose subject matter addresses a wide range of learning objectives appropriate for future teachers.
Mathematics for Secondary School Teachers can also be used in licensing programs; as a supplement to mathematics methods courses; as a text for graduate courses for in-service teachers; and as a resource for faculty development.
The word trigon refers to a three-sided figure, while metry means measurement. Thus trigonometry is the measurement of triangles, which is tantamount to studying the measurement of the relationships among side-lengths and angles. There is no doubt that trigonometry is useful. Human beings have been using triangles to make measurements (e.g., the height of Everest, the circumference of the earth, the distance of from the earth to the sun) for thousands of years. But why is trigonometry nontrivial? After all, we know how to measure angles and line segments, so what can be so hard about trigonometry?
As the authors state early on, this book is intended in part as a response to the 2001 report from the Conference Board of the Mathematical Sciences on the mathematical education of future teachers.
There is a very definite need for books like this one. While courses and textbooks on mathematics for elementary teachers are common across America, few schools offer a course for which this book would be a good fit, and that's probably regrettable. There is a good argument to be made for offering prospective secondary teachers the same kind of course — in which they consider the math they expect to teach from an advanced perspective and with some attention to how to teach it — that we routinely require of prospective elementary teachers. Continued... | 677.169 | 1 |
Give students a solid high school introduction to math with Horizons Algebra 1. The colorful student workbook reviews all foundational pre-algebra concepts before covering topics such as: common factors, exponents, radical expressions, writing linear equations from graphs, polynomials, binomials, FOIL method, quadratic inequalities, and more. Grade 8. The student workbook includes a set of lesson review boxes accompanied by questions that provide practice for previously taught concepts and the concepts taught in the lesson. Exploring Math Through... sections help students understand how ordinary people use algebraic math, providing concrete examples of how math is useful in life. Students will need to supply paper to work the problems. 333 pages, softcover. The teacher's guide includes...
Less
Introduce your 8th graders to Algebra 2 with AGS Publishing: Algebra 2 from Pearson Learning. The content provides learners of all abilities with essential preparation in problem solving, calculator usage, and application lessons that demonstrate how algebra is integrated with measurement Algebra 2 chapters will...
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Provide students with a college-prep Algebra II course that will allow them to easily progress onto even more difficult mathematical challenges. Saxon Algebra 2, 4th Edition prepares students for calculus through explicit embedded geometry instruction. Trigonometry concepts, statistics, and applications for other subjects such as physics and chemistry are also included. Incremental lessons include a Warm Up activity; New Concepts section that introduces new concepts through examples with sidebar hints and notes; and Lesson Practice questions with lesson reference numbers underneath the question number. Online connections are given throughout for additional help. Real-world applications and continual practice & review provide the time needed to master each concept, helping students to...
Less
Provide students with a college-prep math course that will give them the foundation they need to successfully move into higher levels of math. Saxon Algebra 1, 4th Edition covers all of the traditional first-year algebra topics while helping students build higher-order thinking skills, real-world application skills, reasoning, and an understanding of interconnecting math strands. Saxon Algebra 1 focuses on algebraic thinking through multiple representations, including verbal, numeric, symbolic, and graphical, while graphing calculator labs model mathematical situations. Incremental lessons include a Warm Up activity; New Concepts section that introduces new concepts through examples with sidebar hints and notes; and Lesson Practice questions with lesson reference numbers underneath the...
Less
Make math matter to students in all grades using Math Tutor: Algebra Skills! This 80-page book provides step-by-step instructions of the most common math concepts and includes practice exercises, reviews, and vocabulary definitions. The book covers factoring, exponents, variables, linear equations, and polynomials. It aligns with state, national, and Canadian provincial standards.
Introduce your 5th graders to Pre-Algebra with AGS Publishing: Pre-Algebra from Pearson Learning. The content provides learners of all abilities with essential preparation in problem solving, calculator usage, and application lessons that demonstrate how algebra is integrated with data Pre-Algebra chapters will...
Less skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations. Explanations and practical examples that mirror today's teaching methodsRelevant cultural vernacular and referencesStandar d For Dummies materials that match the current standard and design Basic Math & Pre-Algebra For Dummies takes the intimidation out of tricky operations and helps you get ready for algebra
Repetition, drills, and application exercises ensure mastery of computational skills with Lifepac Math. Students will progress to higher-level cognitive reasoning and analysis as their problem solving ability increases. Perfect for students who flourish in a self-paced, individualized learning format, each consumable Lifepac combines lessons, exercises, projects, reviews and tests. This set includes 10 Lifepac Workbooks and the Teacher's Guide. Grade 7. 2014 Edition. Subjects covered include: Integers Fractions Decimals Patterns and equations Ratios and proportions Probability and graphing Data Analysis Measurement and Area Surface Area and VolumePlease Note: This set is part one of a two-part curriculum series, with the second half of the Pre-Algebra and Pre-Geometry curriculum being...
Less
Make math matter to students in all grades using Math Tutor: Pre-Algebra Skills! This 80-page book provides step-by-step instructions of the most common math concepts and includes practice exercises, reviews, and vocabulary definitions. The book covers factoring, positive and negative numbers, order of operations, variables, exponents, and formulas such as perimeter, area, and volume. It aligns with state, national, and Canadian provincial standards.
Step-by-step, to-the-point lessons will help both students looking to incorporate supplemental teaching into their Saxon curriculum, as well as struggling students who need additional help. With one session per Saxon lesson, Art Reed presents the concepts taught in that chapter with clarifying examples. The actual problems from the textbook aren't used; students should read the text, watch the video, and work the workbook problems with a more fully-understood knowledge of the process. Set up with Mr. Reed using a whiteboard to illustrate examples in a classroom, students can easily pause or rewind if needed. Algebra 8/7, 2nd or 3rd Edition. DVDs come in plastic clamshell case.
Expand your students' mathematical knowledge with Learning Resources' Math Kit for Grade Two. The tools in this kit keep kids engaged and help them understand addition and subtraction, working with large numbers, measurements and | 677.169 | 1 |
Education
Language
"Waiting for Superman": What it Means for You and Your Child. Every president, from Johnson to Obama, has made big promises when it comes to "fixing" education in America.
And almost every parent, from then until now has asked themselves an essential question—"Is my child getting a good education?" Regardless of neighborhood or income, it's a concern that keeps parents up at night, and the answer rests at the heart of a national firestorm brewing over education. Fanning the flames is a new and controversial documentary, Waiting for Superman, which paints a grim picture of the education system in the United States today. Director and writer Davis Guggenheim, best known for his film An Inconvenient Truth, picks apart the issues holding American school children back; from government bureaucracy, to bad teachers who can't be fired, to a system that is out of touch with the needs of the global economy.
Waiting for Superman has raised a lot of fear and anger.
Ted
This is the first lecture in MIT's course 18.06, linear algebra, and I'm Gilbert Strang. The text for the course is this book, Introduction to Linear Algebra. And the course web page, which has got a lot of exercises from the past, MatLab codes, the syllabus for the course, is web.mit.edu/18.06. And this is the first lecture, lecture one. So, and later we'll give the web address for viewing these, videotapes.
The fundamental problem of linear algebra, which is to solve a system of linear equations. So let's start with a case when we have some number of equations, say n equations and n unknowns. So an equal number of equations and unknowns. That's the normal, nice case. And what I want to do is -- with examples, of course -- to describe, first, what I call the Row picture. So in a minute, you'll see lines meeting. The second picture, I'll put a star beside that, because that's such an important one. And maybe new to you is the picture -- a column at a time.
Okay, so can I do an example?
Fun
Cell Size and Scale. Some cells are visible to the unaided eye The smallest objects that the unaided human eye can see are about 0.1 mm long.
That means that under the right conditions, you might be able to see an ameoba proteus, a human egg, and a paramecium without using magnification. A magnifying glass can help you to see them more clearly, but they will still look tiny. Smaller cells are easily visible under a light microscope. It's even possible to make out structures within the cell, such as the nucleus, mitochondria and chloroplasts. To see anything smaller than 500 nm, you will need an electron microscope. How can an X chromosome be nearly as big as the head of the sperm cell? No, this isn't a mistake. The X chromosome is shown here in a condensed state, as it would appear in a cell that's going through mitosis.
A chromosome is made up of genetic material (one long piece of DNA) wrapped around structural support proteins (histones). Adenine.
Science
Physics. Astrophysics I « Physics Made Easy. Luminosity: Fλ is the radiative flux at the stellar surface.
Energy may be lost due to neutrinos or direct mass loss. Flux: At the Earth's surface, observed flux is Stellar flux Apparent magnitude, m, is based on the flux received at the Earth's surface, fν (flux at frequency ν). Fainter star has larger magnitude. Absolute magnitude, M, is defined using the flux we would see from a star if it was 10 parsecs distant, Fν. Bolometric magnitude is calculated using the total flux f integrated over all frequencies. apparent absolute Distance modulus, d: by taking the difference between the apparent and absolute magnitudes, a measure of the distance to a star can be found.
Distance measurements using trigonometric parallax: 'close' stars appear to move against the background of fixed stars, such that their positions appear different to an observer on Earth when observed at the two extremes of the Earth's orbit. Small angle know this, everything should be easier. d=distance to binary Kepler's Laws 3<ν<5 or iii)
18.06 Linear Algebra, Spring 2010. | 677.169 | 1 |
A Course in Linear Algebra with Applications / Edition 2…
See more details below
Overview is related to solutions of linear system of equations. Also a simplified treatment of Jordan normal form is given.
Numerous applications of linear algebra are described: these include systems of linear recurrence relations, systems of linear differential equations, Markov processes and the Method of Least Squares. In addition, an entirely new chapter on linear programming introduces the reader to the Simplex Algorithm and stresses understanding the theory on which the algorithm is based. The book is addressed to students who wish to learn linear algebra, as well as to professionals who need to use the methods of the subject in their own fields.
About the Author: Derek J.S. Robinson held positions at the University of London, the National University of Singapore and the University of illinois at Urbana-Champaign, where he is currently Professor of Mathematics | 677.169 | 1 |
If you are looking for a quick nuts-and-bolts overview, turn to Schaum's Easy Outlines! Schaum's Easy Outline of Mathematical Handbook of Formulas and Tablesis a pared-down, simplified, and tightly focused review of the topic. With an emphasis on clarity and brevity, it features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form. Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give you quick pointers to the essentials. Expert tips for mastering math formulas Last-minute essentials to pass the course Complete index to all topics Appropriate for the following courses: College Mathematics, Numerical Analysis, Calculus, Calculus II, Calculus III, Differential Equations, Probability and Statistics Clear and concise explanations of all procedures Formulas and tables for elementary to advanced topics Complete index to all topics | 677.169 | 1 |
Mableton Precalculuset B.
...When students get too submerged into problems, they tend to forget basic stuff. That is where reminding them of basic concepts becomes important. One of the examples is when students see a really long equation, they automatically think it involves some complex formulas and techniques while it can be just broken down with simple concepts | 677.169 | 1 |
What is Algebra II?
Algebra II is an extension of Algebra I. We'll still look at functions, graphs, and variables, but this time we'll step it up a notch.
One of the big boys in Algebra II is the quadratic equation. This guy seems pretty mean and tough at first, but it's really a teddy bear. It is, at least, once we use the magical quadratic formula on it. Poof. Done.
We'll also learn the ins and outs of the natural number and the natural log. By the end, dare we say, working through these types of problems will come naturally. | 677.169 | 1 |
Registration: Interim
Interim 2007
Mathematics
W81 Graph Theory. This course is an introduction to graph theory, a relatively "young" (after 1890) branch of pure mathematics that has many applications to other disciplines, most notably computer science. The course is primarily a problem-solving class with student work consisting of homework problems, in-class group work, and student presentations of solutions (both written and oral). There is some direct instruction but the most of the learning will be through active participation in problem-solving. Problems will range from relatively concrete (though sometimes difficult) counting and enumeration problems to quite abstract theorems that require careful proofs. Graph theory is a good area in which to develop theorem-proving skills as the objects, though abstract, are finite and relatively easy to understand and the subject matter does not have any particular mathematical prerequisite. Applications of graph theory may be introduced but are not the major focus of this course. Students are evaluated based on their problem solutions and presentations. The prerequisite for this course is at least one of Mathematics 301, 312, 351, or 361. The course satisfies the requirement in the mathematics major of one interim course. M. Stob. 8:30 a.m. to noon.
W82 Curricular Materials for K-8 School Mathematics. This course examines and evaluates K-8 mathematics curricula in the context of the NCTM Principles and Standards for School Mathematics . Although the emphasis this year will be on grades 6-8, curricula at all grade levels will be examined. Some of the curricula to be discussed are Everyday Mathematics, Investigations, Math TrailBlazers, Connected Mathematics, MathScape, MathThematics and Mathematics in Context. Students are expected to complete assigned readings, to participate in and lead sample activities and lessons, and to contribute to small-group and whole-class discussions of the materials under consideration. Evaluation is based on in-class participation, presentation of grade-level lessons, several written quizzes, and written projects. Optional K-8 classroom observations can be arranged for the morning hours. Students should arrange their schedules so that they can spend some additional hours in the Curriculum Center . Prerequisite: Mathematics 222. This course may replace Mathematics 110 in the elementary education mathematics minor for students who have completed four years of high school mathematics and who have received permission from their mathematics advisor. J. Koop . 2:00 p.m. to 5:00 p.m.
MATH-160 Elementary Functions and Calculus (core). This course is a continuation of Mathematics 159. Topics include applications of derivatives, integrals, the fundamental theorem of calculus, and applications of integrals. Grades are based on problem sets, tests, and a final exam. Prerequisite: Mathematics 159. T. Jager. 8:30 a.m. to noon and 2:00 p.m. to 5:00 p.m. | 677.169 | 1 |
This algebra unit from Illuminations includes three lessons which highlight Pick's Theorem. Students will be introduced to the theorem, determine the coefficients of the equation, and explore the concept of rates of... has students study equivalence and systems of equations while identifying and using related functions. It includes an interactive applet that can be used in class. Discussion questions are... | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Paul Wilmott introduces quantitative finance.
Paul Wilmott Introduces Quantitative Finance, Second Edition is an accessible introduction to the classical side of quantitative finance specifically for university students. Adapted from the comprehensive, even epic, works Derivatives and Paul Wilmott on Quantitative Finance, Second Edition, it includes carefully selected chapters to give the student a thorough understanding of futures, options and numerical methods. Software is included to help visualize the most important ideas and to show how techniques are implemented in practice. There are comprehensive end-of-chapter exercises to test students on their understanding.Read more... | 677.169 | 1 |
"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books
£41The writing is both rigorous and thorough, and the authors use compact presentations to support their explanations and proofs. Highly recommended. (N. W. Schillow, CHOICE)
--This text refers to an alternate Paperback edition.
About the Author
Ian Stewart is Professor of Mathematics at Warwick University, and Director of the Mathematics Awareness Centre at Warwick. An active research mathematician, he is also a well-known popularizer of mathematics and related areas of science.
Most Helpful Customer Reviews
'The Foundations of Mathematics' is essential reading for anyone starting a mathematics degree. Its clear and precise style give an excellent insight into areas discussed in many different areas of mathematics, and so makes good preparatory reading. This book would also be suitable for anyone studying 'A' level maths who needed further inspiration on topics with which they were sruggling, or were contemplating studying maths at university. However, they might find many of the topics covered were superflous to their needs, as a lot of the book it not contained in the 'A' level syllabuses.
I will admit that this review is based only on the Kindle sample, but that was enough for me. The style is very old-fashioned with an over-insistence on explaining every aspect of why we might need to learn maths through a progressive expansion of concepts. It seems tedious beyond belief and really needs to be rewritten in a more modern style with greater emphasis on examples. | 677.169 | 1 |
Trigonometry: A Unit Circle Approach, Books a la Carte Edition
by Michael Sullivan Publisher Comments
NOTE: This edition features the same content as the traditional text in a convenient, three-hole-punched, loose-leaf version. Books a la Carte also offer a great value—this format costs significantly less than a new textbook. Before purchasing... (read more)
Multivariable Calculus 3RD Edition
by William G Mccallum Publisher Comments
The new edition exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. * The conceptual and modeling problems, praised... (read more)
Concepts in Calculus, II
by Miklos Bona Publisher Comments
From the University of Florida Department of Mathematics, this is the second volume in a three volume presentation of calculus from a concepts perspective. The emphasis is on learning the concepts behind the theories, not the rote completion of... (read more)
Geometrical Methods in Variational Problems
by N. A. Bobylev Review
"... the book is a valuable contribution to the literature. It is well-written, self-contained and it has an extensive bibliography, especially with regard to the literature in the Russian language." (Mathematical Reviews, 2001a)... (read more)
First Course in Calculus 5TH Edition
by Serge Lang Publisher Comments
This fifth edition of Lang's book covers all the topics traditionally taught in the first-year calculus sequence. Divided into five parts, each section of A FIRST COURSE IN CALCULUS contains examples and applications relating to the topic covered. In... (read more)
The Riemann Zeta-Function: Theory and Applications
by Aleksandar Ivic Book News Annotation
In this reprint of his 1985 text, Ivic (U. of Belgrade) presents a
comprehensive account of the theory and applications of the Riemann
zeta- function, a type of function that is important for analytic
number theory. Starting with elementary theory... (read more)
Calculus (Rea's Problem Solvers)
by Rea Publisher Comments
Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More... (read more)
Precalculus : Functions and Graphs (3RD 09 - Old Edition)
by Mark Dugopolski Publisher Comments
Making the transition to calculus means being prepared to grasp bigger and more complex mathematical concepts. Precalculus: Functions and Graphs is designed to make this transition seamless, by focusing now on all the skills that you will need in | 677.169 | 1 |
MAS2104 : Introduction to Vector Calculus
Semesters
Aims
To present the basic mathematical methods needed in the formulation of both physical and mathematical problems involving vector and scalar quantities in 3D space.
Module Summary
The development of mathematical representations of physical and thought models, and their solutions, requires some basic mathematical tools. This course (with MAS2105) introduces the various ideas that are needed in order to describe and formulate problems in three dimensions. Thus MAS2104 provides the important links between the calculus and (3D) vectors: the vector calculus.
Jointly Taught With
Code
Title
MAS3104
Introduction to Vector Calculus
Teaching Rationale And Relationship
Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students' abilities at applying the theory to solving problems. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Office hours provide an opportunity for more direct contact between individual students and the lecturer90
1
A
90
unseen
Exam Pairings
Module Code
Module Title
Semester
Comment
MAS3104
Introduction to Vector Calculus
1
N/A
Other Assessment
Description
Semester
When Set
Percentage
Comment
Prob solv exercises
1
M
10
Written assignments and computer based assessments
Assessment Rationale And Relationship
A substantial formal unseen examination is appropriate for the assessment of the material in this module. Coursework assignments (approximately 5 assignments of approximately equal weight) allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this is thus formative as well as summative assessment. The coursework assignments may be written assignments, computer based assessments or a combination of the two, and in the case of combined assessments the deadlines for the two parts will not necessarily be the same | 677.169 | 1 |
College Algebra
Browse related Subjects ...
Read More help students learn to think mathematically and to develop genuine problem-solving skills. Comprehensive and evenly-paced, the text has helped hundreds of thousands of students. Incorporating technology, real-world applications, and additional useful pedagogy, the sixth edition promises to help more students than ever build conceptual understanding and a core of fundamental skills.
Read Less
Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9781111990169-4Very good. Hardcover. Has minor wear and/or markings. SKU: 9781111990169-3Fair. 1111990166 Used text may NOT contain supplemental materials-Book may contain some stickers on cover from previous owner(s), highlighting, underlining, rough edges, slight page curling and general signs of a used book-all books hand packed-Free tracking information on all orders | 677.169 | 1 |
GeoGebra is a free interactive math program: geometry, algebra, and calculus. One of the things I
like about it is the ability to
easily export things to interactive html pages. For information about GeoGebra, visit the
GeoGebra home page.. Go to Learning Materials (item 3 under first steps last time I looked at the page). There
you will find downloads and lots of examples various people have created.
Below are a few things I have created. At the moment I haven't gotten around to loading them on the GeoGebra wiki. The Creative Commons license applies to them. The interactive worksheets are web pages. You're powers to change things are somewhat limited but they are instantly available. The GeoGebra files (.ggb) you may download and edit with the full power of GeoGebra.
In music, the superposition of waves of similar but not identical frequencies is very important. It
tells musicians whether or not they are in tune. When frequencies differ more, intervals arise - some
pleasant and some less so.
These two sets of files show the behavior as a function of time of voltages and other
quantities in a circuit consisting of
a sinusoidal applied voltage connected to a resister, a capacitor, and an inductor in series. The first
set shows the behavior as a function of time illustrating how the phases change as inductance, capacitance,
and driving frequency are varied.
These are intended for use after basic ray-tracing has been covered. In each the object can be moved and its height changed by dragging the dot at the top of the arrow labeled object. In addition, the focal point f can be dragged along the axis. In the case of the mirror, the object can be dragged through the mirror so that the connections between concave and convex mirrors may be easily explored. In the case of the lens, the focal point can be dragged through the lens which will cause the lens to switch between converging and diverging.
Disclaimer: The views and opinions expressed in this page are strictly those of the page author.
The contents of this page have not been reviewed or approved by SIUE. Copyright 2009-2011 Lenore Horner. NOTE: IE 6 does not display these pages correctly. Please use another browser. | 677.169 | 1 |
Daily Warm-Ups: Algebra II Retail Price: $20.00 CBD Price: $15.49
( In Stock ) Never waste a single minute with Daily Warm-Ups! Give your students the confidence to solve problems and help prepare them for standardized tests with these quick and fun Algebra II problems. Students will hone their problem-solving skills to determine the years when various historical events; famous firsts; discoveries, inventions, and accomplishments; and notable birthdays occurred. Includes a glossary of terms and an answer key. 180 reproducible warm-ups.
Geometry, Level 2 Thomas Campbell Retail Price: $20.00 CBD Price: $15.49
( In Stock ) Never waste a single minute, when you fill in down time with Daily Warm-Ups. Give your students the skill to become confident at solving problems, and helps prepare them for standardized tests. Contains 180 warm-ups, from converting distance to tesselations and everything in between. Spice up your geometry class with this book and your kids will thank you for it!
Daily Warm Ups Economics Economics problems. Supply and demand, price control, the market, government systems and more are covered in these short exercises, giving your students a chance to think critically about the broad spectrum that Economics encompasses. 180 reproducible warm-ups, answers in the back. 212 pages, softcover.
Daily Warm-Ups Math in Real Life Real Life Math problems. Measurement, trends and records, converting temperatures, surveys and more are covered in these short exercises, giving your students a chance to think critically about the broad spectrum that math encompasses. 192 reproducible warm-ups, answers in the back. softcover. 180 warm-ups.
Daily Warm-Ups SAT Prep: Math, Level II Retail Price: $20.00 CBD Price: $14.99
( In Stock ) Build math skills--and confidence--with these 180 quick activities. Your students will review and practice math problems included on the SAT tests; review math concepts; and learn strategies for fearless test-taking. Great for "extra credit", class starters and transition times...whenever you have unused minutes! Includes answer key. Ages 12 and up. 183 reproducible pages, softcover. Please note that this text is identical to the sold-separately Daily Warm-Ups SAT Prep Math with companion CD-ROM; the difference lies in that the other book contains a companion CD-ROM which this one does not, and the cover is different.
Algebra: NCTM Standards with CD-ROM Betsy Berry Ph.D. Retail Price: $26.00 CBD Price: $18.99
( In Stock ) Reinforce lessons with this collection of 180 Algebra problems for before, in-between, or after classes. Focusing upon the NCTM Standards, this Daily Warm-Ups book covers data analysis, probability, algebraic symbols, number sense, number theory, measurements, and more. The included CD-ROM is fully searchable with all 180 problems and answer key, perfect for printing activities as needed or projecting from the computer. 232 pages, softcover with CD-ROM. Grades 9-12.
Daily Warm-Ups: NCTM Standards: Pre-Algebra with CD-Rom Retail Price: $26.00 CBD Price: $23.40
( Expected to ship on or about 11/03/15. ) Expand your Daily Warm-Ups with this collection of 180 pre-algebra problems for before, in-between, or after classes. Focused upon the most important mathematics as described in the NCTM standards, this book covers: number and operations; measurement; algebra; and data analysis and probability. The included CD-ROM is fully searchable with all 180 problems and answer key, perfect for printing activities as needed, or projecting from the computer. 232 pages, softcover with CD-ROM. Grades 6-8.
Shop hundreds of supplemental publications in English, Math, Science, Social Studies, Art, Music and Special Education. Choose from materials to supplement your current curriculum with current learning practices.
Walch publishing has provided high quality, supplemental teaching resources in English, Math, Science, Social Studies, Art, Music and Special Education for over 77 years. | 677.169 | 1 |
McGraw-Hill Education Announces Interactive E-books with ALEKS 360
McGraw-Hill Education has introduced ALEKS 360, a mathematics solution that combines an artificial intelligence and personalized learning program with a fully integrated, interactive e-book package. ALEKS 360 delivers assessments of students' math knowledge, guiding them in the selection of appropriate new study material, and recording their progress toward mastery of course goals. Through adaptive questioning, ALEKS accurately assesses a student's knowledge state and delivers targeted instruction on the exact topics a student is most ready to learn.
The e-books featured within ALEKS 360 are interactive versions of their physical counterparts, which offer virtual features such as highlighting and note-taking capabilities, as well as access to multimedia assets such as images, video, and homework exercises. E-books are accessible from ALEKS Student Accounts and the ALEKS Instructor Module for convenient, direct access.
The initial e-books to be offered in ALEKS 360 include: Introductory Algebra, Second Edition, by Julie Miller and Molly O'Neill; Intermediate Algebra, Second Edition, by Miller and O'Neill; College Precalculus, Second Edition, by John W. Coburn; and College Algebra, Second Edition, by Coburn. | 677.169 | 1 |
1 Ekim 2011 Cumartesi
Geometric Algebra for Computer Science
Right up until recently, all of the interactions amongst objects in virtual 3D worlds have been based on calculations carried out employing linear algebra. Linear algebra relies greatly on coordinates, even so, which can make many geometric programming duties extremely certain and intricate-usually a lot of work is essential to carry about even modest functionality enhancements. Although linear algebra is an effective way to specify reduced-stage computations, it is not a suitable large-level language for geometric programming. Geometric Algebra for Pc Science presents a compelling choice to the restrictions of linear algebra. Geometric algebra, or GA, is a compact, time-powerful, and performance-enhancing way to signify the geometry of 3D objects in computer packages. In this guide you will locate an introduction to GA that will give you a powerful grasp of its romantic relationship to linear algebra and its significance for your work. You will discover how to use GA to signify objects and carry out geometric operations on them. And you will get started mastering established techniques for producing GA an integral aspect of your applications in a way that simplifies your code with no slowing it down.
 
Explains GA as a organic extension of linear algebra and conveys its importance for 3D programming of geometry in graphics, vision, and robotics.
 
Systematically explores the concepts and strategies that are important to representing elementary objects and geometric operators utilizing GA.
 
Handles in detail the conformal design, a handy way to put into practice 3D geometry utilizing a 5D representation room.
Involves many drills and programming exercises useful for both pupils and practitioners.
 
Companion world wide web internet site contains backlinks to GAViewer, a method that will let you to interact with several of the 3D figures in the e-book, and Gaigen 2, the platform for the instructive programming work outs that conclude each chapter. | 677.169 | 1 |
Introductory Modern Algebra: A Historical Approach / Edition 1
Overview traces the historical development of modern algebra from the Renaissance solution of the cubic equation to Galois's expositions of his major ideas.
Professor Saul Stahl gives readers a unique opportunity to view the evolution of modern algebra as a consistent movement from concrete problems to abstract principles. By including several pertinent excerpts from the writings of mathematicians whose works kept the movement going, he helps students experience the drama of discovery behind the formulation of pivotal ideas. Students also develop a more immediate and well-grounded understanding of how equations lead to permutation groups and what those groups can tell us about multivariate functions and the 15-puzzle. To further this understanding, Dr. Stahl presents abstract groups as unifying principles rather than collections of "interesting" axioms.
This fascinating, highly effective alternative to traditional survey-style expositions sets a new standard for undergraduate mathematics texts and supplies a firm foundation that will continue to support students' understanding of the subject long after the course work is completed. | 677.169 | 1 |
Write Clearly
Homework in mathematics should always be completed with the
same care as in other subjects: write clearly. Some questions require
short answers only, but if written sentences are expected, then
you should use correct English grammar, spelling and punctuation. In most cases
(just as in writing English essays), the submitted version will probably not be
the first draft.
It is not necessary to rewrite the questions when you write up homework
solutions. If you choose to do so, however, you should clearly separate the
question from your answer.
Time to spend on the assignments
I
do expect you to spend at least as much time working outside class as we spend
in class for this course. In fact, I recommend that you spend about two hours
working on your own for each hour we spend in class if you aim to get a good
grade.
Homework Numbers
Always indicate your name and the homework number (as HW#1,
HW#2, etc.) on each homework assignment so that it will not be misplaced or
delayed (until I have time to figure out the correct homework number).
Submit Your Own Work
Remember to write up homework yourself. Although you are
encouraged to work together with others, the submitted homework should be your
own work and not a copy of someone else's.
Submit Your Homework On Time
I expect you to submit each completed homework assignment on
the specified due date. I will accept homework one or two days late unless it
seems to me that this is becoming a chronic or widespread occurrence. However,
I will not accept late homework after I have already returned the graded
assignment to the rest of the class.
If you arrive at class without your homework completed, I would prefer that
you participate in the class rather than spending that time writing up your
homework. In this situation I would ask you to hand in your homework a few
hours later, without penalty. | 677.169 | 1 |
PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! Glencoe Pre-Algebra is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments | 677.169 | 1 |
FUNDAMENTALS OF MATHEMATICS, 9th Edition offers a comprehensive review of all basic mathematics concepts and prepares students for further coursework. The clear exposition and the consistency of presentation make learning arithmetic accessible for all. Key concepts are presented in section objectives and further defined within the context of How and Why; providing a strong foundation for learning. The predominant emphasis of the book focuses on problem-solving, skills, concepts, and applications based on "real world" data, with some introductory algebra integrated throughout. The authors feel strongly about making the connection between mathematics and the modern, day-to-day activities of students. This textbook is suitable for individual study or for a variety of course formats: lab, self-paced, lecture, group or combined formats. Though the mathematical content of FUNDAMENTALS OF MATHEMATICS is elementary, students using this textbook are often mature adults, bringing with them adult attitudes and experiences and a broad range of abilities. Teaching elementary content to these students, therefore, is effective when it accounts for their distinct and diverse adult needs. Using Fundamentals of Math meets three needs of students which are: students must establish good study habits and overcome math anxiety; students must see connections between mathematics and the modern day-to-day world of adult activities; and students must be paced and challenged according to their individual level of understanding. | 677.169 | 1 |
9780195219173Signal Processing and Linear Systems
This as well as new chapters on analog and digital filters and digital signal processing. Lathi emphasizes the physical appreciation of concepts rather than the mere mathematical manipulation of symbols. Avoiding the tendency to treat engineering as a branch of applied mathematics, he uses mathematics to enhance physical and intuitive understanding of concepts, instead of employing it only to prove axiomatic theory. Theoretical results are supported by carefully chosen examples and analogies, allowing students to intuitively discover meaning for themselves | 677.169 | 1 |
Here are two books which can be warmly recommended. Kutzler's
introduction to DERIVE is a model of its kind and cannot be
faulted as an introduction to DERIVE 3 use for the absolute
beginner, whether at school or university. It is suitable for
self-study and will not tax non-mathematical readers. Sjostrand's
book can be recommended with some reservations. It attempts an
important task; to show how a widely-used spreadsheet, in this
case Microsoft Excel 5.0 for Windows or for Macintosh, can be
used to explore mathematics at secondary school level and beyond.
It would be suitable for the school or third-level student who
is well supervised and has directions to a suitable mathematics
text-book.
As to Kutzler, the book consists of ten short chapters together
with an integrated summary and index. Each chapter ends with a
command summary. Almost every page contains numerous DERIVE
screen captures to illustrate algebraic and graphical examples.
Format and layout are exceptionally clear. All elementary
features of DERIVE use are thoroughly explored, and particular
attention is paid to what can go wrong. The book does not
explore DERIVE features such as the utility files, or other
advanced features treated in the DERIVE user manual.
Chapter 1 introduces DERIVE as an alternative to a scientific
calculator, and covers basic DERIVE commands and conventions. The
next three chapters deal with polynomials and their graphs,
expanding and factoring expressions and sub-expressions, and the
important topic of numerical precision and its significance. How
DERIVE simplifies expressions, particularly expressions involving
trigonometric , exponential and logarithmic functions is next
explained at length. Then vectors and matrices are introduced,
followed by DERIVE's programming facility (the treatment here is
very brief), before a treatment of two-and three-dimensional
plots. The final chapters treat simple calculus and analytic
geometry, and introduce further applications of DERIVE. As an
illustration of the hundreds of examples in the book I
reconstruct one only in Figure 1.
Figure 1: A nice example from Kutzler, chapter8; one among many
This is derived from a TIF file screen capture produced by DERIVE itself. You
can see a 16-colour image in the on-line version of this Review.
This book is exceptionally well-written. It is an
easy introduction to DERIVE and the DERIVE user manual. It
prepares the reader for more advanced mathematical investigations
such as those in the excellent Johnson and Evans (1992). If the
price seems steep for a short paperback, it may be a relief to
know that it is bundled with DERIVE Classic Student at ś29
plus VAT plus p&p for both. I cannot imagine a book of this sort
being better done.
Large numbers of students at school and university have access
to and use a spreadsheet without scratching the surface of the
mathematical and statistical insights that such a program
can provide. Berkey and Blanchard (1992) is an excellent
1300-page college calculus text with extensive materials for
Pascal, Basic, Mathematica, DERIVE, and graphing scientific
calculators. In a review of that text-book (Murphy (1994)) I
regretted that the opportunity had not been taken to use the
power of spreadsheets as a learning tool, and gave some
elementary examples in numerical integration. Sj"strand attempts
the important task of showing how a spreadsheet can be of use in
exploring, learning and understanding mathematics. This is
welcome, even if all the problems that arise in such an attempt
are not entirely overcome.
Sjostrand's book has twelve chapters on various aspects of
spreadsheet use. Each chapter consists of worked examples, about
half a dozen in number, of mathematical problems that can
be presented or solved with Excel, together with a larger number
of exercises, usually not elementary, to test the reader's
knowledge and develop further mathematical ideas. Very detailed
step-by-step instructions, including cell references,
menu commands and screen captures are provided at all stages. The
spreadsheet techniques are well organised, with Visual
Basic user-defined functions and macros introduced at an early
stage, and continual reference back to concepts and methods
already introduced. A large number of clever techniques are
developed to make Excel throw light on mathematical questions.
Particularly ingenious are methods for applying the laws of set
theory to a specified finite set of numbers, array functions for
dynamic matrix inversion, and a method of animating simple
diagrams to display motion and velocity of a particle. The book
is suitable for self-study, provided mathematical guidance
is available from a teacher or a good text-book. A keen student
will have great fun with this book and learn a lot of concrete
mathematics.
There are, however, certain reservations to be made. The book was
apparently written for a Swedish secondary-school audience and
has slight imperfections of expressions in English; these are
very minor. The organising principle is based around Excel
rather than mathematical content, and the logic of spreadsheet
learning and use is not the logic of mathematical learning and
use. As a result, the order and development of mathematical ideas
is not particularly coherent. The level of mathematics is in
principle elementary, but of the sort that is in practice
quite sophisticated, with examples in simple abstract algebra,
set logic, analytic geometry, number theory, calculus (including
integrals and differential equations), and matrices. There is
also the fact that a spreadsheet has limitations as a tool for
mathematical investigation, and that other software applications
exist for this purpose. This is recognised by the inclusion of
a chapter on using Excel to prepare input for DERIVE files, but
sometimes one feels that the author is straining to show that
almost any topic of school or first-year college mathematics is
a natural candidate for spreadsheet treatment. Curiously, several
topics in which spreadsheets still have a substantial advantage
over systems like DERIVE, such as linear and non-linear
programming, the mathematics of finance, and statistical
estimation and inference, are not explored in detail.
To repeat: these are two good books; So"strand attempts a
difficult task with tremendous enthusiasm and ingenuity; Kutzler
is a model of exposition and clarity and the best starting
point for any DERIVE beginner. | 677.169 | 1 |
Designed to enrich mathematics teaching in grades 4-10, each volume in this series stands alone and follows the same format. All books contain ten chapters; each chapter devoted to a particular historical personality: Euclid, Fibonacci, Maria Agnesi, George Polya, etc. A portrait introduces the person, followed by a brief historical sketch of their life and work, including some notable quotes and anecdotes. This concise but informative introduction is then followed by several, one page, learning activities. Illustrations are sharp and clear and the worksheets are readily reproducible. The activities have been collected from various sources and rephrased within the appropriate historical contexts. Their selection and presentation make them appealing and classroom useful. While the discussions span cultures, gender and chronological time periods, they are all within a western context emphasizing the Ancient Greece to Europe evolution of mathematical thought. The only "nonwestern" mathematician considered is the Indian genius, Ramanujan, a product of the British colonial education system. It would have been nice to have also seen some consideration given to the traditional mathematics of individuals such as: Liu Hui, 3rd century, China; Brahmagupta, 6th century, India or Omar Khayyam, 11th century, Persia.
Despite this cultural shortcoming, these books provide a valuable and easily accessible resource in the history of mathematics for any teacher. I recommend their use. | 677.169 | 1 |
Abstract Algebra: An Interactive Approach / Edition 1/b>/b>/b>/b>/sup>/b>
Overview problems that utilize GAP and Mathematica to explore groups and rings.
Although the book gives the option to use technology in the classroom, it does not sacrifice mathematical rigor. It covers classical proofs, such as Abel's theorem, as well as many graduate-level topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik's Cube®-like puzzles, and Wedderburn's theorem. He also incorporates problem sequences that allow students to delve into interesting topics in depth, including Fermat's two square theorem.
This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area. | 677.169 | 1 |
Boswell, Laurie
Geometry by Ron Larson(
Book
) 21
editions published
between
2001
and
2012
in
English
and held by
244 WorldCat member
libraries
worldwide
Essentials of geometry -- Reasoning and proof -- Parallel and perpendicular lines -- Congruent triangles -- Relationships
within triangles -- Similarity -- Right triangles and trigonometry -- Quadrilaterals -- Properties of transformations -- Properties
of circles -- Measuring length and area -- Surface area and volume of solids
Geometry : concepts and skills by Ron Larson(
Book
) 9
editions published
between
2003
and
2010
in
English
and held by
91 WorldCat member
libraries
worldwide
This book has been written so that all students can understand geometry. The course focuses on the key topics that provide
a strong foundation in the essentials of geometry. Lesson concepts are presented in a clear, straightforward manner, supported
by frequent worked-out examples. The page format makes it easy for students to follow the flow of a lesson, and the vocabulary
and visual tips in the margins help students learn how to read the text and diagrams. Checkpoint questions within lessons
give students a way to check their understanding as they go along. The exercises for each lesson provide many opportunities
to practice and maintains skills, as well as to apply concepts to real-world problems. - p. ii
Big ideas math : a common core curriculum by Ron Larson(
Book
) 6
editions published
between
2012
and
2015
in
English
and held by
73 WorldCat member
libraries
worldwide
The Big Ideas Math program balances conceptual understanding with procedural fluency. Embedded Mathematical Practices in grade-level
content promote a greater understanding of how mathematical concepts are connected to each other and to real-life, helping
turn mathematical learning into an engaging and meaningful way to see and explore the real world
Holt McDougal Larson pre-algebra(
Book
) 2
editions published
in
2012
in
English
and held by
18 WorldCat member
libraries
worldwide
Holt McDougal Larson Pre-Algebra provides clear, comprehensive coverage of the new Common Core State Standards, including
the Standards of Mathematical Practice. The program gives students a strong preparation for Algebra 1, and may be used with
advanced students enrolled in the Accelerated Pathway. - Publisher | 677.169 | 1 |
Please note: Due to internal system enhancements, this site will be unavailable on Saturday, October 17, 2015 between 8am – 5pm PST. We apologize for the inconvenience.
Patterns and Problem Solving - MAT-914 purpose of this course is to introduce the teacher to the important role patterns play in mathematics. Patterns unlock the world of mathematics. This course provides useful resources and strategies for teachers who would like to help students recognize patterns and use them to make discoveries. | 677.169 | 1 |
SED559 Problem Solving for Mathematics Teachers
Problem solving is a major focus of the current upper elementary and secondary mathematics curricula. A study and investigation of methods for solving problems in mathematics, and teaching these problem solving skills to our students.
Credits
3
Restrictions
Must be enrolled in the following level: Graduate
Must be enrolled in one of the following majors: Adolescence Ed: Mathematics (107A), Adolescence Ed: Mathematics (037A), Adolescence Ed:Mathematics (107B), Adolescence Ed:Mathematics (037B) | 677.169 | 1 |
Numerical methods for ordinary differential equations
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals. Many differential equations cannot be solved analytically; however, in science and engineering, a numeric approximation to the solution is often good enough to solve a problem | 677.169 | 1 |
7MathZone Access Card for Math in Our World
Math In Our World, Media Update
ConnectPlus by ALEKS Access Card 52 Weeks for Math in Our World
Math in Our World
Student Solutions Manual to accompany Math in Our World
Summary
The author team of Dave Sobecki and Allan Bluman created an engaging text and digital program aimed at meeting the needs of today's liberal arts math students, resulting in the third edition of Math in Our World. This revision focused on further development of critical thinking skills through several hundred revised exercises and examples, still presented within the hallmark style of the Math in Our World program. Carefully chosen questions help students to form a connection between relevant examples and the mathematical concepts of the chapter. Using the engaging writing style characteristic of the text, the authors support concepts through abundant examples, helpful practice problems, and rich exercise sets. The author team was also deeply engaged in the development of the Connect and LearnSmart online content
to help ensure a consistent experience for students regardless of medium. The result is an exceptionally engaging program that is able to both effectively and creatively convey the fundamental concepts of a liberal arts math curriculum to even the most hesitant student.
Table of Contents
Mathematics in Our World, Third Edition
Chapter 1: Problem Solving
1-1 The Nature of Mathematical Reasoning
1-2 Estimation and Interpreting Graphs
1-3 Problem Solving Strategies
Chapter 1 Summary
Chapter 2: Sets
2-1 The Nature of Sets
2-2 Subsets and Set Operations
2-3 Using Venn Diagrams to Study Set Operations
2-4 Using Sets to Solve Problems
2-5 Infinite Sets
Chapter 2 Summary
Chapter 3: Logic
3-1 Statements and Quantifiers
3-2 Truth Tables
3-3 Types of Statements
3-4 Logical Arguments
3-5 Euler Circles
Chapter 3 Summary
Chapter 4: Numeration Systems
4-1 Early and Modern Numeration Systems
4-2 Tools and Algorithms in Arithmetic
4-3 Base Number Systems
4-4 Operations in Base Number Systems
Chapter 4 Summary
Chapter 5: The Real Number System
5-1 The Natural Numbers
5-2 The Integers
5-3 The Rational Numbers
5-4 The Irrational Numbers
5-5 The Real Numbers
5-6 Exponents and Scientific Notation
5-7 Arithmetic and Geometric Sequences
Chapter 5 Summary
Chapter 6: Topics in Algebra
6-1 The Fundamentals of Algebra
6-2 Solving Linear Equations
6-3 Applications of Linear Equations
6-4 Ratio, Proportion, and Variation
6-5 Solving Linear Inequalities
6-6 Solving Quadratic Equations
Chapter 6 Summary
Chapter 7: Additional Topics in Algebra
7-1 The Rectangular Coordinate System and Linear Equations in Two Variables
7-2 Systems of Linear Equations
7-3 Linear Inequalities
7-4 Linear Programming
7-5 Functions
7-6 Quadratic, Exponential, and Logarithmic Functions
7-7 Available Online: Solving Systems of Linear Equations Using Matrices | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Mathematica for Teaching and Education
Learn how to improve your classroom experience with Mathematica. This course gives a tour of functionality relevant to teaching and learning, along with case studies and best-practice suggestions for course integration. Topics include making your classroom dynamic with interactive models and a survey of computation and visualization capabilities useful for teaching practically any subject at any level. | 677.169 | 1 |
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This lesson plan involves comparing different methods to determine the shortest route when traveling from Cleveland to Boston. Students will be given the opportunity to interpret data presented in table and graph format...
This series of videos contains 180 Worked Algebra I examples (problems written by the Monterey Institute of Technology and Education). You should look at the "Algebra" playlist if you've never seen algebra before or if...
This math unit from Illuminations asks students to inscribe and circumscribe regular polygons. They will then analyze the area of the polygons, using either an applet or a graphing calculator. The unit includes two...
This lesson helps students understand financial topics (interest rates, FICO scores and loan payments) in a mathematical context. Students will calculate monthly payments for a car or home based on the best interest...
This series of videos, created by Salman Khan of the Khan Academy, features topics covered from very basic algebra all the way through algebra II. This is the best algebra playlist to start at if you've never seen... | 677.169 | 1 |
History of Mathematics
9780673380395
ISBN:
0673380394
Publisher: Addison-Wesley Educational Publishers, Incorporated
Summary: One of the leading historians in the mathematics field, Victor Katz provides a world view of mathematics, balancing ancient, early modern, and modern history. Egypt and Mesopotamia, Greek Mathematics to the Time of Euclid, Greek Mathematics from Archimedes to Ptolemy, Diophantus to Hypatia, Ancient and Medieval China, Ancient and Medieval India, The Mathematics of Islam, Mathematics in Medieval Europe, Mathematics in... the Renaissance, Precalculus in the Seventeenth Century, Calculus in the Seventeenth Century, Analysis in the Eighteenth Century, Probability and Statistics in the Eighteenth Century, Algebra and Number Theory in the Eighteenth Century, Geometry in the Eighteenth Century, Algebra and Number Theory in the Nineteenth Century, Analysis in the Nineteenth Century, Statistics in the Nineteenth Century, Geometry in the Nineteenth Century, Aspects of the Twentieth Century For all readers interested in the history of mathematics.
Katz, Victor J. is the author of History of Mathematics, published under ISBN 9780673380395 and 0673380394. Ten History of Mathematics textbooks are available for sale on ValoreBooks.com, four used from the cheapest price of $50.00, or buy new starting at $365.53 | 677.169 | 1 |
80 Activities to Make Basic Algebra Easier / Edition 2
With this sourcebook of reproducible puzzles and practice problems, you can successfully reinforce first-year algebra skills. Now revised to meet NCTM standards, this book contains more teaching tips, new calculator activities, and additional "outdoor math" activities. Secret codes, magic squares, cross-number puzzles, and other self-correcting devices provide…
See more details below | 677.169 | 1 |
Everyone needs to know math and algebra to be able to solve daily life problems. Algebra-1 gives needed skills to move on, and it is needed even for Calculus. Algebra-2: More real life related math | 677.169 | 1 |
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Review: Principles of Mathematical Analysis
User Review - Hououin Kyouma - Goodreads
It's pretty much great, but the last 2 chapters are not for a beginner. The first chapter on mutlivariable analysis sounded fine, but the others were pretty painful to read. However, for someone with ... | 677.169 | 1 |
College Algebra (3rd Edition)
9780321466075
ISBN:
0321466071
Edition: 3 Pub Date: 2007 Publisher: Addison Wesley
Summary: These authors have created a book to really help students visualize mathematics for better comprehension. By creating algebraic visual side-by-sides to solve various problems in the examples, the authors show students the relationship of the algebraic solution with the visual, often graphical, solution. In addition, the authors have added a variety of new tools to help students better use the book for maximum effecti...veness to not only pass the course, but truly understand the material.
Judith A. Beecher is the author of College Algebra (3rd Edition), published 2007 under ISBN 9780321466075 and 0321466071. Two hundred ninety two College Algebra (3rd Edition) textbooks are available for sale on ValoreBooks.com, one hundred fifteen used from the cheapest price of $0.60, or buy new starting at $29Cathedral City, CAShipping:Standard, ExpeditedComments:ALTERNATE EDITION: New Annotated 3rd Edition. Ships from CA day of order, or next business day. Complete Satisfactio... [more]ALTERNATE EDITION: New Annotated 3rd Edition. Ships from CA day of order, or next business day. Complete Satisfaction Assured! [May NOT include supplemental materials such as CDs and access codes. May include some highlighting or writing. Ships next business day! Choose expedited shipping for delivery [more]
May NOT include supplemental materials such as CDs and access codes. May include some highlighting or writing. Ships next business day! Choose expedited shipping for delivery in 2-6 business days! Excellent customer service | 677.169 | 1 |
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Unizor is an open source mathematics and physics Web site that seeks to promote intellectual strength, creativity and analytical abilities. Unizor's founder, Zor Shekhtman, does this through a series of lectures on mathematics and physics designed to help high school students exercise the mind just as one would exercise his or her body in a gymnasium. The abilities strengthened by using Unizor can readily be applied to real life. Another great strength of Unizor is that parents and other responsible adults are placed in charge of their students' education.
Zor discusses Permutations.
So, how does Unizor do this? A parent/supervisor goes to Unizor's Web site and creates an account for him or herself. The parent/supervisor then has two roles. The first is to enroll his or her students into instructional programs. The second is to manage the learners' progression through the programs. Each student has an account created by the parent/supervisor which makes this possible. From here, students' progress can be monitored, including exam scores, and they can be passed on to the next level within the course.
Zor discusses possible outcomes of a given set of options.
How does Unizor work? Each topic is presented by Zor Shekhtman as a video lecture. Far from being dry, Zor conveys his enthusiasm for the curriculum through his instruction, which makes the lectures very engaging. Zor also utilizes visual aids and examples to enhance the learning experience. The educational experience is further augmented by the use of other media and materials. Furthermore, teachers are not left out. To quote from the Unizor home page, "The function of a Unizor teacher is to provide quality educational materials. Control over educational process is not a function of a Unizor teacher, this is supposed to be provided by parents/supervisors." Teachers can modify both instructional content and exams as well.
Unizor has a very different approach to mathematics education than the more prevalent principles utilized by many schools. These principles have an emphasis on formulas and procedures and the memorization of these. The problem with this is that students, not finding immediate real-world application for this information, will soon forget it once the assessment is passed. Unizor focuses on a logical and analytical approach to mathematics education, encouraging problem-solving, proving theorems, axiomatic foundation and rigorousness of educational material. This approach is conducive to the development of students' minds, something that will be of value in any occupation.
Unizor is ideal for learners who have been identified as gifted and talented. The opportunities for academic and intellectual growth abound here. Be aware, that Unizor is a work in progress. However, there are more than 400 lectures available with more to come. You should also know that the physics component is still predominantly under development. So take control of your student's learning and create a supervisor account on Unizor's Web site today.
What would be better for your school system: a Student Information System or a School Management System? How's about having both tools in one software package? In this installment, I'd like to introduce you to Gibbon, an open source suite that can be utilized to organize your facilities and to record and analyze data on students. In short, Gibbon is a SIS, a SMS and VLE functionality all in one place. If you are involved in your school district's decision-making, you should really check this suite out before shopping for proprietary software.
The Planner window
The secret to Gibbon's functionality is modules which can be added to make it do what you need it to do. There are a set of core modules that are installed by default. I'm going to take a look at some of those now. The first of these is Planner. Planner is a lesson planner that allows you to incorporate multimedia integration into your lessons. Because these lesson plans are stored on a server, they can be readily shared with parents and students. Homework can be assigned and collected online. These lesson plans can even be organized into units and used to generate curriculum maps. Assignments can be graded and the grades recorded. Other cool features include students being able to comment on peers' work, guests being able to take classes and students being able to "like" lessons by awarding them gold stars.
A planning session with Timetable
Timetable allows for the creation of timetables for students and faculty alike. Resources is a database which allows users to catalog and share resources school-wide. Library empowers users with the capacity to catalog, lend and track items such as books, movies or even electronics. The Individual Needs module allows for the creation of lessons designed to meet the needs of individual learners. These can be archived as the student progresses and used to measure progress. Activities facilitates the creation and management of school activities, including Web-based notifications and registration as well as attendees lists. Markbook allows educators to record and track students academic progress. This data can, if so desired, be easily shared with students and parents. The Rubrics module not only allows teachers to create rubrics, but the rubrics are actually interactive via mouse-clicks.
You too can track student performance using Gibbon.
There is one aspect of education that I haven't addressed yet and it's the most important component -people. Students is the SIS module. Student information from across the system can be accessed here. Academic, behavioral and medical alerts can be generated and accessed by those who need them. Students' attendance history can be tracked in Attendance and behaviors, both positive and negative, can be tracked and analyzed using the Behavior module. Data Updater assures that the right people, parents for example, have current data on certain students. Other modules in this category can empower users with the ability to easily create groups of staff and students alike as well as to generate staff directories. Finance is a tool for the tracking of finances, for example in a fund-raiser. Messenger, an internal messaging system that can be used to set up groups of specific recipients and supports such formats as email, SMS and Message Wall. The last of the default modules addresses administration, both in terms of school and in terms of the system as well as user and timetable administration.
What could be better than having all of these components in one suite? Being able to add to them. There are far more expansion modules than I could possibly address here, but they include, and certainly are not limited to, assessment administration, query building, Moodle integration, an integrated Help Desk and Free Learning integration as well. There are also proprietary modules available, should the open source modules not meet your needs.
As an experienced educator, I can tell you that if your school or district is looking for a school platform, the search should begin, and will end, here.
Thanks to Ross Parker for permission to use the screenshots and for his work on this extremely worthwhile project.
It's been a while, too long in fact, since I've written anything here. Hard to believe that the summer is winding down. You probably have lots of photos and videos taken this summer. What better way to share them than in a movie that you've made yourself? You don't need Microsoft Windows Movie Maker either. Let me introduce you to OpenShot, the open source alternative to Movie Maker.
OpenShot logo (copyright 2014 OpenShot Studios LLC)
Upon initial launch, OpenShot bears a striking resemblance to its proprietary counterpart and it works in a similar fashion. What I find extremely appealing about this software, personally, is the WYSIWYG interface. The interface is very straightforward, which is a big deal to me as I believe new users will have a tendency to return to an application if they have a pleasant first experience. ( I've just discovered another strength of this software: the user's manual jumps right into using the software via a piece entitled Learn OpenShot in 5 Minutes, rather than to present the application and its features. The manual addresses these topics, but after guiding the reader through initial use of the software. How cool is that?)
OpenShot ready to go.
Looking at the screenshot at left, the Main Toolbar is at the top of the window, under the menu. Below this are the Function Tabs, which allow users to toggle between files, transitions and effects. The Project Files pane below this shows all media files that have been added. The Preview Window to the right displays video playback. Just below these two panes is the Edit Toolbar (left) and the Zoom Slider, which allows users to tweak the time-scale. Below this is the Play-Head/Ruler. The Ruler displays time-scale and the Play-Head shows the current position of the movie on the time-scale (appears in red when in use). Finally, the Timeline is at the bottom of the window and displays each component of the movie.
OpenShot at work
Adding media is easy. Once you've added media to the Project Files pane, simply click and drag them to the Timeline. You can add a wide variety of audio, video and image media to your video. Once media has been added to the Timeline, it can be repositioned by clicking and dragging. You can also add effects, such as transitions, special effects and sounds. Finished videos can be exported to such video formats as AVI, MOV, MP4 and MPEG, If you really want to see something cool created using this software, let your kids or students run wild (well, not that wild) with OpenShot. They'll show you what thinking outside of the box is all about.
OpenShot is available for Fedora Linux and Ubuntu Linux and also as a Live version run from DVD so that you don't have to install it to try it.
One of the great benefits of mailing lists is that you have the opportunity to learn about new things. In this case, the new thing that I have learned about via the schoolforge-discuss mailing list is FisicicaLab, an open source educational application developed to solve physics problems. FisicaLab handles all of the mathematics related to physics, giving the user the ability to focus purely on physics. So, without further adieu, let's take a closer look at this thought-provoking piece of software.
FisicaLab running on the MATE Desktop Environment.
The graphical interface is similar to that of GIMP, incorporating multiple windows. Unlike GIMP, FisicaLab utilizes only two windows initially. The main window is called the Chalkboard and the other window is entitled Modules and Elements. The Modules and Elements tool enables users to add items to the Chalkboard and to modify those items. Buttons at the top of the Modules and Elements window allow users to toggle between different types of elements. (See the screenshot for a typical session). Additional windows open as needed.
FisicaLab allows users to manipulate virtual objects such a blocks, pulleys and forces. These can be handled and allowed to interact in a variety of ways, including, but not limited to, relative motion, collision and explosion. Other factors that can be adjusted include friction and force, among others. FisicaLab gets a high level of expandability via additional modules which users can install. These modules include, but are not limited to, kinematics of particles, dynamics of particles and calorimetry, ideal gas and expansion.
This brief article is written merely to inform and cannot do this wonderful application justice. If you teach physics, FisicaLab is designed with both instruction and learning in mind.
Arguably, one of the greatest strengths of open source software is that it can add new life to old hardware. For example, I have a Dell laptop built for the now unsupported Microsoft Windows XP. The lack of support from Microsoft doesn't bother me, because that laptop is now running Xubuntu 14.04 LTS. With this in mind, I've chosen to take a look at Partimus, an organization that refurbishes computers, installs open source software on them and then distributes the computers to students and schools that need them.
Partimus Mission Statement: Provide educational opportunities through open technology to educators and students.
Partimus (Latin for "we share") is non-profit and currently serves schools in the San Francisco Bay area. This project was co-founded by Cathy Malmrose and Maile Urbancic. These two ladies share a passion for helping children succeed and for open source technology. They also share a background working in education. The organization is now run by a board the members of which share the passions that led to Partimus being established.
Students using the new computer lab at the ASCEND School. (Photo from Robert Litt of ASCEND School)
So, what kind of projects has Partimus been involved with? One program that they implemented that is somewhat close to my heart (see my blog of February 25, 2015, An Old Laptop Made New) is the Laptops for Linux Users (LALU) program. They accept donated laptops (better they should go to people who need them than to sit on a closet shelf forgotten). The people at Partimus then talk to the person who needs the laptop and they install the free and open source software needed to meet the user's requirements. For example, on the Partimus site, they mention helping an elderly Washington state woman, Sky, who was a retired system administrator. Sky likes to help others, especially elderly friends, get into computing. She could not afford a new computer, so the people at Partimus matched her up with a laptop that fulfilled her needs. Now Sky provides elderly friends with laptops running Puppy Linux and helps them get started in computing.
The new computer lab in the library at the International Studies Academy.
Partimus has also provided used computers running the Linux OS to schools in the San Francisco Bay area. Partimus donated over 20 networked, standalone Ubuntu Linux desktop computers to the International Studies Academy in San Francisco. This school has 420 students in grades 6-12 who are pursuing the study of foreign cultures, languages and geography. These computers provide Internet access using Mozilla Firefox and productivity via OpenOffice. Other schools that have received Linux computers and ongoing support from Partimus include the KIPP San Francisco Bay Academy in San Francisco, the ASCEND School in Oakland and the Computer & Technology Resource Center in Novato, among others. All of these organizations are non-profit.
So you're thinking "This is a wonderful organization, Chris, but what can I do to help?" There are a variety of things that you can do to help Partimus bring technology to those in need. They accept the following hardware: flatpanel monitors, laptops and desktops with at least 1 GB of RAM and CPUs at 2 Ghz (at least), optical mice and USB/PS2 keyboards. You can also give the gift of funds through monetary donations or through the patronage of such companies as AmazonSmile and Boutique Academia. For more information about how you can help or to ask them to help your non-profit organization with its computer needs, check out their Web site (link below).
It's been over a month since I published my last article, so I just wanted to check in with my readers. I have every intention of continuing this blog indefinitely, or as long as possible. My hiatus has been due to a recent lack of time. I have enrolled in courses to become an educational consultant, which has tapped into my time for this project. This project is one of the reasons that I am (eventually) leaving the field of teaching for that of being a consultant. This new career will provide me with the increased income that I need to develop this project adequately and to give it the chance to thrive that it so deserves.
In short, though my blogs will be less frequent until my courses are done, this blog, like my enthusiasm for the Energize Education project, will carry on. Thanks for reading.
I've recently come across a piece of open source technology that will not only take the struggle out of getting your kids to do schoolwork at home, but will also put them in control of this work while helping them to self-monitor and develop independence at the same time. AutoTeach will do all of these things. So, what is AutoTeach and how can it do all of this?
With AutoTeach Parent Tool, students earn credits which can be used to "purchase" time on the Internet. There are three components that make this happen. the first is the Credit Meter. The student logs onto this via his or her wi-fi device (tablet, game system, etc.). While this is running, the student can access the Internet. The next component is a Raspberry Pi. For those unfamiliar with Raspberry Pi, they are open source palm-sized computers consisting of a motherboard with a CPU and RAM, as well as audio, video, SD card (used as hard drive), USB and LAN ports. In short, they are fully functioning computers. The Raspberry Pi serves as router, Credit Meter and firewall. By default, the firewall only allows the students to access the third component of AutoTeach, the Credit Reader Web site.
Raspberry Pi mini computer
The AutoTeach Web site allows parents to set up school-based work for their young academicians, as well as a schedule for completing this work. This site, by default, is the only one white-listed on the Raspberry Pi firewall, so it's the only one kids can access until they have credits. Topics covered include reading, mathematics, music, vocabulary building and research. These are all available as JavaScript plugins to be loaded onto the user's Credit Reader account.
AutoTeach Student Meter
The activities are just as engaging as they are educational. By completing activities to a predefined skill level, students can earn credits toward free Internet time. You may actually find your kids begging to do schoolwork. A really neat feature of AutoTeach is that credits can be awarded manually. This means that you could use it as a reward for completing chores or other activities that you would like to encourage. In terms of personal growth, students will have greater control over their learning and a greater enjoyment of it. Through monitoring their own progress and working independently, young people will develop a sense of independence as well as one of self-reliance.
TuxMathScrabble via AutoTeach
Arguably one of AutoTeach's greatest strengths is its capacity for growth. Developers will be constantly creating new plugins. There is also a development suite available, AutoTeach's Development Sandbox, which will allow developers to create plugins on their own. The result is that potential for more plugins is limitless.
As you can see, I am very enthusiastic about this technology. So how do you acquire AutoTeach? It is available as a subscription. To learn more, check out some of the resources below to which I've provided links.
I'd like to look at a nifty little media converter entitled WinFF. WinFF bills itself as an open source video converter. I refer to it as a media converter as it is capable of converting audio files as well. Truth be told, WinFF is actually a graphical front-end program for FFmpeg, which is a program run from the comand line that does the actual converting. WinFF simply allows users unfamiliar with the command line to run conversions using a graphic interface.
Before I begin, I want to say that I am running WinFF in Ubuntu Linux. Although WinFF is available for both Linux/UNIX and Microsoft Windows, the screenshots in this article will show WinFF being run on Linux and will also show a file manager, equivalent to Windows Explorer, displaying the contents of a CD, or as Linux refers to them, an Audio Disc, which is similar to a CD's/DVD's folder in Windows. That having been stated and without further adieu, let's get to the point of this blog: converting media files using WinFF.
The WinFF window
When initially launched, you will be presented with a window that should look like the screenshot at left. WinFF's interface is WYSIWYG and you can see from the screenshot that the buttons on the toolbar (referred to as the Buttons in the WinFF documentation, which can be downloaded from their Web site) offer tooltips to provide more information about their function. Above the Buttons is a menu bar. Below these is an empty field where information is displayed during the conversion process. Below this are tabs pertaining to various types of media and a place to specify your output, or destination, folder.
Let's convert these .WAV files on this CD to MP3s.
I will be converting the CD shown in the screenshot at right. It was a gift from my niece (Love you, Ayla) and I'm anxious to add it to my MP3 collection. Place the audio CD in your DVD drive. A window will open asking you what to do with the disc that has been placed in the drive. Choose to open it in file manager. This way you can view the files on the CD. WinFF and FFmpeg support WAV files, but not CDA (Compact Disc Audio) files. Your CD is likely to have files in either format. Once you've checked the contents of the CD or, as WinFF refers to this type of media, Audio Disc, you can start WinFF.
The Select Video Files window
Once WinFF has opened, click the Add button. This will open the Select Video Files window (shown at left), which looks like a file manager. Select the files that you wish to convert (Tip: You can press the CTRL and A keyboard keys simultaneously to select all pf the files) and click the Open button. This window will close and return you to the main WinFF window.
The video conversion process
Now, set your desired conversion format and preset. For this type of action, Convert to should be set to Audio and the Preset to MP3. Next, choose your destination folder. This is important as it will make finding your converted files easier if you place them in a precise location. Clicking on the ellipsis (…) will allow you to select a more specific location than the default, which in Linux/UNIX is your home directory. When you are ready, click on the Convert button and the process will begin. The actual conversion process occurs in a terminal window as shown in the screenshot at right.
When the process is completed, you will be instructed to Press Enter to Continue (as shown in the screenshot above), which will close the terminal window and return you to the main WinFF window. The process is complete and you can now close WinFF. You may still want to rename your new MP3 files as they will still possess the generic name (e.g. "Track 1.mp3") that they had while on the CD, but that can be done in a file manager at your leisure.
I want to look at a fun application entitled LinCity-NG. As the name would imply, it is an open source clone of Electronic Artis' (EA) SimCity. LinCity-NG has evolved quite a bit since my first encounter with it ten years ago. It has an aesthetically appealing interface and is highly customizable in terms of features and game play.
LinCity-NG is also a wonderful way for students to learn about both economics and ecology. My reasoning for this is that this game requires users to build a civilization. In order for a civilization to grow it must first survive and then expand. Surviving means that you must have a successful economy with employment, resources and trade. These things fluctuate during the game and to succeed, you must be able to compensate for them. In terms of ecology, as you expand, you will encounter various types of terrain, such as wetlands, that you must work around as removing them is very expensive. You must also be aware that civilizations generate pollutants. These pollutants must be dealt with responsibly in order to avoid repercussions. Keeping these factors in mind, let's take a closer look at LinCity-NG.
LinCity-NG Main Menu
When initially launched, LinCity-NG presents the user with a straightforward interface. The screenshot at left displays the main menu. Clicking the New option opens a menu allowing users to select a scenario. Available options include Beach, good times and bad times, among others. Personally, I like to start with an empty board and when I create my LinCity-NG academic unit (forthcoming), this will be required so that all students start at the same level in the game. If you're experimenting with LinCity-NG, by all means try different scenarios. The titles are self-explanatory.
Once your game starts, you will be presented with a map of the terrain upon which a civilization must be built. There is a panel on the upper left-hand side of the screen that provides access to available actions and structures. In the lower left corner is what looks like the control buttons on a DVD player. These allow users to accelerate and pause the simulation or to run it at normal speed. Users can also access the main menu from here. In the lower right-hand corner, is a panel offering a map, some buttons below it and several tabs. Both tabs and buttons allow you to view various information about your civilization, such as economic standing and resource availability, among other things. The map is laid out in a rhomboid shape. Check out the screenshot at right for an idea of the initial layout.
An empty map upon which to build your civilization
In the beginning, users can create only the bare minimum in terms of structures for their civilization. As your civilization grows, more options become available. This is what would make LinCity-NG an ideal platform for learning. All learners start at the same level. Each could be provided with a rubric identifying what their society must have in terms of services and industry at specified points in game time. For example, "By simulation year 60, your civilization should have Residences and Farms powered by Windmills."
Looking at the panel in the upper-left corner, each button represents a category. The top button allows you to toggle between the Query tool (mouse pointer),the Bulldozer and Water. Clicking on anything with the Query tool will provide information about that item in the little map window in the lower-right corner. The next tool on the panel allows you to iniitally build Residential areas. You can choose from one of three options, each of which affects the population levels differently. The button below this could best be described as basic resources. These include at outset Market (where jobs are created and goods exchanged), Farm (for food) and Water well.
One of the author's bustling communties
The next button opens a menu that could be best described as social services. Initially, Monument (something to give the citizens pride in their community) is the only option available, but others include School, Fire Department and Sport (like a basketball court). Transportation is the next category. The only option available is Track (like a trail) at first, but others such as Road and Port can quickly be unlocked. Power sources are next and none of these are available at start up time. Windmills however can be readily earned to provide power to Residences and Farms, as I indicated above.
You can zoom in and out with the mouse wheel. Cool, huh?
Resource sources are next. The options available at the beginning include Commune (a place where such goods as coal and steel are produced), Ore mine and Rubbish tip (landfill). Other choices that become available are Coal mine and Recycle (recycling center). Industries make up the final menu. Pottery is the only option available at outset (like all industries in the game, Pottery converts resources into goods). As the game progresses, users have access to Blacksmith, Mill, Light Industry and Heavy Industry. If you haven't got all of that committed to memory, don't worry. One of LinCity-NG's greatest strengths is its integrated help. Just right-click on any of these options for more information about them.
I could write more on this stimulating application, but I leave it to you to explore LinCity-NG for yourself. Your students will be enrapt. There is one more academic aspect of LinCity-NG that I neglected to mention and that is creativity. Though you can use it to teach students about economics and ecology, one fun aspect for the educator is the opportunity to observe the worlds that students will create and how they vary. Student creativity is often one of the greatest rewards that educators can enjoy.
I'm going to walk you through the process I followed to update an old laptop into a useful computer. My thinking is that if I could do this, so could anyone else, especially educators in poorer school districts or even homeschoolers.
IBM ThinkPad T40
A friend of mine gave me an old IBM ThinkPad T40 (shown at right). It has an ebay value of around $40 (U.S.). It was built for Microsoft Windows XP and worked decently, save for the outdated software and hardware and the fact that the onbooard wifi didn't work. I decided to make it into something a little more useful for today's computing.
Here are some specs:
Intel Celeron M processor at 1.5 GHz
512 MB SO-DIMM SDRAM
80 GB ATA Hard Drive
BIOS Date: 2006-06-02
6x DVD-ROM
2 USB ports
2 Sound Ports (1 in, 1 out)
On-board modem, NIC and wifi
The panel covering the RAM module.
As I saw it, the best and easiest course of action was to upgrade the RAM. This can arguably be said to be the most approachable upgrade for any laptop or desktop PC as it's a fairly easy procedure and a fairly inexpensive as well. I bought my 200-pin SO-DIM SDRAM replacement module on Amazon, but there are many other places online where this can be procured. For those of you thinking "Chris, I'm not a computer repair technician. It's too complex. I can't do this," I say "Yes, you can." Internally computers are compartmentalized and therefore easy to work on, so long as you're careful about electricity. Laptops are no exception. If you turn the laptop over, you will see various panels held in place with Phillips screws. The screenshot at left indicates the panel we need to open.
T-40 RAM module
Once you have removed the panel (after taking out the screw, I had to use a small knife to lift the plate up), you will see the RAM module. You will notice two clips, one on the right of the module and one on the left (if you look at the screenshot on the right, you can see that the right clip is just to the left of the screw hole). These are simply pulled out away from the RAM module. The module will pop up and can then be slid out. Make a note of its positioning to make sliding the new RAM into place easier. For my RAM update, I chose a module offering 1.2 GB of RAM, which is a tremendous upgrade for this system and the maximum supported RAM for this system. Slide the new RAM module into place, push down on it until it lies flat in its cavity and the clips will lock it down. Put the plate back on and we're ready for the next step.
I chose to install Xubuntu Linux 12.04 LTS (Long-Term Support) 64-bit as Xubuntu 14.04 LTS presented a warning during installation stating that Celeron M processors do not support PAE (Physical Address Extension). If you have the time and the interest, however I have provided a link below to an article on working around this problem. In either case, I have provided a link to Xubuntu's download page. Once you have downloaded the ISO file, you'll need to burn it to a DVD as an image then place the disc in the DVD-ROM drive. You'll also need to enter the T-40's BIOS and set the DVD-ROM as the primary boot device. To enter the BIOS, you'll need to press the Access IBM key (located in the upper left corner above the keyboard) when you first turn the laptop on and choose Start setup utility from the menu. When done, save your settings and restart the computer. Installation will begin on reboot.
IBM ThinkPad T-40 running Xubuntu Linux
The reason I chose to install Xubuntu is that it is lighter than Ubuntu in terms of hardware demands, which makes it an ideal OS for older computers. You will be guided through through the installation process. When it is completed, you will be prompted to remove the disc and restart the computer. When it restarts, you can log in using the account created during installation. If all goes as it should, you should be greeted with a desktop like the one shown in the screenshot.
Regarding the on-board wifi, you will find many discussions online about this topic. Rather than to resolve it, I went on to Amazon and purchased a Panda Mini Wifi 150 Mbs Wireless-N 24 GHz USB Adapter. It plugs right into one of the USB ports and starts working immediately. It provides a reliable connection and after a year and a half, I still have no complaints. You may want to install additional software, depending on your needs, but other than that, you now have a perfectly good computer for education, Web surfing, productivity, gaming or whatever your needs may be. | 677.169 | 1 |
In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration (...) of a string and heat conduction. The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit–reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science. (shrink)
The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions (...) with the related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematicalphysics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)
It is commonly thought that before the introduction of quantum mechanics, determinism was a straightforward consequence of the laws of mechanics. However, around the nineteenth century, many physicists, for various reasons, did not regard determinism as a provable feature of physics. This is not to say that physicists in this period were not committed to determinism; there were some physicists who argued for fundamental indeterminism, but most were committed to determinism in some sense. However, for them, determinism was often (...) not a provable feature of physical theory, but rather an a priori principle or a methodological presupposition. Determinism was strongly connected with principles of causality and continuity and the principle of sufficient reason; this thesis examines the relevance of these principles in the history of physics. Moreover, the history of determinism in this period shows that there were essential changes in the relation between mathematics and physics: whereas in the eighteenth century, there were metaphysical arguments which lent support to differential calculus, by the early twentieth century the development of rigorous foundations of differential calculus led to concerns about its applicability in physics. The thesis consists of six papers. In the first paper, "On the origins and foundations of Laplacian determinism", I argue that Laplace, who is usually pointed out as the first major proponent of scientific determinism, did not derive his statement of determinism directly from the laws of mechanics; rather, his determinism has a background in eighteenth century Leibnizian metaphysics, and is ultimately based on the law of continuity and the principle of sufficient reason. These principles also provided a basis for the idea that one can find laws of nature in the form of differential equations which uniquely determine natural processes. In "The Norton dome and the nineteenth century foundations of determinism", I argue that an example of indeterminism in classical physics which has attracted attention in philosophy of physics in recent years, namely the Norton come, was already discussed during the nineteenth century. However, the significance which this type of indeterminism had back then is very different from the significance which the Norton dome currently has in philosophy of physics. This is explained by the fact that determinism was conceived of in an essentially different way: in particular, the nineteenth century authors who wrote about this type of indeterminism regarded determinism as an a priori principle rather than as a property of the equations of physics. In "Vital instability: life and free will in physics and physiology, 1860-1880", I show how Maxwell, Cournot, Stewart and Boussinesq used the possibility of unstable or indeterministic mechanical systems to argue that the will or a vital principle can intervene in organic processes without violating the laws of physics, so that a strictly dualist account of life and the mind is possible. Moreover, I show that their ideas can be understood as a reaction to the law of conservation of energy and to the way it was used in physiology to exclude vital and mental causes. In "The nineteenth century conflict between mechanism and irreversibility", I show that in the late nineteenth century, there was a widespread conflict between the aim of reducing physical processes to mechanics and the recognition that certain processes are irreversible. Whereas the so-called reversibility objection is known as an objection that was made to the kinetic theory of gases, it in fact appeared in a wide range of arguments, and was susceptible to very different interpretations. It was only when the project of reducing all of physics to mechanics lost favor, in the late nineteenth century, that the reversibility objection came to be used as an argument against mechanism and against the kinetic theory of gases. In "Continuity in nature and in mathematics: Boltzmann and Poincaré", I show that the development of rigorous foundations of differential calculus in the nineteenth century led to concerns about its applicability in physics: through this development, differential calculus was made independent of empirical and intuitive notions of continuity and was instead based on mathematical continuity conditions, and for Boltzmann and Poincaré, the applicability of differential calculus in physics depended on whether these continuity conditions could be given a foundation in intuition or experience. In the final paper, "Determinism around 1900", I briefly discuss the implications of the developments described in the previous two papers for the history of determinism in physics, through a discussion of determinism in Mach, Poincaré and Boltzmann. I show that neither of them regards determinism as a property of the laws of mechanics; rather, for them, determinism is a precondition for science, which can be verified to the extent that science is successful. (shrink)
This paper explores the relationship between Kant's views on the metaphysical foundations of Newtonian mathematicalphysics and his more general transcendental philosophy articulated in the Critique of Pure Reason. I argue that the relationship between the two positions is very close indeed and, in particular, that taking this relationship seriously can shed new light on the structure of the transcendental deduction of the categories as expounded in the second edition of the Critique.
One may discuss the role played by mechanical science in the history of scientific ideas, particularly in physics, focusing on the significance of the relationship between physics and mathematics in describing mathematical laws in the context of a scientific theory. In the second Newtonian law of motion, space and time are crucial physical magnitudes in mechanics, but they are also mathematical magnitudes as involved in derivative operations. Above all, if we fail to acknowledge their (...) class='Hi'>mathematical meaning, we fail to comprehend the whole Newtonian mechanical apparatus. For instance, let us think about velocity and acceleration. In this case, the approach to conceive and define foundational mechanical objects and their mathematical interpretations changes. Generally speaking, one could prioritize mathematical solutions for Lagrange's equations, rather than the crucial role played by collisions and geometric motion in Lazare Carnot's operative mechanics, or Faraday's experimental science with respect to Ampère's mechanical approach in the electric current domain, or physico-mathematical choices in Maxwell's electromagnetic theory. In this paper, we will focus on the historical emergence of mechanical science from a physico-mathematical standpoint and emphasize significant similarities and/or differences in mathematical approaches by some key authors of the 18th century. Attention is paid to the role of mathematical interpretation for physical objects. (shrink)
Sir Arthur Eddington, the celebrated astrophysicist, made great strides towards his own 'theory of everything'in his last two books published in 1936 and 1946. Unlike his earlier lucid and authoritative works, these are strangely tentative and obscure - as if he were nervous of the significant advances that he might be making. This volume examines both how Eddington came to write these uncharacteristic books - in the context of the physics and history of the day - and what (...) value they have to modern physics. The result is an illuminating description of the development of theoretical physics, in the first half of the twentieth century, from a unique point of view: how it affected Eddington's thought. This will provide fascinating reading for scholars in the philosophy of science, theoretical physics, applied mathematics and the history of science. (shrink)
Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, I (...) show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematicalphysics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematicalphysics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink) (...) In contrast to Aristotelian lodestone is an extremely rare form of the mineral magnetite that occurs naturally as a permanent magnet. It therefore attracts metallic iron as well as fragments of ordinary 'inert' magnetite. This 'magic' property was known to many ancient cultures, and a powerful lodestone has always commanded a high price. By the eleventh century AD the Chinese had discovered that a freely suspended elongated lodestone would tend to set with its long axis approximately north–south, and utilized this property in the (...) magnetic compass. They also appear to have discovered that this invaluable characteristic could be handed‐on to a steel needle if the latter were contacted with, or stroked by, a lodestone.The magnetism of the lodestone was scientifically investigated by William Gilbert in the sixteenth century, when he defined its 'poles' and the well‐known rule that 'like poles repel, unlike attract'. He also studied 'inclination' and 'variation', and means to aid the preservation of magnetic power. How to concentrate it by 'arming' the lodestone with caps or pole‐pieces of soft iron was discovered in the same century. These methods have been repeated, confirmed, and improved. The lodestone occupies a vital place in the history of magnetism, but little beyond Gilbert's work can be reached by historical studies because vastly improved steel or alloy permanent magnets, and electromagnets, replaced it before quantitative measurements were developed. These techniques have therefore been applied retrospectively to both museum specimens and contemporary natural lodestones. A good source of the latter was found to be the igneous complex known as Magnet Cove, Arkansas, and this material has been used as the 'type example'. All specimens were discrete, well‐rounded, rusty brown pebbles found near the surface. Their unweathered interiors were black titanomagnetite. No significant trace element or crystallographic differences could be found between the lodestones and the magnetically inert material that always accompanied them. The magnetic moment per unit volume of the 'as‐found' Magnet Cove lodestones varied between 6.5 and 11.6 emu cm−3, which compares poorly with the hundreds of units characteritzing modern permanent magnets. Hysteresis loops gave a saturation intensity of 27–51 emu cm−3, suggesting that intensity has diminished since formation. This agrees with general experience of magnets, especially in the absence of a 'keeper'. The initial volume susceptibility of Magnet Cove magnetite was about 0.18 for low fields, and always remained <1. This means that a normal terrestrial magnetic field with a maximum vector <1 Oe is unable to induce even the low magnetic moments we see today, while a field approaching 1000 Oe is required for saturation. These parameters, and the rare occurrence of lodestones as near‐surface fragments, support suggestions that they are the product of a lightning strike upon an exposure of a suitable magnetite. Transient currents averaging 30 000 A have been measured. This would give rise to a zone of potential magnetic saturation at least 12 cm in diameter, to which some of the ejected fragments would be exposed. An attempt to determine the period elapsed since formation of the Magnet Cove lodestone was made by annealing magnetically saturated specimens at temperatures up to 500°C, and measuring J v at weekly intervals for 100 days. The decay curves visually resembled exponential functions, but mathematical tests proved that they were not strictly so. Interpretation was therefore difficult, but a pragmatic procedure involving excessive extrapolation suggested an 'age' of about 3500 years (...) wave-particle duality and the uncertainty principle, and the action-at-a-distance theory of Wheeler and Feynman. 1961 ed. (shrink)
With over 150 alphabetically arranged entries about key scientists, concepts, discoveries, technological innovations, and learned institutions, the Oxford Guide to Physics and Astronomy traces the history of physics and astronomy from the Renaissance to the present. For students, teachers, historians, scientists, and readers of popular science books such as Galileo's Daughter, this guide deciphers the methods and philosophies of physics and astronomy as well as the historical periods from which they emerged. Meant to serve the lay (...) reader and the professional alike, this book can be turned to for the answer to how scientists learned to measure the speed of light, or consulted for neat, careful summaries of topics as complicated as quantum field theory and as vast as the universe. The entries, each written by a noted scholar and edited by J. L. Heilbron, Professor of History and Vice Chancellor, Emeritus, University of California, Berkeley, reflect the most up-to-date research and discuss the applications of the scientific disciplines to the wider world of religion, law, war, art and literature. No other source on these two branches of science is as informative or as inviting. Thoroughly cross-referenced and accented by dozens of black and white illustrations, the Oxford Guide to Physics and Astronomy is the source to turn to for anyone looking for a quick explanation of alchemy, x-rays and any type of matter or energy in between. (shrink) (...) an apparently empirical debate. (shrink)
Abstract Naturalized metaphysics remains a default presupposition of much contemporary philosophy of physics. As metaphysics is supposed to be about the general structure of reality, so a naturalized metaphysics draws upon our best physical theories: Assuming the truth of such a theory, it attempts to answer the "foundational question par excellence ", "how could the world possibly be the way this theory says it is?" It is argued that attention to historical detail in the development and formulation of physical (...) theories serves as an ever-relevant hygienic corrective to the "sentiment of rationality" underlying the naturalistic impulse to read ontology off of physics. (shrink)In this book physicist Roland Omnès addresses some big questions in philosophy of mathematics. Anyone who reflects on the history and practice of mathematics and the sciences, especially physics, will naturally be struck by some remarkable coincidences. First, often newly developed mathematics was not well understood. But its successful applications and its agreement with intuitive representations of reality promoted confidence in its correctness even absent clear foundations . Later, this confidence is vindicated when a proper setting for the (...) concepts and techniques is discovered . Second, often mathematical concepts designed for one purpose later turn out to have pervasive applications that could not have been imagined by the original practitioners. Third, many of the most important results obtained in physics since the late nineteenth century were driven by the search for precise, comprehensive, consistent theoretical frameworks: the sequence special relativity, general relativity, relativistic quantum mechanics, string theory can be seen as one that increases comprehensiveness by consistent unification. The fundamental theoretical work has little to do with empirical investigation and a lot to do with mathematical and conceptual investigation of invariances and symmetries. Fourth, mathematical principles guarantee existence principles needed by physics . Such coincidences naturally invite questions: Why is confidence in the consistency of a successful piece of mathematics so often vindicated? Why does mathematics turn out to be so comprehensive and fruitful in unexpected …. (shrink)
What were the reasons of the Copernican Revolution ? How did modern science (created by a bunch of ambitious intellectuals) manage to force out the old one created by Aristotle and Ptolemy, rooted in millennial traditions and strongly supported by the Church? What deep internal causes and strong social movements took part in the genesis, development and victory of modern science? The author comes to a new picture of Copernican Revolution on the basis of the elaborated model of scientific revolutions (...) that takes into account some recent advances in philosophy, sociology and history of science. The model was initially invented to describe Einstein's Revolution of the XX century beginning. The model considers the growth of knowledge as interaction, interpenetration and unification of the research programmes, springing out of different cultural traditions. Thus, Copernican Revolution appears as a result of revealation and (partial) resolution of the dualism , of the gap between Ptolemy's mathematical astronomy and Aristotelian qualitative physics. The works of Copernicus, Galileo, Kepler and Newton were all the stages of mathematics descendance from skies to earth and reciprocal extrapolation of earth physics on skies. The model elaborated enables to reassess the role of some social factors crucial for the scientific revolution. It is argued that initially modern science was a result of the development of Christian Weltanschaugung . Later the main support came from the absolute monarchies. In the long run the creators of modern science appeared to be the "apparatchics" of the "regime of truth" built-in state machine. Natural science became a part of ideological state apparatus providing not only scientific education but the internalization of values crucial for the functioning of state. -/- . (shrink)
Fundamental notions Husserl introduced in Ideen I, such as epochè, reality, and empty X as substrate, might be useful for elucidating how mathematicalphysics (...) case study of the construction of noncommutative geometry. The perspective in Ideen I about geometry and mathematicalphysicsphysics. (shrink)
The aim of the paper is this: Instead of presenting a provisional and necessarily insufficient characterization of what mathematicalphysics is, I will ask the reader to take it just as that, what he or she thinks or believes it is, yet to be prepared to revise his opinion in the light of what I am going to tell. Because this is precisely, what I intend to do. I will challenge some of the received or standard views about (...)mathematicalphysics and replace them by a more sophisticated picture, which takes into account the methodological and philosophical roots of mathematicalphysics in Göttingen. (shrink)
On December 10th, 1947, John von Neumann wrote to the Spanish translator of his Mathematical Foundations of Quantum Mechanics: 1Your questions on the nature of mathematicalphysics and theoretical physics are interesting but a little difficult to answer with precision in my own mind. I have always drawn a somewhat vague line of demarcation between the two subjects, but it was really more a difference in distribution of emphases. I think that in theoretical physics the (...) main emphasis is on the connection with experimental physics and those methodological processes which lead to new theories and new formulations, whereas mathematicalphysics deals with the actual solution and mathematical execution of a theory which is assumed to be correct per se, or assumed to be correct for the sake of the discussion. In other words, I would say that theoretical physics deals rather with the formation and mathematicalphysics rather with the exploitation of physical theories. However, when a new theory has to be evaluated and compared with experience, both aspects mix. (shrink) | 677.169 | 1 |
Sunol CalculusAlan W.
...I encourage students to take time to be clear about the details of definitions and concepts, rather than simply memorizing theorems and procedures for solving particular problem types. For linear algebra, I find it especially helpful to use realistic examples that reveal the value of studying wh... | 677.169 | 1 |
The Geometry Junkyard website provides a broad introduction to the specifics of geometry. The site, designed by David Eppstein of UC-Irvine, provides all the basics of geometry for students and teachers alike. Topics...
Educational consultant and textbook author Jill Britton is the author of these metasites listing Web resources for grade 5-8 mathematics. Each metasite revolves around a certain topic. The third metasite listed here...
Geometric coffee tables, an introduction to polyhedra and the jitterbug (a dynamic polyhedron) that draws on the golden ratio, a collection of notes with respect to R. Buckminster Fuller's work, photos of a dome frame... | 677.169 | 1 |
Foundations of Mathematical & Computational Economics
9780324235838
ISBN:
0324235836
Edition: 1 Pub Date: 2006 Publisher: Thomson Learning
Summary: Economics doesn't have to be a mystery anymore. FOUNDATIONS OF MATHEMATICAL AND COMPUTATION ECONOMICS shows you how mathematics impacts economics and econometrics using easy-to-understand language and plenty of examples. Plus, it goes in-depth into computation and computational economics so you'll know how to handle those situations in your first economics job. Get ready for both the test and the workforce with this ...economics textbook.
Dadkhah, Kamran is the author of Foundations of Mathematical & Computational Economics, published 2006 under ISBN 9780324235838 and 0324235836. Two hundred fifteen Foundations of Mathematical & Computational Economics textbooks are available for sale on ValoreBooks.com, fifty five used from the cheapest price of $136.84, or buy new starting at $27.85 | 677.169 | 1 |
Description
The Blitzer Algebra Series combines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum appeal. Blitzer's personality shows in his writing, as he draws readers into the material through relevant and thought-provoking applications. Every Blitzer page is interesting and relevant, ensuring that students will actually use their textbook to achieve success!
Table of Contents
1. Variables, Real Numbers, and Mathematical Models
1.1 Introduction to Algebra: Variables and Mathematical Models
1.2 Fractions in Algebra
1.3 The Real Numbers
1.4 Basic Rules of Algebra
Mid-Chapter Check Point Section 1.1–Section 1.4
1.5 Addition of Real Numbers
1.6 Subtraction of Real Numbers
1.7 Multiplication and Division of Real Numbers
1.8 Exponents and Order of Operations
Chapter 1 Group Project
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Test
2. Linear Equations and Inequalities in One Variable
2.1 The Addition Property of Equality
2.2 The Multiplication Property of Equality
2.3 Solving Linear Equations
2.4 Formulas and Percents
Mid-Chapter Check Point Section 2.1–Section 2.4
2.5 An Introduction to Problem Solving
2.6 Problem Solving in Geometry
2.7 Solving Linear Inequalities
Chapter 2 Group Project
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Test
Cumulative Review Exercises (Chapters 1–2)
3. Linear Equations and Inequalities in Two Variables
3.1 Graphing Linear Equations in Two Variables
3.2 Graphing Linear Equations Using Intercepts
3.3 Slope
3.4 The Slope-Intercept Form of the Equation of a Line
Mid-Chapter Check Point Section 3.1–Section 3.4
3.5 The Point-Slope Form of the Equation of a Line
3.6 Linear Inequalities in Two Variables
Chapter 3 Group Project
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Test
Cumulative Review Exercises (Chapters 1–3)
4. Systems of Linear Equations and Inequalities
4.1 Solving Systems of Linear Equations by Graphing
4.2 Solving Systems of Linear Equations by the Substitution Method
4.3 Solving Systems of Linear Equations by the Addition Method
Mid-Chapter Check Point Section 4.1–Section 4.3
4.4 Problem Solving Using Systems of Equations
4.5 Systems of Linear Inequalities
Chapter 4 Group Project
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Test
Cumulative Review Exercises (Chapters 1–4)
5. Exponents and Polynomials
5.1 Adding and Subtracting Polynomials
5.2 Multiplying Polynomials
5.3 Special Products
5.4 Polynomials in Several Variables
Mid-Chapter Check Point Section 5.1–Section 5.4
5.5 Dividing Polynomials
5.6 Dividing Polynomials by Binomials
5.7 Negative Exponents and Scientific Notation
Chapter 5 Group Project
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Test
Cumulative Review Exercises (Chapters 1–5)
6. Factoring Polynomials
6.1 The Greatest Common Factor and Factoring By Grouping
6.2 Factoring Trinomials Whose Leading Coefficient Is 1
6.3 Factoring Trinomials Whose Leading Coefficient Is Not 1
Mid-Chapter Check Point Section 6.1–Section 6.3
6.4 Factoring Special Forms
6.5 A General Factoring Strategy
6.6 Solving Quadratic Equations By Factoring
Chapter 6 Group Project
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Test
Cumulative Review Exercises (Chapters 1–6)
7. Rational Expressions
7.1 Rational Expressions and Their Simplification
7.2 Multiplying and Dividing Rational Expressions
7.3 Adding and Subtracting Rational Expressions with the Same Denominator
7.4 Adding and Subtracting Rational Expressions with Different Denominators | 677.169 | 1 |
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This text provides an introduction to the entire modeling process. Throughout the book, students practice key facets of modeling, including creative and empirical model construction, model analysis, and model research. The authors apply a proven six-step problem solving process to enhance a student's problem solving capabilities. Rather than simply emphasizing the calculation step, the authors first ensure that students learn how to identify problems, construct or select models, and figure out what data needs to be collected. By involving students in the mathematical process as early as possible, beginning with short projects, the book facilitates their progressive development and confidence in mathematics and modeling. | 677.169 | 1 |
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The authors show that there are underlying mathematical reasons for why games and puzzles are challenging (and perhaps why they are so much fun). They also show that games and puzzles can serve as powerful models of computation?quite different from the usual models of automata and circuits?offering a new way of thinking about computation. The appendices... more...
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Math is not as daunting as it seems, it's all about following simple rules. Repeated use of these rules builds understanding and confidence. This article will teach you how to use and understand those rules. You also must pay attention and be positive of your answers and your mark.
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Steps
Pay attention to your teacher. If you're in a fast paced math class, they'll probably only teach a concept once and move on. This means that if you don't pay attention, you've just missed your one chance to learn it.
Do your homework. This will help you understand the concepts more clearly. Your homework will probably be the only time you get to practice what you just learned before a test, so you need to make sure you do it! If you can't attend the class for a day, ask your teacher to give you the homework you missed as soon as possible.
3
Talk to your teacher when you don't understand something. A math class is not a class you want to only think you know what you're doing.
Learn the terminology. Knowing what is asked of you is half the problem. Math is its own language (solve, expand, simplify, etc.). Becoming accustomed to this will really benefit you. Highlight key words in the question to ascertain what you need to find out. Attempting past exam papers will give you a feel of what to expect; many of these are available for download from your examination board's website..
Buy a good calculator depending on the class you're taking. If you're taking a basic algebra class, a scientific calculator should suffice. If you're taking a calculus class, a graphing calculator will probably be necessary. Your teacher or professor should have a good idea of what's necessary as well, so asking them at the beginning of the term would be a good idea.
Sit next to friendly, positive people. These should be people you can turn to when you have a question about something, or don't know what to do for a certain problem. However, make sure they won't just give you the answer.
Find extra learning materials. Oftentimes, different sources will be better explaining some things than others. If you can expose yourself to a different explanation of a concept, you may understand something a lot better. Only use "cheat books" to check your answer.
Show all your working. Most of the time, your teachers don't care as much about what you put down as an answer as the working that you show. Many will only give partial credit for the correct answer; the rest must be earned by showing your working correctly.
Be organized. If you aren't organized, doing all that homework will do nothing to help you. Many times, you think you're organized but to gain a true perspective on whether you are organized or not, ask someone you know that will give you an honest opinion.
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Vine someone you know has taken the class before you, try talking to them for tips or notes.
Consider taking notes on the lesson if you can keep up with what the teacher is saying. This will help you remember the material better and let you go back if you forget something in time.
Do as many questions as possible, you will eventually see the steps you need to follow to solve particular problem. Going above and beyond and doing more problems than assigned will not only boost your understanding of the concept, but will put you on your teacher's good side.
Try not to forget the formulas. If you revise formulas frequently, it'll be hard to forget them easily, so keep refreshing your memory by studying math often!
Don't be too shy to ask questions.
Produce formula flash cards. For example on one side you would write the question "What is the area of a triangle" and on the other side you would have "Width x Height / 2 ". Keep a pile with you and go over them when you find some free time, for instance when you're at the bus stop. In addition, if you have many flash cards, you might want to consider hole punching them and putting them on a ring.
Put all of the formulas that you learn onto one sheet of paper to study off of.
Never underestimate yourself - give yourself a chance!
Don't slack around. Always listen to your teacher because they might explain the question even more. You get most of your knowledge from listening in class. Ask your teacher questions before, after, or in class. You could probably be able to leave study hall or homeroom too.
Cheating is not having a good grade it is making you when you are in collage a bully guy or girl.
Warnings
Don't sit in the back or next to the class clown, unless s/he is helpful | 677.169 | 1 |
...The math content on the disks were Algebra 1, Algebra 2, and Geometry. I practiced and mastered them all during my downtime after school. This attributed greatly to my understanding of mathematics today.
...As students progress in math from the basics into preparation for algebra, they need to begin thinking of numbers in more abstract ways. They need to understand different kinds of fractions and negative numbers, which we don't see around us in our everyday world. They need to think of addition and multiplication in new and more advanced ways.
...As an attorney, my law school experience included various courses in criminal law as well as criminal procedure. In my first years of professional practice, I handled different aspects of criminal law; not only street crimes, but other crimes such as financial crimes and fraud. More recently, ... | 677.169 | 1 |
Download
Chapter 8: Exponents and Exponential Functions
Exponential functions occur in daily situations; money in a bank account, population growth, the decay of carbon-14 in living organisms, and even a bouncing ball. Exponential equations involve exponents, or the concept of repeated multiplication. This chapter focuses on combining expressions using the properties of exponents. The latter part of this chapter focuses on creating exponential equations and using the models to predict. | 677.169 | 1 |
Vectors
7 videos
4 skills
We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations like in a more basic algebra course) and defining some basic operations (like addition, subtraction and scalar multiplication). | 677.169 | 1 |
MAT-125-20, Elementary Math Models (SP15)
An introduction to the methods used in applying mathematics to real problems from the physical life and social sciences. Applications will be investigated numerically and graphically using a graphing calculator before the mathematical model is expressed algebraically. Algebraic concepts will be reviewed and developed as needed. This course will help students refresh and strengthen their mathematics and increase their understanding of its usefulness.
REQUIRED MATERIALS:
-Elementary Mathematical Models, Dan Kalman
-Mathematical Association of America: 1997.
-A graphing calculator will be needed. It should be a TI-83 Plus, TI-83, or TI-84.
Please note that cell phones and PDA's may not be used as calculators during an exam.
LEARNING OBJECTIVES:
To enable the student to
explore a variety of basic and useful mathematical models;
strengthen and extend basic algebra skills;
utilize a graphing calculator as a tool in understanding mathematical models;
learn by collaborative work with other students;
develop confidence in her mathematical judgment.
METHODS OF INSTRUCTION:
In general, each class will include time for questions on homework and for guided problem solving as well as for the presentation of new material.
REQUIREMENTS/ASSIGNMENTS FOR THE COURSE:
There will be two tests and a comprehensive final exam.
Homework will be assigned for each chapter we cover. Late homework will be accepted if received by the next class but will be penalized by 20%. Homework that is later than that will not be accepted for a grade.
We will have time during several classes designated for collaborative work. During these classes you will work in groups on problems designed to help you integrate and extend the material you have been studying. If a collaborative assignment is missed, the work is to be completed in a timely fashion; however, full credit will not be given.
Students are required to be registered for the Joule site for this course.
GRADING POLICIES AND COURSE EXPECTATIONS:
Students are expected to attend every class on time and will be held responsible for any material covered, any announcements made in class, and any changes in the schedule whether they are present or not.
If you have a legitimate reason for missing a test, I must be contacted before the testing has taken place. If I cannot be reached, leave message via phone or email. Otherwise a make-up will not be given unless there is a documented emergency. Any make-up must be taken before class within a week of the original test date. Only one make-up test may be taken. Any missed test that is not made up will result in a grade of zero. | 677.169 | 1 |
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Created by David Smith for the Connected Curriculum Project, the purpose of this module is to study data that may be modeled by sinusoidal functions; in particular, to determine average level, period, frequency,...This lesson from Illuminations teaches students the features of sine and cosine graphs. Students use uncooked spaghetti to demonstrate the properties of the unit circle, which they will then represent with graphs. It is...
Created by Lang Moore and David Smith for the Connected Curriculum Project, this is a module to review concepts of inverse functions, and to use those concepts, together with functions defined by integrals, to develop...
In this animated and interactive object, the learner uses a TI-86 calculator to solve for the magnitude of the resultant and of one of the component forces in a right triangle. Target Audience: 2-4 Year College... | 677.169 | 1 |
Doing 2nd yr uni maths after having not done maths in 2 years =O
Hello!
I'm Chloe. I'm doing two 2nd year advanced maths courses at uni (Real and Complex Analysis + Vector Calculus and Linear Algebra) and I've been struggling a lot because it's been two years since I've done uni maths.
I've forgotten basic things like how to integrate trig functions, how to use trig identities to that end, etc. Even when I learn it again, I've lost a lot of computational speed!
I'm about 1.5 months away from final exams and it's not looking too good.
Re: Doing 2nd yr uni maths after having not done maths in 2 years =O
Hi, If you get stuck you can always post on the forums here, and they can guide you but don't expect them to answer your question without you even attempting it, because you're the one who has to learn it so not trying to scare you or anything but try your best and post what troubles you, that's what I've learned since I've used the math help forum, and the people who help you are great, and it's sometimes crazy how they can stand some of the things I've seen lol or even me at times I guess. Another thing, you can always ask your prof if you don't understand something during office hours or whatever, and also some people on youtube have great guides on trig and etc, I recommend Patrickjmt, he's awesome for pre - calc/ linear algebra/ calc, you name it. Good luck in your future studies! | 677.169 | 1 |
The student studies math using the incremental method. This integrated program includes varied content, such as: basic number concepts, estimations, functions, two and 3 digit computations, geometry and even algebra. Saxon uses the continual and incremental practice with word problems, measurement, scale and graph reading, as well as arithmetic.
Math 5/4 helps students extend their understanding of critical skills and concepts such as number sense, numeration, numerical operations, measurements and geometry, patterns, relationships, math functions, and data manipulation through charts.
Softcover text has 627 pages.
Softcover Solution Manual has 302 pages.
Softcover Tests and Worksheets has 283 pages. To purchase the Tests and Worksheets separately, please click on the following link: Saxon 54 Tests & Worksheets$78.00 | 677.169 | 1 |
books.google.com - This book includes over 1500 exercises, many with multiple parts, ranging in scope from routine to fairly sophisticated, and ranging in purpose from basic application of text material to exploration of important theoretical or computational techniques. The structure of the book permits instructors and... Algebra
Abstract Algebra
This book includes over 1500 exercises, many with multiple parts, ranging in scope from routine to fairly sophisticated, and ranging in purpose from basic application of text material to exploration of important theoretical or computational techniques. The structure of the book permits instructors and students to pursue certain areas from their beginnings to an in-depth treatment, or to survey a wider range of areas, seeing how various themes recur and how different structures are related. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible. Contains many topics not usually found in introductory texts. Students are able to see how these fit naturally into the main themes of algebra.
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User Review - Ming - Goodreads
Used this book for Group Theory, Ring Theory, Module Theory and Galois Theory for the Honors Algebra Sequence in UChicago. Pretty good book - some very basic exercises, some suitably challenging ones ...Read full review
Review: Abstract Algebra
User Review - Justin - Goodreads
Wow. Another page turner. This is a wonderful text for anyone looking to learn abstract algebra. No wonder it's called the algebra bible! David S. Dummit weaves in tons of juicy examples for basically ...Read full review | 677.169 | 1 |
The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic.Algebraic Number Theory takes the reader from unique factorisation in the integers through to the... more...
This book develops the method of "algebraic patching" to realize finite groups and, more generally, to solve finite split embedding problems over fields. The method succeeds over rational function fields of one variable over "ample fields". more...
Using a computational algebra approach, this work addresses the center and cyclicity problems as behaviors of dynamical systems and families of polynomial systems. The self-contained text contains exercises as well as historical notes and algorithms. more...
Decomposing an abelian group into a direct sum of its subsets leads to results that can be applied to a variety of areas, such as number theory, geometry of tilings, coding theory, cryptography, graph theory, and Fourier analysis. Focusing mainly on cyclic groups, Factoring Groups into Subsets explores the factorization theory of abelian groups.... more...
The concept of factorization, familiar in the ordinary system of whole numbers that can be written as a unique product of prime numbers, plays a central role in modern mathematics and its applications. This exposition of the classic theory leads the reader to an understanding of the current knowledge of the subject and its connections to other mathematical... more...
Field Arithmetic explores Diophantine fields through their absolute Galois groups. The treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. more...
The book presents foundations of the micropolar continuum mechanics including a short but comprehensive introduction of stress and strain measures, derivation of motion equations and discussion of the difference between Cosserat and classical (Cauchy) continua, and the discussion of more specific problems related to the constitutive modeling, i.e.... more...
Mathematical algorithms are a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. This book provides a bridge between algebraic geometry and geometric modelling algorithms, formulated within a computer science framework. Apart from the algebraic geometry topics covered, the entire book is based on the unifying concept... more...
Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resouce for those already grounded in it, consists of 21 expository surveys... more... | 677.169 | 1 |
Prealgebra Mathxl Tutorial interactive tutorial CD-ROM provides algorithmically generated practice exercises that are correlated at the objective level to the exercises in the textbook. Every practice exercise is accompanied by an example and a guided solution designed to involve students in the solution process. Selected exercises may also include a video clip to help students visualize concepts. The software provides helpful feedback for incorrect answers and can generate printed summaries of students' progress. | 677.169 | 1 |
Setting linear algebra problems
1.
Setting Linear Algebra Problems
John D. Steele
School of Mathematics
University of New South Wales
Sydney NSW 2052
Australia
j.steele@unsw.edu.au
Abstract: In this report I collect together some of the techniques I have evolved for setting
linear algebra problems, with particular attention payed towards ensuring relatively easy arithmetic.
Some are given as MAPLE routines.
AMS Classification: 15-00, 15-02, 15-04, 15A36.
1 Introduction
When I first began teaching Linear Algebra at the University of New South Wales, I was
disappointed by the level of arithmetic competence of my students, and annoyed to find that
problems were being set where this lack of competence meant one could never really see
whether students understood the material and techniques or not. I began to think through
these problems backwards, and ask "How can I set these problems so that the arithmetic
does not obscure the mathematics?"
I am sure many people have gone through the same process and evolved their own ideas
and ad hoc methods of conjuring up the appropriate matrix or set of vectors, but I looked
in vain for anything on paper. This report is intended to set down some of the useful tricks
I came up with while setting linear algebra problems. The point of this exercise is to pay
careful attention to having the arithmetic as easy as can be arranged. This latter requirement
means that vectors and matrices should be integer where possible, or rational numbers with
small denominator, or simple, small surds if that is the best that can be done. Of course,
there is a downside to this, as it is also a useful lesson for students that the numbers do not
always work out, but one can also use these tricks to make sure that they do not.
I will only be concerned with Rn , given that any other finite-dimensional vector space
over R can be isometrically mapped to a suitable Rn . Of course, R can usually be replaced
with C.
1
2.
Most of these tricks are essentially routine applications of basic linear algebra, and there-
fore most of the references given are to textbooks. A few use slightly more sophisticated
ideas to ensure the methods work. I expect no result given here is either deep or new —
most are very simple and/or very old. I would expect most readers reaction to them would
be along the lines of "Well, obviously!" A few are perhaps not as well known, or as fully
appreciated, as they should be.
Throughout this report I will refer to MAPLE procedures that I have written to use some
of these ideas. These procedures are given in the file MakeMatrix , available at
and are based around the LinearAlgebra package introduced in Maple 7. Similar routines
based on the older linalg package are also available at
In one case the method is given as a semi-MAPLE algorithm. The routines are very
simple, and readers will find it easy to write their own versions in whatever language they
are most comfortable.
2 Vectors of integer norm
In R2 , a vector will have an integer norm if and only if its components are the first two
numbers in a Pythagorean triple. All these have been known since antiquity, and are given
by the following well-known result:
Theorem 1 (Pythagorean triples). The three numbers x, y, z form a Pythagorean triple
if and only if there are three integers u, v and k such that
x = k(u2 − v 2 ) y = 2kuv z = k(u2 + v 2 )
(or the same with x and y exchanged).
Typical examples of such triples are, of course, (3, 4, 5), (5, 12, 13) etc, leading to vectors
like (−3, 4), (12, −5) etc.
In R3 the situation is governed by the theorem, see e.g. [3]:
Theorem 2. The square of an integer n can be decomposed non-trivially into the sum of
two or three squares if and only if n is not a power of 2.
Typical examples of these that are useful in linear algebra are, for example
32 = 12 + 22 + 22
72 = 22 + 32 + 62
92 = 12 + 42 + 82 = 42 + 42 + 72
112 = 62 + 62 + 72 = 22 + 62 + 92
122 = 42 + 82 + 82
2
3.
as well as "degenerate" cases arising from Pythagorean triples.
For higher dimensions, we have Lagrange's theorem (see, e.g. [3])
Theorem 3 (Lagrange's Theorem). Any integer can be decomposed into the sum of at
most four squares.
One of the most useful examples of this is that 22 = 12 + 12 + 12 + 12 .
3 Matrices of a given determinant
One of the standard results of linear algebra is the P LU decomposition (see e.g. [4]):
Theorem 4. Any matrix can be decomposed into the product of a permutation matrix P , a
unimodular lower triangular matrix L and an upper triangular matrix U .
There are two uses for this theorem. Firstly, the choice of L will govern the number (and
difficulty) of the row operations for reducing the product A = LU back to echelon form U :
choosing any vector x then gives us the right hand side for a set of linear equations Ax = b.
Secondly, we can use this theorem in reverse to create a matrix of a given determinant. All
that is required is to take the product of a unimodular lower triangular matrix and an upper
triangular matrix with the determinant desired: the MAPLE routine RandomMatrix has
options that create such matrices. A MAPLE procedure MakeDetMat to make the required
matrices is given in the file MakeMatrix . The columns (or rows) of a matrix of non-zero
determinant give bases of the appropriate space of course.
A most useful way of using the P LU decomposition is to create matrices of determinant
±1. These are particularly good for problems involving calculating the inverse of a matrix
(there will be no fractions in the solution), and for similarity transformations. The following
easy theorem illustrates why this is so:
Theorem 5. Let A be a square integer matrix. Then A−1 is an integer matrix if and only
if det A = ±1.
Proof. Since det A−1 = (det A)−1 , the only if part is easy. The if part is a consequence of the
well-known result that A−1 is the (classical) adjoint (the transpose of the matrix of minors,
which will be an integer matrix) divided by det A. Or, alternatively, it follows from Cramer's
Rule.
4 Matrices with a given kernel
There are several ways of doing this. One is to use the normal form [6] directly:
3
4.
Theorem 6. Any n × m matrix A can be written as the product QN P −1 where P ∈
GL(m, R), Q ∈ GL(n, R) and the normal form N is of the form
Ir O
O O
with rank(A) = r, Ir the r × r identity matrix and O signifying zero matrices.
In order to find P in this theorem, one finds the kernel of A and extends it to a basis of
m
R . This basis then forms the columns of P , with the the last m − r columns of P spanning
the kernel of A. For Q one takes the images of the extension vectors and extends them to a
basis for Rn .
So, to get a matrix with a given kernel, extend the basis for the kernel by adjoining
random vectors as desired. Then premultiplying P −1 by a matrix of the form of N in the
theorem and of the required size will give the matrix A with given kernel. A premultiplication
by any invertible Q can be applied as a disguise.
A second way is to jump the need for extending a basis by using the fact that (W ⊥ )⊥ = W
for finite dimensions. Take a matrix A whose rows are a basis for the desired kernel. Then
a matrix whose row space is ker(A) has the desired kernel.
The procedure MakeKernel in MakeMatrix does this and it is designed so that the output
will always be an integer matrix with the minimum number of rows: premultiplication by
any matrix of full rank and more rows will not alter the kernel.
Alternatively, one can build a suitable matrix directly from row reduction. Suppose W
is a linearly independent set of r vectors in Rn , and is to be a basis for the kernel. Then
create the matrix whose first r rows are the members of W and whose last row is a general
vector (v1 , v2 , . . . , vn ) in Rn . This is easily done with Matrices and Vectors in MAPLE. Then
row reduce the matrix to echelon form. There will be n − r entries in the echelon form that
are linear equations in the entries vi and give conditions for a vector to be in the span of
W . The coefficients of these equations are the rows of a (n − r) × n matrix whose kernel
is therefore W . Any matrix whose rows are linear combinations of the rows of this latter
matrix will have kernel containing W , so one can multiply on the left by any matrix of full
rank to get any size of matrix required.
5 Leontieff Input-Output Matrices
The problem with creating Leontieff matrices is to ensure that we have a consumption (or
technology) matrix A, a demand vector d and an output vector x all of which have non-
negative entries and for which (I − A)x = d. A consumption matrix for which each entry of
(I − A)−1 is non-negative is called productive, and such a consumption matrix will always
do the job.
One simple way of making a productive consumption matrix is to rely on the theorem
(e.g. [1], p. 615) that non-negative A is productive if each row (or column) sum of A is less
than 1.
4
5.
More generally, a non-negative matrix A is productive if and only if there is a non-
negative x such that x − Ax is non-negative ([1] p. 615). So create an initial non-negative
matrix A and non-negative output vector x and calculate y = A x; we need to make the
entries of y less than the corresponding entries of x. One can fiddle a little with the entries
of A , if it is small and only one problem is being set to make this the case. Alternatively,
scale A by a suitable factor: if r ≥ max{yi /xi }, let A = A /r. Then the demand vector is
set to be (I − A)x, which will have non-negative entries.
6 Orthogonal matrices
By orthogonal matrices, I mean (not necessarily square) matrices Q whose columns are
an orthonormal set, so that QT Q = I. Orthogonal 3 × 3 matrices with determinant 1
are rotations, and are dealt with separately below. More general orthogonal matrices are
useful for building sets of vectors for applying the Gram-Schmidt process, or matrices to be
factorised as QR, i.e. orthogonal times upper triangular, by the Gram-Schmidt process.
To make a square orthogonal matrix of any size one can rely on the Cayley transform:
if A is any anti-symmetric n × n matrix, then the matrix (I + A)(I − A)−1 is square and
orthogonal, and can be cut down to size by deleting columns if necessary. The problem with
this method is that is is hard to control the numbers that come out.
Another way to get these matrices is to generate a set of mutually orthogonal vectors
by hand, and then normalise, giving the orthonormal set that forms a basis: of course, this
is what the Gram-Schmidt process is designed to do in the first place. This can easily be
done by trial and error (possibly using one or more columns from the Cayley transform)
for two vectors. But for more than that trial and error becomes tedious if one tries to
make the numbers come out. Instead, one can expedite matters by using the Hodge dual
of the outer product in Rn , which gives a generalised cross product. This dual is given by
Σ i...k...l vk . . . wl , where i...l is the alternating symbol — i...l is the sign of the permutation
(i . . . l) or zero if (i . . . l) is not a permutation — and the sum is over the components of the
vectors v, . . . , w. Each column in the Hodge dual is then orthogonal to each of the vectors
v,. . . ,w.
For example, if one begins with the orthogonal vectors v = (5, 3, 1, 1) and w = (1, −3, 5, 1)
found by trial and error, then my procedure
GenCross in MakeMatrix applied to these
0 −6 0 18
6 0 −6 −24
vector yields the matrix , and taking the vectors v, w and the second
0 6 0 −18
−18 24 18 0
√
5 1 − 2
1 3 −3 0
column of this matrix as our orthogonal set we get the orthogonal matrix 6 √ .
1 5 2
√
1 −1 4 2
5
7.
There is little freedom here: the equations are unpleasant to solve directly except for the
2 × 2 case. Anything else should be designed to be tackled with a QR factorisation, with A
set up to be as easy to factorise as possible. Of course, one is constrained by the fact that
in fitting a polynomial of degree n, A will of necessity consist of the first n + 1 columns of a
Vandermonde matrix so there is no freedom of choice except in the second column.
In the case of lines, while with integer points the solution is always rational, it is possible
to arrange for the solution to be integer, as follows: Suppose we are looking for the line of
best fit to the data points (xi , yi ), where the xi are known and the yi are to be chosen to make
the final answer integer. Let x ∈ Rn be the vector (x1 , . . . , xn ) and A the coefficient matrix of
the least squares problem, with first column 1s and second column x. Define ∆ = det(AT A)
and M = ∆(AT A)−1 to be the classical adjoint of AT A. Now calculate ker M in Z∆ and
form the matrix B by augmenting AT with any vector in kerZ∆ (M ). Then for any integer
vector w in ker B, the first n components can be used as the yi in the set of data points,
since then AT y is ∆ times an integer vector. Of course, the vector y chosen should not be in
ker AT or the problem is trivial. The procedure MakeLeastSquares in MakeMatrix takes
the vector x and returns a matrix M such that any vector in the image of M is a suitable y.
If a more general least squares problem is to be set, the easiest course is to build a matrix
A whose QR factorisation is easy and use that.
The interesting general question is: can one start with the normal equations M x = z for
any symmetric M with full rank (so the solution is unique) and get a matrix A and a vector
y for which these equations are the appropriate normal equations?
Thus, given symmetric invertible M and z ∈ Rn , we want to set up a problem Ax = y,
so that M = AT A, A is m × n and z = AT y. Clearly y ∈ Rm .
Firstly, the rank of A: any vector in ker A is in ker M and ker M is trivial so ker A is
trivial. Therefore A has rank n and m ≥ n.
The next point to make is that M must positive definite, since
M u, u = AT Au, u = Au, Au ≥ 0,
and M has no zero eigenvalues.
Finally, if we have such an A, we can solve AT y = z iff z ∈ (ker A)⊥ (see e.g. [5] p. 198).
As ker A is trivial, we can always find a suitable y given z.
These necessary conditions are in fact sufficient. To prove this, we make use of the
Singular Value decomposition, see e.g. [4] section 6.2. The matrix A can be decomposed as
QΛP T , where Q is m × m orthogonal, P is n × n orthogonal and Λ is m × n and of the form
D
, where D is n × n diagonal and O is an appropriately sized zero matrix. It follows
O
that M = P D2 P T , and so the diagonal entries of D are square roots of the eigenvalues of
M (giving a second proof of the necessity of positive definiteness). Note that if Λ+ is the
D−1
matrix , so that A+ = P Λ+ QT is the Moore-Penrose generalised inverse, the solution
O
to the least squares problem is x = A+ y [4].
7
8.
Thus, given positive definite M , we can find D and P , and then we need only extend D
to Λ and rig a suitable Q to get A. Whether this can be done so that A is integer is another
matter, as is thinking of a problem for which A arises as required.
9 Householder matrices and QR by Householder
By the Householder matrix for a vector v ∈ Rn of norm M , I mean the matrix of the
reflection that swaps v with w = ±M e1 , where e1 is the first member of the standard basis
for Rn and the sign is chosen to be the opposite to that of v · e1 , see [4]. Since this matrix is
given by I − 2uuT , where u is the unit vector in the direction of v − w, the vector v must
be of integer norm if the calculation is to be possible without introducing nested surds.
Creating a matrix that can be QR factorised by hand with Householder matrices (see [4])
is best done backwards. Since Q is the product of Householder matrices, it is square and R
is the same dimension as the matrix to be factorised. I illustrate one way of doing this with
a 4 by 3 matrix.
The last matrix to be transformed, A3 , will be 2 × 1, and should be chosen to be a vector
of integer norm, for example (4, 3). The second last matrix to be transformed, A2 , will be
a b
3 × 2, and after multiplication by a Householder matrix will look like R2 = 0 4 for our
0 3
choice of A3 . It follows that the first column of A2 will be a vector of norm a, so choose a
and the column c2 appropriately. Then A2 is the matrix Q2 R2 , where Q2 is the Householder
matrix of c2 . The constant b is chosen to make A2 as simple as possible (integer if possible).
−1 −1 1
For example, with a = 3, c2 = 2 and b = −1, A2 is 2 3.
−2 −2 4
c d e
0 −1 1
Finally, the matrix A will after the first transformation look like R1 = , so
0 2 3
0 −2 4
as before choose c and appropriate first column c1 of A of norm c, then A = Q1 R1 where Q1
is the Householder matrix of c1 , with d and e chosen to give as simple a matrix as possible.
For example, with c = 7, c1 = (−2, 0, 6, 3)T , d = 3 and b = 1 we get
−2 0 4 −30 30 −96 3 7 3 1
0 −1 1 1 0 −35 −14 −98 0 3 −1
A= =
6 4 1 105 90 50 −13 −16 0 0 −5
3 −1 3 45 −80 −38 34 0 0 0
Note that the product of the norms of the columns will occur as the factor before the matrix
Q.
8
9.
10 Matrices with a given Jordan form
Firstly, to create a diagonalisable matrix, the simplest way is to create a matrix P with
determinant ±1 and then perform a similarity transformation P DP −1 , where D is a diagonal
matrix with the desired eigenvalues. The columns of P then form the eigenvectors.
A simple extension of this idea gives non-diagonalisable matrices with a given Jordan
canonical form [6]. Rather than performing a similarity transformation on a diagonal matrix,
perform it on the appropriate Jordan form. Alternatively, use the matrix formed from blocks
given by companion matrices, since the companion matrix of the polynomial (t − λ)n has
Jordan form consisting of a single Jordan block. The routine MakeJordan in MakeMatrix
takes the first of these approaches. It can also be used with a given transition matrix.
11 Orthogonal matrices in R3
I have three ways of calculating these matrices, which geometrically are rotations in R3 . If
one wishes to control the axis of rotation and the angle, use the following parameterisation
of SO(3):
0 −u3 u2
Theorem 7. If u = (u1 , u2 , u3 )T is a unit vector in R3 and U = u3 0 −u1 then
−u2 u1 0
the matrix
Q = I3 cos θ + uuT (1 − cos θ) + U sin θ
is the matrix of the rotation about the direction of u through angle θ.
Proof. That QT Q = I3 can be checked with a direct calculation. Since det Q is a continuous
function from R to {−1, 1} that clearly has value 1 at 0, it has value 1 for all θ, and so
Q ∈ SO(3). Then since U u = 0, and uT u = 1, Qu = u, so u is the direction of the axis.
Finally Tr Q = 1 + 2 cos θ, so that θ is the angle of rotation.
The MAPLE procedure MakeRot in the file MakeMatrix can use this method.
In order for the final matrix in the first method to have no (or few) surds, it is preferable
to match up any surds in the direction and the angle of rotation. For example, with u =
2 −1
√ √ √ 2
1
(1/ 3, 1/ 3, 1/ 3) the rotation about u through angle π/3 is −1 2 2 .
3
2 −1 2
The other two methods of creating a rational rotation matrix can be used if one is not
concerned with the axis and angle. One is to use the Cayley transform, as mentioned in
section 6. The other is to take, say, two orthogonal vectors of unit norm as the first two
columns and their cross product as the third. For example 1 (2, 3, 6)T and 1 (3, −6, 2)T are
7 7
orthogonal, and their cross product is 1 (6, 2, −3)T .
7
The question then arises as to when this latter method can be used: in other words, given
a vector v = (a, b, c) in Z3 , is there a second vector whose entries are some permutation of a, b
9
10.
and c — with sign changes — that is orthogonal to v? There are three possibilities, excluding
re-arrangements and changes of overall sign, depending on how many of the components
change position (I am grateful to Peter Brown [2] for the following analysis):
1. If a2 + b2 = c2 , then (a, b, −c) is orthogonal to v. For example (3, 4, 5) and (3, 4, −5).
2. If c2 = 2ab then (b, a, −c) is orthogonal to v. For example (1, 2, 2) and (2, 1, −2).
3. If a−1 = b−1 +c−1 then (c, a, −b) is orthogonal to v. For example (2, 3, 6) and (6, 2, −3).
The Diophantine equation in (3) has been studied since antiquity. Given any a it is
possible to find a b and c to satisfy the equation: factorise a2 = pq where p ≥ q, and then
b = (q + a) and c = (p + a) solve the equation.
√
The first of these cases will leave a 2 in the matrix, and is therefore not suitable. The
other two methods both lead to vectors of rational norm, and thence to rational orthogonal
matrices. In case 2, the vectors (a, b, c) etc. will all have norm a + b, and in case 3 the vectors
will have norm |c − bc2 − bc + b2 |, as can easily be checked. In both cases we get the nice
form "rational times integer matrix".
Similar methods could be used in higher dimensions. For example, in R4 , the vector
(a, b, c, d) is orthogonal to (a, b, c, −d) if a2 + b2 + c2 = d2 etc.
12 Symmetric Matrices with given eigenvalues
What we wish to do here is ensure that the matrix we end up with has small integer entries.
At first sight this is relatively simple: take a rational orthogonal matrix and use it in a
similarity transformation of a diagonal matrix with the desired eigenvalues. Then multiply
the final matrix by the appropriate integer to clear all fractions. Unfortunately, I have found
this simple idea of somewhat limited usefulness. The eigenvalues end up being n2 times the
ones you began with (assuming they were integers to start with). In order to try to "clear
up" some (at least) of the fractions, it is necessary to be careful in the choice of eigenvalues.
a
The 2 × 2 case is very easy: if one eigenvector is ∆−1 , ∆ = 1/(a2 + b2 ), then the
b
−b
other is ∆−1 . We take the eigenvalues to be 0 and k∆ (which can be shifted by adding
a
λI), and then the matrix is
b2 k bka
.
bka a2 k
For the 3 × 3 case, proceed as follows:
1
Let Q = n P , where P is an integer matrix, be orthogonal. Define A = QΛQT , where
Λ is the diagonal matrix with entries λi and define M = n2 A. We want to solve a linear
system in integers, so use the MAPLE command
syss:=seq(M[1,i]-n^2*d[i]=0,i=1..3),seq(M[2,j]-n^2*d[2+j]=0,j=2..3),
10
11.
M[3,3]-n^2*d[6]=0;
where the d[i] are dummy variables, to set up a system of 6 equations in 9 unknowns. The
command
G:=GenerateMatrix([syss],[seq(lambda[i],i=1..3),seq(d[k],k=1..6)])[1];
extracts the matrix of this system, and we want the kernel of G: set kk:=NullSpace(G);
Then if one takes an arbitrary linear combination of the vectors in this kernel, the first
three components will give you three eigenvalues that will ensure that A is an integer matrix.
For example, with the matrix of a rotation through π/3 around (1, 1, 1) (see previous
section), the kernel of the matrix G is found to be spanned by
{(3, −3, 0, 0, 2, −2, −1, 0, 1), (−5/2, 2, 2, 0, −1, 2, 3/2, 1, 0), (−2, 4, 1, 1, −2, 2, 2, 0, 0)}
and taking xv1 + 2yv2 + zv3 , say, the eigenvalues are 3x − 5y − 2z, −3x + 4y + 4z and 4y + z
for any choice of x, y and z, so for example, with eigenvalues a = −5, b = −2 and c = 1, A
is an integer matrix.
Naturally, all the eigenvalues can be shifted by λ by the addition of λI. And of course
this all works for higher dimensions, suitably modified. The procedure MakeSymmetric in
MakeMatrix will provide the matrix A with given eigenvectors.
It is also possible to cook up symmetric matrices more directly, with less control over the
eigenvalues and eigenvectors. For example the matrix
1 b c
b b2 bc
c bc c2
has eigenvalues 0, 0 and 1 + b2 + c2 .
13 Matrices with easily calculated exponential
The n × n matrices A whose exponential is easiest to calculate are either those with only one
eigenvalue, diagonalisable matrices whose eigenvalues are ±1, or matrices with eigenvalues
0 and any one other non-zero, non-deficient eigenvalue µ.
In the first case, since (A − λI)m = O for some m ≤ n, the exponential can be expressed
as eλt times a polynomial in t(A − λI) of degree m.
In the second case, A2 = I, since the minimal polynomial is λ2 − 1, and one can calculate
the exponential directly from the series definition.
In the third case, Am+1 = µAm , where m is the size of the largest Jordan block for
eigenvalue 0, since the minimal polynomial is λm (λ − µ), and again one can calculate the
exponential directly from the series.
All these types can easily be created using the above methods.
Failing that, ensuring that there is a basis of generalised eigenvectors that is as easy to
invert as possible will simplify any calculations.
11
12.
14 Systems of linear differential equations with con-
stant coefficients
The basic philosophy here is to ensure that the vectors that occur are (generalised) eigen-
vectors. So to solve, for example, y (t) = Ay(t) + b(t), by the formula y = exp(tA)z(t) with
z = exp(−tA)b, one would want b(t) to be a (generalised) eigenvector, so that exp(−tA)b
is an easy calculation with the following theorem, which follows from basic properties of the
exponential
Theorem 8. Let A be an n × n matrix; let λ be an eigenvalue of A and v ∈ ker(A − λI)k+1
be a generalised eigenvector for eigenvalue λ. Then
1 1
etA v = eλt v + t(A − λI)v + t2 (A − λI)2 v + · · · + tk (A − λI)k v .
2! k!
The point here is that one calculates a finite sum, all terms of which are of the form
matrix×vector. One would also need to rig the initial condition so that the arbitrary constant
from the integration is a (generalised) eigenvector. If b is of the form eλt v where v is a
(generalised) eigenvector for eigenvalue λ, this latter part is easy, as the constant vector is
the initial condition, otherwise one needs to be more careful.
To make these problems slightly more involved, rather than use (generalised) eigenvectors
one can use a simple combination of two or more (generalised) eigenvectors.
15 Acknowledgements
I wish to record my thanks to Peter Brown, Peter Donovan, Dennis Trenerry and Rod James
of the University of New South Wales for several useful discussions over the past several years.
MAPLE is a registered trademark of Waterloo Maple Inc.
References
[1] Howard Anton and Chris Rorres, Elementary Linear Algebra Applications Version,
Eighth edition, Wiley (2000)
[2] Peter Brown, Private communication
[3] H. Davenport The Higher Arithmetic, fourth edition Hutchinson University Library
(1970)
[4] Richard O. Hill, Jr Elementary Linear Algebra with Applications, second edition, Har-
court Brace Jovanovich (1991)
[5] Terry Lawson, Linear Algebra, Wiley (1996)
[6] Michael O'Nan, Linear Algebra, Harcourt Brace Jovanovich (1976)
12 | 677.169 | 1 |
Understanding mathematics means understanding how a great deal of the world works. A Survey of Mathematics with Applications, Eighth Edition, introduces students to the practical, interesting, accessible, and powerful world of mathematics today.CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book. | 677.169 | 1 |
Study Skills
Solving and understanding a mathematical problem, at any level, provides a great sense of personal achievement. Unfortunately, it can also be an extremely frustrating process, often leaving people doubting their ability to "get it". This section provides various study tips and techniques that may help overcome many of the barriers that often develop when studying mathematics.
Everyone has various study methods that work for them. The following are guides, not set rules, but they may help you to build upon and develop your existing techniques. Some general tips are provided along with more specific techniques (for maths) on reading, writing, problem solving, revision and exam skills on subsequent pages. For further tips and advice please feel free to contact the Maths Support Service. | 677.169 | 1 |
View Career Major Course
Course Information
Division
Science/Technology/Engineering/Math
Course Code
ST00062
Course Title
AP Calculus BC
Suggested Course Hours
120.00
Course Description
The AP Calculus BC covers the same differential and integral calculus topics that are included in Calculus AB , plus additional topics in differential and integral calculus, and polynomial approximations and series. This is material that would be included in a two-semester calculus sequence at the college level. Graphing calculator use is an integral part of the course. Students cannot take both the Calculus AB and Calculus BC exams during the same year.
--Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.
--Understand the meaning of the derivative in terms of a rate of change and local linear approximation and they should be able to use derivatives to solve a variety of problems.
--Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems.
--Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
--Communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.
--Model a written description of a physical situation with a function, a differential equation, or an integral.
--Use technology to help solve problems, experiment, interpret results, and verify conclusions.
--Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
--Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.
Accountability Measures, Assessments, or Evaluation Procedures
Referenced Standards
Principles and Standards for School Mathematics (4th ed.). (2005). National Council of Teachers of Mathematics, Reston, VA | 677.169 | 1 |
Practical Math
Chenier's Practical Math Application Guide is the kind of math I wish they'd taught me in high school. Each section uses real-world situations (with a homebuilding bent) to illustrate different math principles, so you simultaneously learn trigonometry and how to lay out stairs, geometry and how to level with a plumb-bob. It has worksheets to practice what you've learned as well as "trade tricks" for DIYers that you wouldn't find in a math textbook.
Many makers know this stuff intuitively already, but if you're new at this kind of thing, the book is a great resource. By the end, you'll know how to deal with unequal roof pitches, but you might just be a bit more of a math geek, too. | 677.169 | 1 |
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Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.
Editorial Reviews
Review
From a review of the German edition: "The book covers all the topics which could be necessary later for learning higher level differential geometry. The material is very carefully sorted and easy-to-read." -- Mathematical Reviews ---- Mathematical Reviews
From a review of the first edition: "It is extraordinarily welcome that this comprehensive textbook containing all the high points of differential geometry is now available in an English translation. Required reading for all mathematicians!" ---- translated from International Mathematical News
Most Helpful Customer Reviews
This is a very fast moving book, covering a huge amount of material at a fairly sophisticated level in under 380 pages. For example, differential forms are introduced in about 2 pages so that the Maurer-Cartan structural equations can be defined. The first 4 chapters makes up a very concise course in curves and surfaces, while the last 4 chapters cover Riemannian geometry. In comparison, do Carmo's two books take 500 pages for the former and 320 pages for the latter.
For this reason I think the claim that this could be used as an undergraduate text is overly optimistic. For that I would use a more self-contained text like Millman & Parker (ISBN: 0132641437). But it would make an excellent text for a graduate survey, or as a second text for someone wanting to make the transition from classical theory (learned from, say, one of the Dover books like Struik, ISBN: 0486656098) to more modern methods. Also, you'll probably want to supplement with a gentler introduction to differential forms.
Of interest to students of physics, the book covers curves and surfaces in Minkowski space, as well as Einstein spaces.
While there is exist many classic texts on differential geometry, I have particularly appreciated this book for its up-to-date treatment, numerous well-done figures, broad coverage, elegant type-setting, and clear expositions. The book covers all the basics expected from an introduction to differential geometry, including curves and 2-D surfaces, but with a look towards the more advanced material in the second half of the book. It alternates between Ricci style notation and Koszul style notation, often carefully explaining the relation between the two and giving examples (I found this particularly helpful). There are, however, some sections where the english is a bit rough (perhaps the fault of the translator). It is also quite brisk throughout, often mentioning advanced topics before they are treated in detail. For example, it already mentions submanifolds, tangent spaces, and tangent bundles in the first chapter on "Notations and Prerequisites from Analysis." It will require serious attention, especially if one has not encountered a good dose of abstract mathematics before. Nonetheless, I have found myself returning to it over several years as an excellent reference and source of many additional topics that I skipped on a first reading. For example, the final chapter on Einstein spaces is a valuable, though demanding, bonus. Thanks to the AMS for publishing a fine edition of a top-notch German author's work.
I feel, the book is an excellent introduction to differential geometry. It moves on gradually, from curves to Einstein manifolds. Much emphasis is placed on visualizing the concepts. While mostly using the 'coordinate free' approach, all important statements are also given in the 'Ricci calculus' version. The mathematical prerequisits are modest, I feel, an undergratuate background is more than sufficient. ( I am only a hobby mathematician, and my background was adequate). Virtually all statements are proven, and the proofs are not too difficult to follow - without any sacrifice in rigor. Nice for self study: the book is almost typo free. I found less than 20, almost all of them harmless. | 677.169 | 1 |
wouldn't read any book like this. you don't need a guide for studying university mathematics, you just do it. You are just wasting the time in which you could do actual mathematics (no offence, its merely my opinion).
–
Alexander GrothendieckFeb 16 '14 at 14:02
1 Answer
1
"Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs."
If you follow the link above, you'll be able to "preview" the book, including its table of contents.
Aside: One feature I personally love about this book is that it is both "about math" and "is math": You learn about more mature approaches to math and proofs by doing mathematics, proofs and all.
I'd suggest previewing Mason, Burton, and Stacey's Thinking Mathematically, as well, since many students find it to be a great aid in "transitioning" to a more mature appreciation of and approach to mathematics.
Another classic which many highly recommend is Polya's How to Solve It. Again, you can "preview" this book if you follow the given link. | 677.169 | 1 |
MATH 025 – Elementary Algebra
This course is a review of elementary algebra and requires previous experience in algebra. The course is intended for students who need to take MATH 151 or MATH 161. The topics include linear equations and inequalities, functions and function notations, graphs and equations of linear functions, systems of linear equations, polynomial and exponential expressions, factoring, quadratic equations, rational and radical expressions and equations. Problem solving is stressed throughout the course. Problems are approached from a variety of perspectives, including graphical, numerical, verbal, and algebraic. A graphing calculator is required – the specific model is determined by the department. This is a developmental course in the basic skills and will not be counted towards degree requirements. NOTE: Students taking MATH 025 may not enroll simultaneously in any other math course. | 677.169 | 1 |
This course, presented by MIT and taught by Professor Alar Toomre, provides an introduction to numerical analysis. The material looks at the basic techniques for the efficient numerical solution of problems in science...
This lesson uses the example of successive discounts at a retail store to demonstrate numeric, algebraic, and graphical representations of compositions of mathematical functions. Students will get the opportunity to...
This interdisciplinary lesson ties earth science concepts in with algebra. The forest-fire danger rating index is applied to a mathematical model. Students will learn real-world meaning of the intercepts and slope in... series of videos, created by Salman Khan of the Khan Academy, features topics covered from very basic algebra all the way through algebra II. This is the best algebra playlist to start at if you've never seen... | 677.169 | 1 |
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