text stringlengths 8 1.01M |
|---|
Trigonometry : Graphing Approach - 4th edition
Summary: As part of the market-leading Graphing Approach Series by Larson, Hostetler, and Edwards, Trigonometry new edition, intended for tri...show moregonometry courses that require the use of a graphing calculator includes a moderate review of algebra to help students entering the course with weak algebra skills.
Enhanced accessibility to students is achieved through careful writing and design, including same-page examples and solutions, which maximize the readability of the text. Similarly, side-by-side solutions show algebraic, visual, and numeric representations of the mathematics to support students' various learning styles.
New! The Library of Functions thread throughout the text provides a definition and list of characteristics for each elementary function and compares newly introduced functions to those already presented to increase students' understanding of these important concepts. A Library of Functions Summary also appears inside the front cover for quick reference.
New! Technology Support notes provided at point-of-use throughout the text guide students to the Technology Support Appendix, where they can learn how to use specific graphing calculator features to enhance their understanding of the concepts presented. These notes also direct students to the Graphing Technology Guide on the textbook web site for keystroke support.
New! Technology Tips, also provided at point-of-use, call attention to the strengths and weaknesses of graphing technology. Some of these tips offer alternative methods for solving or checking a problem using technology.
New! Because students are often misled by the jagged nature of graphs generated by graphing calculators, this text frequently highlights the path of a function in color on the calculator image. This unique design feature enables students to visualize the mathematical concepts clearly and accurately and avoid common misunderstandings.
Study Tips at point-of-use throughout the text reinforce concepts and help students learn how to study mathematics.
New! Checkpoint questions appear after each worked-out solution, directing students to work a similar exercise for further practice or concept reinforcement. These can be used by instructors in class to quickly check student understanding or by students to practice and study concepts.
Chapter Review exercises, Chapter Tests, and periodic Cumulative Tests offer students frequent opportunities for self-assessment and help them to develop strong study- and test-taking skills.
The Student Success Organizer is a valuable note-taking guide that helps students organize their class notes and create an effective study and review tool.
New! Text-specific Tutorial Support is provided in numerous additional resources designed to help students succeed. These resources include live online tutoring, instructional DVDs and videos, and algorithmic tutorial support and self-assessment available on CD-ROM and the web.
Explorations provided at point-of-use throughout the text can help instructors provide a quick introduction to concepts or reinforce student understanding.
Modeling Exercises are integrated throughout the text to motivate students and allow them to see the usefulness of the concepts being presented.
Assignments are easily customized to the difficulty level of the instructor's choice. Exercises are carefully graded in difficulty from mastery of basic skills to more challenging. For example, Review Exercises in each section reinforce previously learned skills in preparation for the next lesson. Synthesis Exercises combine skills and check for conceptual understanding. For those instructors wanting to incorporate more theory, Proofs in Mathematics, allowing instructors to incorporate more theory, are provided for selected theorems in Appendix B
A variety of exercise types is included in each exercise set. Questions involving skills, writing, critical thinking, problem solving, applications, and real data sets are included throughout the text. Exercises are presented in a variety of question formats, including free response, true/false, and fill-in the blank.
New! Vocabulary questions at the beginning of every exercise set help students learn proper mathematical terminology.
New! Houghton Mifflin's Eduspace online classroom management tool offers instructors the option to assign homework and tests online, provides tutorial support for students needing additional help, and includes the ability to grade any of these assignments automatically.
New! Digital Lessons and Digital Figures in PowerPoint provide instructors editable electronic instructional resources. These pre-created lessons and textbook figures make it easier than ever for instructors to present in-class examples and graphics. In addition, for instructors with limited office hours, the full-color presentation helps promote better understanding among students, who can access these slides online to review lectures and prepare for exams.
The Instructor Success Organizer includes suggested lesson plans for each section of the text and is an especially useful tool for larger departments that want all sections of a course to follow the same outline.
The Instructor's Edition of the Student Success Organizer can serve as a lecture outline for every section of the text and includes additional examples for classroom discussion. This is another valuable resource for schools promoting consistent instruction or to support less-experienced instructors.
06183945285.52Ex |
Justification for the Course Proposal: This is a
fundamental course designed for all science and engineering students.
Course Objectives: The sequence Math 119-120 is the
standard complete introduction to the concepts and methods of calculus,
taken by all engineering students. The emphasis is on concepts, solving
problems, theory and proofs. All sections take uniform midterm and
final exams. Students develop their reading, writing and questioning skills
in mathematics.
Exams and Grading: Course grades are determined by (online) homework,
short exams (given in recitations), two (non-cumulative) midterm exams,
and a cumulative final exam, as well as a
small number of bonus points awarded on the basis of attendance, class participation,
and/or project completion.
Homework:
There will be 7 online homeworks and 3 written homeworks. The online homeworks are
assigned and graded weekly using the online
WeBWork system.
The written homeworks cover epsilon-delta proofs, optimization, and curve sketching.
The final written homework (curve sketching) includes a MATLAB component.
Suggested Problems:
Due to the limitations of WeBWork, complete mastery of subject material will require
solving additional theoretical problems. For each lecture, the assistants will announce
additional suggested problems from the textbook. Their list of problems is available
on the course website. It is important for students to solve these problems regularly;
however the solutions will not be collected or graded.
Textbook:Calculus, James Stewart,
6th metric international ed., 2009.
(available at the bookstore)
Reference Books:
George B. Thomas et. al., Thomas' Calculus 11th ed.
Robert A. Adams, Calculus, A Complete Course 5th ed.
Howard Anton, Calculus with Analytic Geometry 5th ed.
Make-up Policy:
In order to be eligible to enter the
make-up examination for a missed examination, a student must have a
documented or verifiable and officially acceptable excuse. It is not possible
to make up multiple missed exams. The make-up examination for all
exams will be after the final exam, and will include all topics.
Math Help Room:
The
mathematics help room
in T-103 is a room staffed by mathematics faculty and
teaching assistants where students may gather to ask questions, work on homework,
and view exams.
Students are also invited to seek
out instructors in their offices.
Lectures
Section 1+4
Mon 8:40-10:30 Wed 8:40-10:30
TZ-19
Section 2
Mon 15:40-17:30 Thu 15:40-17:30
TZ-19
Section 3
Wed 15:40-17:30 Fri 8:40-10:30
TZ-20
Section 5
Wed 13:40-15:30 Fri 13:40-15:30
TZ-19
Section 6
Wed 10:40-12:30 Fri 10:40-12:30
TZ-20
Recitations
Recitation 1
Fri 10:40-12:30
SZ-25
Recitation 2
Wed 15:40-17:30
SZ-25
Recitation 3
Tue 13:40-15:30
SZ-25
Recitation 4
Fri 8:40-10:30
SZ-25
Announcements
16.12.2011Short-Exam 4 dates are
20.12.2011, 21.12.2011, 23.12.2011,
in your respective recitation sections.
The students who do not enter the exam in the correct recitation section will not receive any credit.
It will consist of questions starting from 6.2, up to and including 7.4*.
01.12.2011Short-Exam 3 dates are
09.12.2011, 13.12.2011, 14.12.2011
in your respective recitation sections.
The students who do not enter the exam in the correct recitation section will not receive any credit. |
Product Details
Excursions in Classical Analysis by Hongwei Chen
Excursions in Classical Analysis will introduce students to advanced problem solving and undergraduate research in two ways: it will provide a tour of classical analysis, showcasing a wide variety of problems that are placed in historical context, and it will help students gain mastery of mathematical discovery and proof. The author presents a variety of solutions for the problems in the book. Some solutions reach back to the work of mathematicians like Leonhard Euler while others connect to other beautiful parts of mathematics. Readers will frequently see problems solved by using an idea that might at first glance, might not even seem to apply to that problem. Other solutions employ a specific technique that can be used to solve many different kinds of problems. Excursions emphasizes the rich and elegant interplay between continuous and discrete mathematics by applying induction, recursion, and combinatorics to traditional problems in classical analysis. The carefully selected assortment of problems presented at the end of the chapters includes 22 Putnam problems, 50 MAA Monthly problems, and 14 open problems. These problems are not related to the chapter topics, but connect naturally to other problems and even serve as introductions to other areas of mathematics. The book will be useful in students' preparations for mathematics competitions, in undergraduate reading courses and seminars, and in analysis courses as a supplement. The book is also ideal for self study, since the chapters are independent of one another and may be read in any order.
The author's intention in this enlightening and inspiring book is to introduce the reader to advanced problem solving techniques via case studies, twenty-one relatively short independent chapters that...
(read complete review)
The author's intention in this enlightening and inspiring book is to introduce the reader to advanced problem solving techniques via case studies, twenty-one relatively short independent chapters that contain "kernels of sophisticated ideas connected to important current research" and expose the principles underlying these ideas. Chen is a devotee of Pólya's rubric for problem solving and practices what he preaches. |
Do the Math: Secrets, Lies, and Algebra
In the eighth grade, 1 math whiz < 1 popular boy, according to Tess's calculations. That is, until she has to factor in a few more variables, like:
1 stolen test (x),
3 cheaters (y),
and 2 best friends (z) who can't keep a secret.
Oh, and she can't forget the winter dance (d)!
Then there's the suspicious guy Tess's parents know, but that's a whole different problem—
Sydney (Fair Oaks Ranch, TX)
Do the Math: Secrets, Lies, and Algebra by Wendy Lichtman was awesome! I really enjoyed the various chacters and the fast paced plot! I might have learned some math along the way too! :) I rate this book an 8/10!
—
Allie (Forest Hill, MD)
This was a very interesting book. It had a new way of looking at life: through math. As the main character discovers, math is so logical that it can often help to solve problems in real life--and she has some big ones. Any math lover would instantly love this book, and anyone else would love it also for its unique perspective on life. I would highly recommend it to anyone, even those who think math is useless (maybe this will change their minds).
—
Molly (Agua Dulce, CA)
This wonderful, witty book puts things in a refreshingly new perspective, relating everyday things to math in a way that will have you thinking. This book evokes an interest in math without being a textbook and also allows us to enter the world of a typical teenage girl. This book combines typical teenage life and math in a way that will make you excited for math class.
Do the Math #2: The Writing on the Wall |
This monograph is a bridge between the classical theory and a modern approach via arithmetic geometry. The authors aim to provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form.' L'enseignement mathematique 'The quality of exposition is exemplary, which is not surprising, given the brilliant expository style of the elder author.' Yuri Bilu, Mathematical Review 'Bombieri and Gubler have written an excellent introduction to some exciting mathematics ... written with an excellent combination of clarity and rigor, with the authors highlighting which parts can be skipped on a first reading and which parts are particularly important for later material. The book also contains a glossary of notation, a good index, and a nice bibliography collecting many of the primary sources in this field.' MAA Reviews '...a fundamental and pioneering standard text in the field, which will undoubtedly serve as a basic source for the future development of number theory and arithmetic geometry as a whole.' Werner Kleinert, Zentralblatt MATH '... remarkable ...' European Mathematical Society NewsletterRead more... |
Frequently Asked Questions
What is the Math Center?
The Math Center is a place to drop in and get help from tutors in your Math, Physics, Chemistry, or Computer Science course when it's convenient for you. You are also free to do your own work at one of our tables or computers, or to meet with a study group. Our computers have all of the required software for all DCC Math, Physics, Chemistry, and Computer Science courses. We also have calculators and textbooks for use while in the lab and a few calculators for students to borrow to take a test.
Where is the Center?
The Center is in Washington 224/226. Go in the front door of the Washington building, go up one flight of stairs (the stairs are on the right when you enter the building) or take the elevator to floor 2. The lab will be on your left.
How do I get help?
Just come to W224/226 during the hours the lab is open, sign in with your myDCC ID and course, then ask a tutor (someone wearing a tutor badge) for help.
What kind of help do you offer?
We offer help on a drop in basis in Math, Physics, Chemistry, and Computer Science.
Is it free?
Yes, help from the tutors, use of the computers, and use of the printers are all free.
Do I need an appointment?
No, just come when it's convenient for you! The Math Center is also a good place to do your homework or study, even if you don't require the help of a tutor.
Who are the tutors?
The tutors are DCC students who have mastered the material as well as two professional tutors.
What tutors do:
· Help students begin homework and projects · Answer questions about how to approach problems · Encourage students to learn problem- solving techniques
No, an individual tutor will devote an entire hour to helping you only. Our tutors help many students during the course of an hour. If you feel you need individual tutoring, go to Academic Services in Hudson 315 and apply for a tutor. (See their website by following the "tutoring" link under the MyCourses tab in MYDCC.) If you do have an individual tutor, you may still use the Math Center.
When can I get help with my particular course?
Go to the Math Center website (under the MyCourses tab in MyDCC), and click on the "Hours help is available by subject" link, or look at our bookmark (available in the lab). This will tell you when help is available for each course in the Math Center.
May I use the printers?
Yes, the printers may be used for printing assignments (such as papers, graphs, presentations, etc.). If you wish to print a PowerPoint faculty lecture, guidelines which are posted in the lab must be followed. There is a color printer which may be used for the final draft of an assignment if color is required.
Do you have textbooks?
Yes, textbooks and graphing calculators may be borrowed for use in the lab. TI-84 calculators may be borrowed for tests.
Why do I have to log in?
Logging in allows us to keep track of how many people use the lab and for which courses. Your individual information is not used by the Center. |
Access Kit for the Trigsted MyMathLab eCourse plus the eText Reference. Kirk Trigstedrevolutionized the way this course is taught when he created College Algebra, a completely clickable ebook that was written from the ground up within MyMathLab. Recognizing that today's students start with the homework instead of reading the text, Trigsted created an online learning environment that is a seamless mix of exposition, videos, interactive animations, tutorials, and assessment. This approach leverages the power of MyMathLab... MORE and leads students to interact with course materials in a way that is proving to be more effective. With the Second Edition, Trigsted continues to innovate with a revised design that improves navigation and usability, expanded videos, and increased animation coverage. The eText Referenceis a spiral-bound, printed version of the eText that provides a place for students to do practice work and summarize key concepts from the online videos and animations. In addition to the benefits it provides students, the Summary Notebook also provides portability for those instructors that prefer a printed text for class preparation.
4.5 The Zeros of Polynomial Functions; The Fundamental Theorem of Algebra
4.6 Rational Functions and Their Graphs
4.7 Variation
5. Exponential and Logarithmic Functions and Equations
5.1 Exponential Functions
5.2 The Natural Exponential Function
5.3 Logarithmic Functions
5.4 Properties of Logarithms
5.5 Exponential and Logarithmic Equations
5.6 Applications of Exponential and Logarithmic Functions
6. Conic Sections
6.1 The Parabola
6.2 The Ellipse
6.3 The Hyperbola
7. Systems of Equations and Inequalities
7.1 Systems of Linear Equations in Two Variables
7.2 Systems of Linear Equations in Three Variables
7.3 Inconsistent and Dependent Linear Systems in Three Variables
7.4 Partial Fraction Decomposition
7.5 Systems of Nonlinear Equations
7.6 Systems of Inequalities
8. Matrices
8.1 Matrix Operations
8.2 Inverses of Matrices and Matrix Equations
8.3 Determinants and Cramer's Rule
9. Sequences and Series; Counting and Probability
9.1 Introduction to Sequences and Series
9.2 Arithmetic Sequences and Series
9.3 Geometric Sequences and Series
9.4 The Binomial Theorem
9.5 Mathematical Induction
9.6 The Theory of Counting
9.7 An Introduction to Probability
Appendix A. Conic Section Proofs
Kirk Trigsted teaches mathematics at the University of Idaho and has been Director of the Polya Mathematics Center since its inception in 2001. Kirk has taught with MyMathLab for many years, and has contributed to the videos for several Pearson books. Kirk is also actively involved with the National Center for Academic Transformation (NCAT). |
Calculus:
Understanding Its Concepts and Methods
Calculus:
Understanding Its Concepts and Methods is a revolutionary CD-based calculus textbook that represents
a dramatic change in the teaching and learning of calculus. It opens up a mathematical adventure with
explanations, examples, explorations, problems, and self-tests that present calculus
concepts and methods within a dynamic Scientific Notebook environment.
The Latest in Software Technology
The textbook is provided on CD and includes Version 5.5 of Scientific
Notebook. With the powerful computer algebra system included in Scientific
Notebook, students can interactively explore examples and carry out experiments that reveal
what calculus is all about. The book exploits software technology to help students discover
concepts interactively. Live calculations, animated plots, and manipulatable graphics and examples
help clarify concepts, enhance the presentation of calculus concepts, and illustrate methods. The book
is thoroughly indexed and hyperlinked to provide easy navigation and access to relevant information.
An Ideal Teaching Tool
Calculus: Understanding Its Concepts and Methods
contains the information normally taught in a three-semester calculus sequence. Each chapter presents both basic
concepts and computational details, and supplements explanations with tools, popup hints and information,
examples, and explorations.
Many examples have animated mathematics that encourage student exploration, and some include step-by-step
discussions of specific problem solutions. Others are interactive, so students can experiment by defining their
own functions and parameters. The explorations challenge students to write detailed solutions
to open-ended problems; some model solutions are included. Learning is further reinforced with extensive
problem sets and algorithmically generated self-tests that allow students to
measure their level of understanding.
With its focus on mathematical problem solving, experimentation, verification, and
communication of results, Calculus: Understanding Its Concepts and Methods
is an ideal teaching tool for classroom use, independent study, and distance learning.
Support for Instructors and Students
Everything takes place in a Scientific Notebook
environment, which combines a scientific word processor with an easy-to-learn interface to an integrated
computer algebra system. Students are encouraged to think and communicate like mathematicians, entering
mathematics into documents using natural notation, not obscure programming syntax. The mathematics can be edited, evaluated, plotted, simplified, factored,
and expanded, and the output, also in natural notation, can be used in subsequent computations.
In addition to illustrated explanations, animated examples, and opportunities for exploration,
the textbook supports the instructor with hints to the student about how to approach problems, extensive problem
sets with complete answers, and additional resource material. Self-tests in each chapter are randomly generated
from problem prototypes, providing a virtually unlimited number of distinct quizzes to test students'
understanding.
Students with questions about calculus can participate in
an unmoderated discussion forum for users of
Calculus: Understanding Its Concepts and Methods.
And they can get fast answers to questions about using the tools and features
in Scientific
Notebook from the help files in each chapter and the program's online
Help system. |
Master discrete mathematics and ace your exams with this easy-to-use guide that reinforces problem-solving skills and reduces your study time! Students of discrete mathematics love Schaum's--the first edition of this book was a major bestseller--and this edition will show you why! Schaum's Outline of Discrete Mathematics lets you focus on the problems that are at the heart of the subject. It cuts your study time by eliminating the extraneous material that clutters up so many textbooks. ...The guide that helps students study faster, learn better, and get top grades More Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores! This Schaum's Outline gives you
Master beginning statistics with Schaum's-- ...
Master matrix operations with Schaum's If you don't have a lot of time but want to excel in class, this book helps you: ... |
Maths the Wacky Way
This book is for anyone studying Maths or wanting to assist others studying Maths. If you, your family or friends are sitting GCSEs shortly and you want to be able to assist them using fun, memorable methods, this book is ideal.
Parents have been buying the book so they can help their child without having to hire a private Maths tutor or buy study texts that get left on the shelf. This book also appeals to teachers who want to try wacky methods in their class.
It doesn't matter which category you fall into, Maths the Wacky Way for Students...By a Student is the ultimate guide to getting your head around all areas on the Maths syllabus: algebra, trigonometry, equations, vectors, histograms, etc.
The book is full of colour, shows fully worked examples, practice questions at the end of each section and fully worked answers at the back of the book.
The great thing with this book is that it explains all areas of Maths in a modern and 'wacky' way - honestly Maths can be fun! Let Claire help you build on what you already know and help you crack those mental blocks to get that grade average up a notch or two. |
More About
This Textbook
Overview
The Practical Problems in Mathematics series offers students of specific trades useful help in basic mathematics and opportunities to practice math principles on problems applied to their area of interest. Practical Problems in Mathematics for Carpenters, seventh edition, contains 43 instructional units progressing from the simplest basic arithmetic operations to compound problems applied in light frame construction. Each of the 43 units begins with a brief review of the math principal to be applied in that unit. The book contains more than 800 carpentry problems, including two comprehensive tests.
Editorial Reviews
From the Publisher
"This text has many strengths. Good information, quality assignments, and it is easy for my students to use and understand."Kenneth N. Bowling, Associate Professor, Industrial Technology, Southeast Kentucky Community & Technical College, Harlan, Kentucky
"… the text addresses the objects of my course better than any other text I have reviewed."Earl T.TorgersonReviewer's school/affiliation: Associate Professor, Transportation and Construction, Bismarck State College, Bismarck, North Dakota |
Photoshop
Adobe Photoshop is a graphics-editing software program, part of the Adobe Creative Suite. A variety of tools with multiple image-editing functions are the key elements of this software. These tools typically fall under the categories of drawing; painting; measuring and navigation; selection; typing; and retouching. Photoshop is a powerful tool which can be used to do much more than retouch photographs. The user has the ability to manipulate graphic images and type. Files can be saved in a wide variety of formats.
Algebra 1Algebra 2
Algebra 2 expands and builds on the concepts learned in Algebra 1. Algebra 1 is the foundation for all mathematics that comes after it. It is essential for students to master concepts, properties and rules, but more important for them to learn to think mathematically and think critically. I take a problem solving approach, teaching students how to ask questions. Learning how to ask the right questions is one of the best paths to becoming a successful student of mathematics.
Elementary Math
Elementary mathematics consists of mathematics topics frequently taught at the primary school level. The most basic topics in elementary mathematics are number sense, arithmetic and geometry. Students learn basic operations and are expected to memorize math facts such as common sums or products. Long division is learned during the primary years and an introduction to basic problem solving is also included. Elementary mathematics is used in everyday life in activities such as making change, cooking, and buying in a store. It is also an essential step on the path to understanding science. In elementary mathematics, the student begins to learn how to think mathematically and correctly interpret data.
English
Mastering the basics of using the Englishlanguage is essential for any student, or any speaker of another language who hopes to be gainfully employed.
Geometry
Geometry is the branch of mathematics that deals with shapes such as rectangles or circles (plane geometry) or 3-D solids in space (solid geometry) and their properties. It also includes learning deductive proof – which requires the student to be able to think linearly and employ supporting statements when justifying an argument. Example: demonstrating that two triangles are congruent (or similar) using given information, plus supporting postulates or theorems.
Grammar
Grammar is the structure and system of a language. In English, there are eight parts of speech (nouns, verbs, adjectives, adverbs, pronouns, prepositions, conjunctions and interjections). Grammar also includes what is called syntax, which is a set of rules related to the way words and phrases are arranged in a sentence, in order for it to make sense.
Macintosh
I am very experienced in use of a Macintoshcomputer, having used one for over 20 years. I am an expert in understanding and use of the operating system, OS X. I can effectively guide any new users, or users switching from Windows. I am also qualified in supporting users in the use of all built-in Apple applications (Mail, Contacts, Preview, Calendar, Dictionary, iPhoto, iTunes, System Preferences, etc.).
Microsoft Word
Microsoft Word is the most commonly used "word processor" software in the world. Word These include things like tables, columns, and inserting graphics. You can even use Word to do what is called a "Mail Merge" – personalizing form letters with each recipient's name, address and other information.
PrealgebraSAT Math
The SAT Math Level 1 test is a one-hour multiple choice test given on algebra, geometry, basic trigonometry, algebraic functions, elementary statistics and a few miscellaneous topics. A student chooses whether to take the test depending upon college entrance requirements for the schools in which the student is planning to apply. The SATMath Level 2 test covers more advanced content.
Study Skills
To be an effective student, it is important to have a complete set of tools in your student "toolbox." These tools include keeping your materials organized and always appearing in class with all the necessary requirements. Tools also include an organized system of study. This can differ greatly with the subject matter being studied. In all subjects, knowing key terms is essential. Paraphrasing key ideas is also a must. Studying should begin with getting a quick overview of the section at hand, then going back and reading the material section by section. Use of a highlighter or taking notes is helpful 09/10 |
Mathematics Learning Domain
At Dromana College, the Mathematics Learning Domain focuses on helping our students develop a range of mathematical skills and understandings that can be applied in many future pathways. Whilst compulsory for Years 7 to 10, students are encouraged to continue with Mathematics in both Years 11 and 12.
Mathematics teachers work with students to modify their program to meet individual needs. All students at Dromana College are extensively assessed in Mathematics, and this assessment forms the basis for the student's future learning program. These individual needs are addressed within mixed ability classes at Years 7 to 9 and more focused options at Years 10 to 12.
At Year 10, students have options for their Mathematical pathway. They have a choice of:
Year 10 Numeracy which caters for students that will follow a VCAL course in Years 11 and 12
Year 10 Mathematics which caters for those students wanting to follow a General/Further Mathematics and Mathematical Methods course in Years 11 and 12
Year 10 Extension Mathematics which allows higher achieving students to be well prepared for a Mathematics Methods/Specialist course in Years 11 and 12. |
Math Coaches and Mathematica
When I attended this year's National Council of Teachers of Mathematics conference in San Diego, I met many "math coaches". All teachers are coaches of their classrooms, but I'm referring to teachers whose titles are "coach". These coaches spend time with at-risk or struggling students, trying to help the students gain further success in their education.
Coaches spend time working one on one or in small groups with these students to help them achieve a higher level of knowledge. They are looking for interactive ways to get students excited about all of their homework as well as to prepare them for standardized tests—especially in math—in new ways, relevant to the students and the topics.
However, very few of these math coaches have computer programming backgrounds. Quite often, their main technology tool has been the basic calculator. These coaches were interested in a tool that would not cost them hours of time to learn.
I was asked more often about Algebra I than any other subject. Coaches were impressed with Demonstrations that covered basic topics, such as a line through two points:
They also appreciated more advanced Demonstrations that highlight difficult topics. For instance, the coaches I met particularly liked an Algebra II example on ellipses and hyperboles with the same focal points:
The coaches were excited to see how easy it is to show how the change in eccentricity affected the shape and location of curves in the graph. This is not an easy idea to communicate to high school students. The coaches realized that seeing this parameter changing in real time added much value and weight to the lesson. No longer was eccentricity just another value to find after plugging meaningless numbers into some formula the Algebra II book gave them.
Coaches also liked Demonstrations that can take the place of hand-held manipulatives, like this "virtual paper" one on the sum of interior angles of a triangle:
The coaches I met at the conference who work with challenged students on a daily basis see Mathematica as a great way to show students vital math skills. Please email us for more information about how you can get started using Mathematica in your own classroom. |
Hi, my high school classes have just started and I am stunned at the amount of holt algebra 1 textbook table of contents homework we get. My concepts are still not clear and a big assignment is due within few days. I am really worried and can't think of anything. Can someone help me?
I'm also using Algebrator to assist me with my math projects . It really does help you quickly understand certain topics like proportions and quadratic inequalities which would take days to understand just by reading tutorials. It's highly recommended program if you're looking for something that can help you to solve algebra problems and display all essential step by step solution. Highly recommended software .
adding functions, conversion of units and function range were a nightmare for me until I found Algebrator, which is truly the best algebra program that I have ever come across. I have used it frequently through many math classes – Remedial Algebra, Algebra 2 and Algebra 1. Just typing in the algebra problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I really recommend the program. |
Task Maths
Task Maths, written by Barbara and Derek Ball, was published in the early 1990s by Thomas Nelson to provide a mathematics programme for secondary schools covering the revised 1991 National Curriculum.
The course consists of five student books, one for each year, and parallel teacher resource books. Task Maths provides one set of books which are intended for all students. There are not different tracks for students judged to be at different levels of ability. The student's books for the Key Stage Three course are written so as to be manageable by students working at level three and yet make demands on students working at level eight (or beyond). The books for Key Stage Four similarly cover the full spread of attainment for GCSE.
The course is organised into tasks. Each task is presented in the form of several activities (which are graded) and the specific activities and depth to which they are taken is intended to provide the necessary differentiation.
The tasks are not 'normal' textbook chapters but are 'themed'. This encourages the integration of the mathematics content. The material in some tasks may be useful to supplement work in current schemes. The books also contain many helpful sets of review exercises.
The student material is colourfully presented with frequent links to the type of software available at the time - particularly Logo, BASIC programs and the mini-spreadsheet 'Spread'. While not identified as investigations, many tasks leave opportunities for extension and creative student input.
The teacher resource books contain answers and some teaching suggestions, for example discussion opportunities. The is also much cross referencing to the 1991 National Curriculum which is no longer relevant. Apart from the cover, these books are in black and white only – which has the advantage of making it easy to highlight.
Note
This collection of resources was published between 1991 and 1993. The mathematical content is still very relevant but some of the teaching approaches may be at variance with those supported by National Strategy materials. Some of the software used, especially Spread, is unlikely to be available to most teachers although the ideas can be transferred to other, more recent, software Year Seven textbook from Task Maths, there are 28 |
About This Book
This book is
intended for all the mathematicians, engineers, and physicists who have
to know, or who want to know, more about the modern theory of
quaternions. Primarily, as the title page suggests, it is an exposition
of the quaternion and its primary application in a rotation operator.
In a parallel fashion, however, the conventional or more familiar matrix
rotation operator is also presented. This parallel presentation affords
the reader the opportunity for making comparative judgments about which
approach is preferable, for very specific applications.
This book readily divides into three major areas of concern:
The first three or four chapters present introductory material which
establishes terminology and notation to be used later on.
The mathematical properties of quaternions are then presented, including
quaternion algebra and geometry. This is followed by more advanced special
topics in spherical trigonometry, along with an introduction to quaternion
calculus and perturbation theory, required in certain situations involving
dynamics and kinematics.
Lastly, state-of-the-art applications are discussed. A six degree-of-freedom
electromagnetic position and orientation transducer is presented. With
this we end with a discussion of computer graphics, necessary for the
development of applications in Virtual Reality.
The writing of this book was early-on supported by the United States Air
Force, whose objective was to provide a primer on quaternions, suitable for
self-study. Our primary concern was that the book be written at a level such
that much of the subject matter would be accessible to those with a modest
background in mathematics. With this in mind, the quaternion is defined and
its algebra is introduced and developed. Several applications of the
quaternion, the quaternion rotation operator, and quaternion rotation
sequences are presented. A preview of the Table of Contents will provide the
reader with a measure of the intent and scope of this book. |
atti and McWaters wrote this series with the primary goal of preparing students to be successful in calculus. Having taught both calculus and precalculus, the authors saw firsthand where students would struggle, where they needed help making connections, and what material they needed in order to succeed in calculus. Their experience in the classroom shows in each chapter, where they emphasize conceptual development, real-life applications, and extensive exercises to encourage a deeper understanding.
With a new addition to the series, Precalculus Essentials, this text offers the best of both worlds: fast-paced, rigorous topics and a friendly, "teacherly" tone. This text is developed with a focus on key topics for calculus preparation. |
Preface Part I. On vectors: 1. Fundamental principles respecting vectors 2. Applications to points and lines in a given plane 3. Applications of vectors to space Part II. On quaternions: 1. Fundamental principles 2. On complanar quaternions 3. On biplanar quaternions Part III. On quaternions: 1. On the interpretation of a product of vectors 2. On differentials and developments of functions of quaternions 3. On some additional applications. |
College Algebra (2nd Edition)
Book Description: Exceptionally accessible and user-friendly, this introduction to college algebra features an abundance of interesting real-world applications that relate to readers' everyday lives. Filled with scenarios, examples, study tips, exercises, etc., it takes the intimidation out of learning algebra, and gets readers up to speed quickly and painlessly. Prerequisites: Fundamental Concepts of Algebra. Equations, Inequalities, and Mathematical Models. Functions and Graphs. Polynomial and Rational Functions. Exponential and Logarithmic Functions. Systems of Equations and Inequalities. Matrices and Determinants. Conic Sections. Sequences, Induction, and Probability. For anyone wanting a user-friendly introduction to college |
ALEX Lesson Plans
Title: Solving Multi-Step Equations
Description:
In Solving Multi-Step Equations Description: In
Title: The Art of Solving One and Two-Step Equations - A WebQuest
Description:
This lesson is on solving one and two-step equations. Students will watch interactive video clips that provide a step by step process to solving equations. Students will also visit interactive websites that provide games for solving equations The Art of Solving One and Two-Step Equations - A WebQuest Description: This lesson is on solving one and two-step equations. Students will watch interactive video clips that provide a step by step process to solving equations. Students will also visit interactive websites that provide games for solving equations.
Title: Wild Water Adventure
DescriptionStandard(s): [MA2010] AL1 (9-12) 17: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3] [MA2010] AL1 (9-12) 20: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. [A-REI6]
Subject: Mathematics (9 - 12) Title: Wild Water Adventure Description
Title: It's a Party! Solving Multi-step Equations
Description:
OftenStandard(s): [MA2010] (8) 9: Solve linear equations in one variable. [8-EE7] [MA2010] AL1 (9-12) 16: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1] [MA2010] AL1 (9-12) 17: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]
Subject: Mathematics (8 - 12) Title: It's a Party! Solving Multi-step Equations Description: Often
Title: Density
Description:
D
Standard(s): [S1] (8) 1: Identify steps within the scientific process. [S1] CHE (9-12) 1: Differentiate among pure substances, mixtures, elements, and compounds. [S1] ENV (9-12) 1: Identify the influence of human population, technology, and cultural and industrial changes on the environment. [MA2010] AL1 (9-12) 4: Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. [N-Q1] [MA2010] AL1 (9-12) 5: Define appropriate quantities for the purpose of descriptive modeling. [N-Q2] [MA2010] AL1 (9-12) 6: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. [N-Q3] [MA2010] AL1 (9-12) 15: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [A-CED4] [MA2010] AL1 (9-12) 17: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3] [MA2010] AL1 (9-12) 28: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* [F-IF4] [MA2010] AL1 (9-12) 34: Write a function that describes a relationship between two quantities.* [F-BF1] [MA2010] AL1 (9-12) 37: Distinguish between situations that can be modeled with linear functions and with exponential functions. [F-LE1] [MA2010] AL1 (9-12) 38: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). [F-LE2] [MA2010] ALC (9-12) 1: Create algebraic models for application-based problems by developing and solving equations and inequalities, including those involving direct, inverse, and joint variation. (Alabama) [MA2010] GEO (9-12) 40: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).* [G-MG2]
Subject: Mathematics (9 - 12), or Science (8 - 12) Title: Density Description: D |
Part II A. Students taking general education or introductory collegiate courses in the mathematical sciences.
A: Students taking general education or introductory collegiate courses in the mathematical sciences
General education and introductory courses enroll almost twice as many students as all other mathematics courses combined [1]. They are especially challenging to teach because they serve students with varying preparation and abilities, many of whom have had negative experiences with mathematics. Perhaps most critical is the fact that these courses affect life-long perceptions of and attitudes toward mathematics for many students'and hence many future workers and citizens. For all these reasons these courses should be viewed as an important part of the instructional program in the mathematical sciences. This section concerns the student audience for these entry-level courses that carry college credit. An important resource for discussions about these courses is Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus, published by the American Mathematical Association of Two-Year Colleges and available in its entirety on the Internet.
At Princeton University, the Math Alive course is designed for those who haven't had college mathematics but would like to understand some of the mathematical concepts behind important modern applications. It consists of largely independent 2-week units in cryptography, error correction and compression; probability and statistics; birth, growth, death and chaos; geometry and motion control; and voting and social choice. Each unit is divided into two parts. For each part students can download lecture notes in pdf or ps format. Each part has a problem set and a corresponding online lab. Links for the lecture notes, online labs, and problem sets through corresponding links are on the course website. Each problem set is posted on the web one week before it is due. Solutions to the problem are available on the web after the submission deadline. The syllabus describes class topics, lists due dates and posts problem sets and labs.
A widely-used text for a similar but somewhat less mathematically demanding course is For All Practical Purposes by COMAP. The companion website contains extensive resources for students and teachers.
At The University of Texas at Austin, the mathematics course developed for the liberal arts honors program is designed to present 'culturally significant and beautiful concepts with concomitant emphasis on potent strategies of discovery and exploration.� The course presents infinity, the fourth dimension, geometric gems, topology, coincidence, chaos, fractals, and other topics. Each topic is intended to illustrate the process of starting with a simple observation and applying techniques of effective thinking that lead to the creation of new ideas, such as making errors and learning from them, breaking complicated questions into simple components, understanding simple things deeply, and finding the essence of an issue. One project directs students to take a non-mathematical issue they care about and apply the methods of analysis, not the mathematics itself, to analyze the issue or produce a creative work about it. The text used is The Heart of Mathematics: An invitation to effective thinking, by E. Burger, and M. Starbird.
The introductory course in Contemporary Mathematics at Virginia Commonwealth University is taken by more than 2000 students each year. It uses the text Excursions in Modern Mathematics Peter Tannenbaum and Robert Arnold and includes topics such as voting and fair division, applications of graph theory/networks, population growth, symmetry, and fractal geometry. In the context of these applications of mathematics, students strengthen their algebraic and graphing skills. The course serves as a prerequisite for the statistics course that is required in most humanities and social science degree programs. Students take three exams and four quizzes; write two papers; participate in making a group presentation; make a poster session presentation; turn in a dozen in-class/homework worksheets; and respond to weekly prompts in a Learning Log. Grades are based on tests (30%), quizzes (20%), presentations (20%), papers (20%), worksheets etc. (20%). Students who complete a learning log may drop the lowest 10% of the grades for their assignments. A detailed instructor's guide, discusses the use of writing-to-learn, group projects, independent study projects, poster sessions and other approaches that expect active student engagement.
Mount Holyoke College offers a variety of courses to incoming students not enrolling in a calculus sequence. The introductory 'explorations� in algebra, number theory, geometry, and fractals and chaos offer a way for students to begin their study of mathematics. These courses emphasize mathematics as an art and as a way of seeing and understanding. The explorations presuppose neither special talent for nor prior strong interest in mathematics. They intend to awaken interest by demonstrating either the pervasiveness of mathematics in nature and its power as a tool that transcends disciplines or its qualities as an art that brings aesthetic pleasure to the participant. Another alternative for students is an interdisciplinary case-study course in quantitative reasoning.
Resources that can be used to introduce students to contemporary topics in general education courses include the AMS website What's New in Mathematics and PLUS magazine, an Internet magazine from the United Kingdom, which aims to introduce readers to the beauty and the practical applications of mathematics. A number of articles in the MAA journal Math Horizons are also appropriate for a general undergraduate student audience.
Precalculus with Applications, based on Functioning in the Real World: A PreCalculus Experience by Sheldon Gordon et al., is taught at Farmingdale State University of New York. It is designed to prepare students for calculus as well as for quantitative courses in the natural and social sciences. The course introduces students to the fundamental families of functions using contextual, tabular, graphical, and algebraic representations. A common theme is the notion of fitting functions to real-world data. Each family of functions is introduced in context and the emphasis throughout is on realistic applications. Matrices and their use in solving systems of linear equations are also introduced, as are the notion of recursion and applications via models involving difference equations. The course requires three class tests, a series of three individual investigatory projects (which count as equivalent to two class tests), and a cumulative final exam.
The Precalculus Weblet consists of an online textbook, syllabi, homework, and exams developed by the members of The Washington State Board for Community College Education. It contains links for exploring precalculus concepts using current information on the Internet. All the material on the website is freely available for personal use.
The article 'Who Are the Students who take Precalculus� by Mercedes McGowen, William Rainey Harper College, examines the numbers of students in precalculus courses, their backgrounds and motivations for taking the courses, and the subsequent mathematics courses in which they enroll.
An example of a calculus course that integrates calculus and precalculus and places special emphasis on active learning is the Workshop Calculus Program. The textbooks Workshop Calculus: Guided Explorations with Review, vol. 1 and vol. 2, and Workshop Calculus with Graphing Calculators Guided Exploration with Review, vol. 1 and vol. 2, developed by Nancy Baxter-Hastings, Dickinson College, seeks to help students develop the confidence, understanding, and skills necessary for using calculus in the natural and social sciences and for continuing their study of mathematics. Lectures are replaced by an interactive teaching format that does not distinguish between classroom and laboratory work. Students are expected to learn by doing and by reflecting on what they have done, and the instructor is expected to respond to students as they learn.
ARTIST stands for Assessment Resource Tools for Improving Statistical Thinking. 'This website, with support from the National Science Foundation, provides a variety of assessment resources for teaching first courses in Statistics:
1. Assessment Builder: a collection of about 1100 items, in a variety of item formats, according to statistical topic and type of learning outcome assessed. This database can be used to generate files to be edited and manipulated by statistics instructors.
2. Resources:
* Information guidelines, and examples of alternative assessments (such as projects, article critiques, and writing assignments)
* Copies of articles or direct links to articles on assessment in statistics. References and links for other related assessment resources. 3. Research Instruments: instruments that may be useful for research and evaluation projects that involve assessments of outcomes related to teaching and learning statistics.
4. Implementation issues: questions and answers on practical issues related to designing, administering, and evaluating assessments.
5. Presentations: copies of conference papers and presentations on the ARTIST project, and handouts from ARTIST workshops.
6. Events: information on past and upcoming ARTIST events.
7. Participation: ways to participate as a class tester for ARTIST materials.�
The article 'An Activity-Based Statistics Course� by M. Gnanadesikan et al. from the Journal of Statistics Education includes examples of types of activities that work well in various classroom settings along with comments from colleagues and students on their effectiveness. Another source for the activity-based approach is Teaching Statistics: A Bag of Tricks, by A. Gelman and D. Nolan. The software Fathom, developed with support from the National Science Foundation, allows users to type in their own data, to use the over 300 data files that come with Fathom, or to import data from text files or directly from the Internet.
Laurie J. Burton, Western Oregon University, reported about a technique used in a general education mathematics survey course aimed at engaging students and improving their communication skills. She incorporated weekly projects by dividing the class into groups of four and requiring the groups to write summaries of their projects On a rotating basis, one student was responsible for the written submission, while the others served as editors. Over the semester each student was responsible for two weekly, typed 'write ups,� each worth 12.5% of the course grade. In the written submissions the students were required to include an introduction, a statement of assumptions, a rewriting of each problem, a display of all steps the mathematics, and a clear sentence reporting he answer. Burton reported that 'The class started off slowly to say the least! I wrote an extensive set of directions for them, but clearly many of them didn't bother to read their packet! The first three weeks of projects were dismal. Eventually they all sort of clued in and by the end of the term students were turning in really nice projects. Clearly they had learned something. I was really impressed and happy as a teacher that the students were making such clear progress.� Information about the nature and use of current projects for the course, Introduction to Contemporary Math, is available through a link on her website.
The University of South Carolina Spartanburg (USCS) developed Project-Based Instruction in Mathematics for the Liberal Arts. The website provides projects and resources for instructors and students who wish to teach and learn college mathematics or post-algebra high school mathematics via project-based instruction. In 1994 a group of faculty members at USCS began to develop and test an innovative pedagogy integrating technology and activity- or project-based instruction in mathematics for liberal arts majors. The group collected, modified, and wrote items for a packet of activities designed to form the core of material that would be used to supplement and eventually replace the textbook in the 'College Mathematics� course. Subsequently, M.B. Ulmer wrote a booklet to lend structure to the use of the activities, which supplanted used of a regular textbook in many sections. Ulmer reports that success rates have risen dramatically for students who have gone through the program and that their subsequent performance in required statistics courses has also shown improvement.
The Mathematical Association of America recently established a Special Interest Group on Quantitative Literacy (SIGMAA QL). Information about previous work of the CUPM subcommittee on Quantitative Literacy Requirements is maintained by Rick Gillman, Valparaiso University. The Quantitative Literacy webpage of MAA Online contains links for information a reports concerning quantitative literacy that were formerly located on the website of the National Council on Education and the Disciplines at the Woodrow Wilson National Fellowship Foundation.
In 'General Education Mathematics: New Approaches for a New Millennium,� Jeffrey O. Bennett, University of Colorado at Boulder, and William L. Briggs, University of Colorado at Denver present some observations regarding the problems of developing appropriate mathematics curricula for non-science, engineering, mathematics (SEM) students, along with recommendations for their solution The authors state that the ways students need mathematics are for college, for career, and for life. When a committee at the University of Colorado examined what mathematics would be appropriate to meet these needs, four content areas emerged: logic, critical thinking, and problem solving; number sense and estimation; statistical interpretation and basic probability; and interpretation of graphs and models. Bennett and Briggs advocate a context-driven approach for instruction in these areas.
The University of Nevada, Reno, established a Mathematics Center with a focus on integrating mathematics across the curriculum. The main goal of the Center is to improve the quantitative and mathematical skills of all students, and to help them better appreciate the importance and utility of mathematics. The Center does this primarily by working with faculty in various disciplines to assist them in enhancing the quantitative and mathematical content of their courses, and then providing them and their students with the necessary support. The plan calls for influencing courses ranging from the natural and social sciences to English and the fine arts. It also calls for bringing applications from other disciplines into the elementary mathematics classes. The Core Curriculum is a high priority for the project.
Stetson University offers students a wide variety of mathematics courses to complete the mathematics requirement. Courses meeting the general mathematics requirement include Finite Mathematics, Mathematical Game Theory, Chaos and Fractals, In Search of Infinity, Great Ideas in Mathematics, Mathematics and Multiculturalism, Geometry, Introduction to Mathematical Modeling, and Cryptology, as well as the calculus courses.
Goucher College also offers a variety of courses to students completing their general mathematics requirement. Available courses include Topics in Contemporary Mathematics, Introduction to Statistics, Problem Solving and Mathematics-Algebra, Problem Solving and Mathematics-Geometry, Functions and Graphs, Discrete Mathematics, various levels of calculus courses and Linear Algebra.
The Math Lab Program at Francis Marion University is designed to give students access to mathematics across a wide range of entry-level courses and to make it possible for students to work at their own pace. The Math Lab features an individualized format that makes it possible for a student to complete the course in more or less time than the regular semester. However, to succeed in the Math Lab program students must have motivation and self-discipline. Francis Marion strongly recommends that students use the available resources including extra help sessions, extensive mini-lab hours, computer tutorials, videotapes, and instructor office hours. The Math Lab Program offers the introductory courses of College Algebra with Analytic Geometry I, College Algebra with Analytic Geometry II, College Trigonometry with Analytic Geometry II, and Calculus I. All but the first course satisfy the General Education Requirement. The first course does, however, earn credit toward graduation. Syllabi for all courses are located at the site indicated above.
Founded in 1996, the Historically Black College and University (HBCU) Consortium for College Algebra Reform developed the Contemporary College Algebra program. Its purpose is to refocus college algebra to address the quantitative proficiencies that students need for mathematics and other disciplines, society, and the workplace. Thus emphasis is placed on trying to empower students as problem solvers in the modeling sense rather than making them try to master lists of algebraic rules. To support the purpose, the course emphasizes developing communication skills, engaging students in small group activities/projects, using technology for doing mathematics, and trying to build student confidence. Discussions with faculty in different disciplines and with people in the workplace influenced the development of the program. In particular, the heavy emphasis placed on data as well as on graphical and numerical analysis reflects these discussions. Data analysis is used to generate the need for functions, which in turn leads to modeling situations in various disciplines using recursive sequences. Creators of the program believe that the ability to understand elementary data analysis, to extract functional relationships from data, and to model real-life situations mathematically is fundamental to the education of every student. The pedagogical environment is focused on student learning, which includes a strong emphasis on small-group in-class activities and out-of-class projects. Technology is used extensively as part of discovery activities. The program has expanded beyond the HBCU Consortium to include majority schools and tribal colleges.
A conference on College Algebra was sponsored by the HBCU College Algebra Reform Consortium in December 2002. Two articles that resulted from a workshop sponsored by the Consortium are 'College Algebra� by Arnold Packer, Johns Hopkins University, 'An Urgent Call to Improve Traditional College Algebra Programs� by Don Small, U.S. Military Academy, and 'Who Are the Students Who Take Precalculus?� by Mercedes A. McGowen, William Rainey Harper College. Conference participants recommended the following as major characteristics of a college algebra program:
* Real-world problem based: a topic is introduced through a real-world problem and then the mathematics necessary to solve the problem is developed. Example problem: Schedule a multi-faceted process.
* Modeling (transforming a real-world problem into mathematics): - using power and exponential functions, systems of equations, graphing, and difference equations ' primary emphasis is placed on creation of a model and interpretation of the results. Example: Model the stopping time versus speed data presented in a driver's manual by plotting the data and fitting a curve to the plot. Interpret how well the resulting stopping time function models reality at small speeds. Revise the model, if necessary, to account for zero stopping time at zero speed. Use the (revised) function to predict stopping times for speeds not given by the data. Revise the model to account for different road surfaces.
* Emphasize communication skills: as needed in society as well as in academia ' reading, writing, presenting, and listening. Example: Students learn how to read, understand, and critique news articles that include quantitative information and to make informed decisions based on the articles.
* Small group projects: involving inquiry and inference. Example: Analyze the soda preference of students by conducting a survey and comparing the results with data from the school's dining hall or a local fast food restaurant.
* Appropriate use of technology to enhance conceptual understanding, visualization, and inquiry, as well as for computation. Example: 'What-if� a model for paying off a credit card debt by changing the monthly payment, interest rate, size of debt, etc. Plot the results to visually compare the different scenarios.
* Use of hands-on activities rather than all-lecture format.
The Texas Southern Consortium for College Algebra Reform, part of Project Intermath, has two goals: (1) to develop a contemporary college algebra course that educates students for the future rather than training them for the past; and (2) to change the culture surrounding the college algebra program. The primary goal of its contemporary college algebra course is to empower students to become exploratory learners. Most of the topics in the course begin with the analysis of data. The course involves the use of small-group projects developed by interdisciplinary faculty teams, incorporates a strong technology component, emphasizes the development of students' communication skills, and attempts to improve students' mathematical self-esteem and confidence in their problem-solving skills. The specific objectives of the goal of changing the college algebra culture are to energize faculty to develop modes of instruction that actively engage students in their learning, instill in faculty a sense of ownership and pride about teaching college algebra, encourage faculty in disciplines that require college algebra to develop a sense of involvement and responsibility for the college algebra program, and obtain administrative support for a reformed college algebra program.
Three faculty members at the University of Houston Downtown, William Waller, Linda Becerra, and Ongard Sirisaengtaksin, wrote a case study about the process of initiating change in their college algebra course. They write that the challenges they believed needed addressing in their previous course were student performance and student preparation, and that traditional methods were not effective in meeting these challenges. Their aims were to provide students with numerous opportunities to learn, lead students to learn fundamental concepts and skills through solving real-world problems, stimulate student interest and increase motivation (thereby improving retention), increase mathematical literacy, use diverse teaching strategies, and offer a technology-dependent curriculum.
'A Research Evaluation of a Reform College Algebra Course� by Joan Cohen Jones, Eastern Michigan University and Andrew Balas, University of Wisconsin Eau Claire, describes how the authors, a mathematics educator and a mathematician, structured a college algebra course with the aim of empowering students by having them construct their own understanding through discussing concepts in small cooperative groups. In the course, students had to apply traditional algebra skills to problems in real-life situations. Research conducted by the authors indicated that the students improved in their attitudes toward mathematics and their confidence in their ability to solve problems, that students attributed their success less to the instructors and more to themselves and their peers, that successful groups bonded well, and that the groups served as a forum to explore and test ideas.
The College Algebra Reform Papers website at the State University of New York ' Oswego contains articles by William Fox, Francis Marion University, Scott Herriott, Maharishi University of Management, and Laurie Hopkins, Columbia College, discussing the appropriateness of college algebra practices and offering suggestions for improvement. William Fox addresses the issue of integrating modeling and problem solving in developing new courses to replace the traditional college algebra course. Scott Herriott compares the traditional college algebra curriculum with more recent reform approaches and also discusses related issues of national and local educational policy. Laurie Hopkins focuses on the role of technology, and specifically the use of handheld computer algebra systems in the college algebra classroom. The website also includes two 'provacateur� responses to the articles.
1. Create mathematical models of abstract and real world situations using linear, quadratic, polynomial, rational, exponential, and logarithmic expressions.
2. Use inductive reasoning to develop mathematical conjectures involving these function models.
3. Use deductive reasoning to verify and apply mathematical arguments involving these models. (Distinguish between the uses of inductive and deductive reasoning.)
4. Represent these functions in graphical, tabular, symbolic and narrative form, and then use mathematical problem solving techniques to solve problems involving these functions.
5. Make mathematical connections to, and solve problems from other disciplines involving these functions.
6. Use oral and written skills to individually and collaboratively communicate about these function models.
7. Apply appropriate technology to enhance mathematical thinking and understanding, solve mathematical problems, and judge the reasonableness of their results.
After examining the student population in the college algebra course and consulting with departments that required that course, the Hiram College Department of Mathematics eliminated the course and replaced it with Mathematical Modeling in the Liberal Arts. In this course, students use data together with linear, quadratic, polynomial, exponential, and logarithmic functions to model naturally occurring phenomena in medicine, economics, business, ecology, and other disciplines. The course uses numerical, graphical, verbal, and symbolic modeling methods.
Bonnie Gold's article 'Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course� describes Monmouth University's mathematics department's experience replacing a single college algebra course taken by almost all students by four courses designed for particular student populations: elementary education majors, biology majors, social science majors, and students who eventually go on to a standard calculus course. The three new courses were designed in consultation with faculty from the relevant departments. In addition, the course that prepares students for calculus no longer satisfies the general education mathematics requirement, whereas the other three courses ' as well as a pre-existing quantitative reasoning and problem solving course ' do satisfy the requirement. For an electronic copy of the article, contact Bonnie Gold.
At American University, Elementary Mathematical Models is a course at the level of college algebra or precalculus that uses simple discrete growth models to provide a context for the study of elementary real functions. The mathematical content has a large degree of overlap with traditional college algebra or precalculus courses and includes properties and applications of linear, polynomial, rational, exponential, and logarithmic functions. The course goals emphasize looking realistically at the methodology of applying mathematics through models, with consistent use of numerical, graphical, and symbolic methods over the entire course. The use of simple difference equation models throughout is intended to provide a unifying theme. The course begins with arithmetic growth and linear functions, and concludes with logistic growth models. A qualitative discussion of how chaos can arise in discrete logistic models is the climax of the course.
At Georgia College and State University a new college algebra course focuses on integrating technology in the form of graphing calculators and providing learning support: strategies for test taking, dealing with math anxieties, mastering mathematical concepts, and developing graphing calculator skills. The article College Algebra, Learning Support, and Technology: What is the Connection? by Margo Alexander briefly describes a study done to compare college algebra students who concurrently took a learning support course against those who did not have additional support.
Paul Dirks, Miami-Dade Community College, developed a course entitled Contemporary College Algebra that incorporated group activities, a heavy use of technology, and outside-of-class group projects. He reported that he was guided by the description below (from a 2002 AMS-MAA-MER session on education reform):
Contemporary College Algebra, a data-driven modeling course, is an example of a reformed college algebra course that serves as a base course for a quantitative literacy program. The course focuses on problem solving in the modeling sense rather than the exercise sense. Communications (reading, writing, presenting), use of technology, small group interdisciplinary projects, analysis of real data sets, graphical analysis, and recursive sequence models are all strongly emphasized. The course is designed to prepare students to be mathematically literate in today's information society. The focus is on preparing students for the future rather than training them for the past.
Dirks stated that he was at first unsure about whether his students had the mathematical and communication skills required to succeed in this course, but after three semesters of teaching it, he reported that they have exceeded his expectations. He said that he has observed improved student engagement in critical thinking (outlining issues clearly, posing non-trivial questions, organizing their discoveries, and presenting results in a variety of forms), increased exercise of creativity and autodidactic activity (learning new mathematics and adapting old, learning and using new technologies, creatively presenting results); and a phenomenon best expressed by the statement, 'The whole is more than the sum of its parts� (group work pushing toward a better solution). Dirks stated that he has forever changed the way he teaches as a result of this experience, that even if this is not the 'final answer,� he feels his teaching is moving in the right direction.
At the University of Arkansas students can enroll in a special section of College Algebra taught in conjunction with the Mathematics Resource and Tutoring Center (MRTC). The course consists of in-class and MRTC activities plus computer work. The computer work consists of eight interactive modules where the student must demonstrate understanding of the concepts and techniques from the text by scoring 90% or above in order to move to the practice problems for the module. Once students complete all module practice problems correctly, they may take the associated test. This purpose of the course is to prepare students for higher-level mathematics courses. As a consequence, the course offers every student as many different opportunities to learn or re-learn fundamental algebraic material as possible.
Tim Warkentin and Mark Whisler, Cloud County Community College, wrote 'Questions about College Algebra� to describe their experience assessing alternative formats for their college algebra course. They conclude that 'The change with the greatest impact is likely to be the change in format that we instituted in the fall of 2002 in College Algebra. We are offering all of our daytime sections of College Algebra as classes that, along with its companion class, College Algebra Explorations, meet every day.�
'A Research Evaluation of a Reform College Algebra Course� was conducted by Joan Cohen Jones, Eastern Michigan University, and Andrew Balas, University of Wisconsin Eau Claire. The research indicated that 'that the students improved in their attitudes toward mathematics and their confidence in their ability to solve problems. They attributed their success less to the instructors and more to themselves and their peers. Successful groups bonded well, and the group served as a forum to explore and test ideas.�
In 'Analysis of Effectiveness of Supplemental Instruction (SI) Sessions for College Algebra, Calculus, and Statistics,� Sandra Burmeister, Patricia Ann Kenney, and Doris L. Nice explore data from 177 courses in mathematics for which SI support was given (1996). The SI sessions are based on theoretical notions of 'metacognition� and aim to help students develop a cognitive monitoring system and make effective use of learning strategies. The data indicate that SI sessions promote student success. There were positive differences in grades for students who participated in SI sessions in college algebra, calculus, and statistics when compared with students who did not participate. Additionally, in 1994 Kenney and James Kallison reported on research studies on the effectiveness of SI in mathematics classes.
In 'Precalculus in Transition: A Preliminary Report� by Trisha Bergthold and Ho Kuen Ng, San Jose State University, the authors discuss their initial investigation of low student achievement in our five-unit precalculus course. We investigated issues related to course content, student placement, and student success. As a result, we have streamlined the course content, we are planning to implement a required placement test, and we are planning a 1'2 week preparatory workshop for students whose knowledge and skills appear to be weak. Further study is ongoing.
An evaluation of the Moravian College integrated calculus and precalculus course by the Fund for the Improvement of Post-Secondary Education examined student persistence rates, the performance of integrated-course students compared to students in the traditional sequence on a set of problems included in the final examinations of both courses, instructor attitudes, and student attitudes. It concluded: 'Uniformly, student persistence through the sequence was higher for the integrated course than for calculus preceded by precalculus. Integrated-sequence students performed at least as well on a set of common problems as the traditional-course students, and sometimes better. In general, both faculty and students liked the integrated sequence better.�
[1] According to the CBMS study in the Fall of 2000, a total of 1,979,000 students were enrolled in courses it classified as 'remedial� or 'introductory� with course titles such as elementary algebra, college algebra, Pre-calculus, algebra and trigonometry, finite mathematics, contemporary mathematics, quantitative reasoning. The number of students enrolled in these courses is much greater than the 676,000 enrolled in calculus I, II or III, the 264,000 enrolled in elementary statistics, or the 287,000 enrolled in all other undergraduate courses in mathematics or statistics. At some institutions, calculus courses satisfy general education requirements. Although calculus courses can and should meet the goals of Recommendation A.1, such courses are not the focus of this section. |
MATH 120 Pre-Calculus Mathematics
Functions and graphs: polynomial, exponential, logarithmic and trigonometric functions; analytic geometry. Emphasis is on problem-solving, mathematical modeling and the use of technology. Designed primarily as preparation for calculus |
Elementary Classical Analysis - 2nd edition
Summary: Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.
Focuses primarily on analysis in Euclidean space with a view toward applications ...show more Written to appeal to students in engineering and the physical sciences as well as pure mathematics
More material on variable calculus
Expanded treatment of metric spaces
Detailed coverage of the foundations of the real number system ...show less
Continuity Images of Compact and Connected Sets Operations on Continuous Mappings The Boundedness of Continuous Functions of Compact Sets The Intermediate Value Theorem Uniform Continuity Differentiation of Functions of One Variable Integration of Functions of One Variable
6. Uniform Convergence
Pointwise and Uniform Convergence The Weierstrass M Test Integration and Differentiation of Series The Elementary Functions The Space of Continuous Functions The Arzela-Ascoli Theorem The Contraction Mapping Principle and Its Applications The Stone-Weierstrass Theorem The Dirichlet and Abel Tests Power Series and Cesaro and Abel Summability
Giving great service since 2004: Buy from the Best! 4,000,000 items shipped to delighted customers. We have 1,000,000 unique items ready to ship! Find your Great Buy today!
$57.78 +$3.99 s/h
Good
southbrooklyntexts Brooklyn, NY
0716721058 |
Wavelet Theory. An Elementary Approach with Applications
A self-contained, elementary introduction to wavelet theory and applications
Exploring the growing relevance of wavelets in the field of mathematics, Wavelet Theory: An Elementary Approach with Applications provides an introduction to the topic, detailing the fundamental concepts and presenting its major impacts in the world beyond academia. Drawing on concepts from calculus and linear algebra, this book helps readers sharpen their mathematical proof writing and reading skills through interesting, real-world applications.
The book begins with a brief introduction to the fundamentals of complex numbers and the space of square-integrable functions. Next, Fourier series and the Fourier transform are presented as tools for understanding wavelet analysis and the study of wavelets in the transform domain. Subsequent chapters provide a comprehensive treatment of various types of wavelets and their related concepts, such as Haar spaces, multiresolution analysis, Daubechies wavelets, and biorthogonal wavelets. In addition, the authors include two chapters that carefully detail the transition from wavelet theory to the discrete wavelet transformations. To illustrate the relevance
of wavelet theory in the digital age, the book includes two in-depth sections on current applications: the FBI Wavelet Scalar Quantization Standard and image segmentation.
In order to facilitate mastery of the content, the book features more than 400 exercises that range from theoretical to computational in nature and are structured in a multi-part format in order to assist readers with the correct proof or solution. These problems provide an opportunity for readers to further investigate various applications of wavelets. All problems are compatible with software packages and computer labs that are available on the book's related Web site, allowing readers to perform various imaging/audio tasks, explore computer wavelet transformations and their inverses, and visualize the applications discussed throughout the book.
Requiring only a prerequisite knowledge of linear algebra and calculus, Wavelet Theory is an excellent book for courses in mathematics, engineering, and physics at the upper-undergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level.
SHOW LESS
READ MORE >
Preface.
Acknowledgments.
1 The Complex Plane and the Space L2(R).
1.1 Complex Numbers and Basic Operations.
Problems.
1.2 The Space L2(R).
Problems.
1.3 Inner Products.
Problems.
1.4 Bases and Projections.
Problems.
2 Fourier Series and Fourier Transformations.
2.1 Euler's Formula and the Complex Exponential Function.
Problems.
2.2 Fourier Series.
Problems.
2.3 The Fourier Transform.
Problems.
2.4 Convolution and B-Splines.
Problems.
3 Haar Spaces.
3.1 The Haar Space V0.
Problems.
3.2 The General Haar Space Vj.
Problems.
3.3 The Haar Wavelet Space W0.
Problems.
3.4 The General Haar Wavelet Space Wj.
Problems.
3.5 Decomposition and Reconstruction.
Problems.
3.6 Summary.
4 The Discrete Haar Wavelet Transform and Applications.
4.1 The One-Dimensional Transformation.
Problems.
4.2 The Two-Dimensional Transformation.
Problems.
4.3 Edge Detection and Naive Image Compression.
5 Multiresolution Analysis.
5.1 Multiresolution Analysis.
Problems.
5.2 The View from the Transform Domain.
Problems.
5.3 Examples of Multiresolution Analyses.
Problems.
5.4 Summary.
6 Daubechies Scaling Functions and Wavelets.
6.1 Constructing the Daubechies Scaling Functions.
Problems.
6.2 The Cascade Algorithm.
Problems.
6.3 Orthogonal Translates, Coding and Projections.
Problems.
7 The Discrete Daubechies Transformation and Applications.
7.1 The Discrete Daubechies Wavelet Transform.
Problems.
7.2 Projections and Signal and Image Compression.
Problems.
7.3 Naive Image Segmentation.
Problems.
8 Biorthogonal Scaling Functions and Wavelets.
8.1 A Biorthogonal Example and Duality.
Problems.
8.2 Biorthogonality Conditions for Symbols and Wavelet Spaces.
Problems.
8.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair.
Problems.
8.4 Decomposition and Reconstruction.
Problems.
8.5 The Discrete Biorthogonal Wavelet Transformation.
Problems.
8.6 Riesz Basis Theory.
Problems.
9 Wavelet Packets.
9.1 Constructing Wavelet Packet Functions.
Problems.
9.2 Wavelet Packet Spaces.
Problems.
9.3 The Discrete Packet Transform and Best Basis Algorithm.
Problems.
9.4 The FBI Fingerprint Compression Standard.
Appendix A: Huffman Coding.
Problems.
References.
Topic Index.
Author Index.
"The book, putting emphasize on an analytic facet of wavelets, can be seen as complementary. to the previous Patrick J. Van Fleet's book, DiscreteWavelet Transformations: An Elementary. Approach with Applications, focused on their algebraic properties." (Zentralblatt MATH, 2011)
"Requiring only a prerequisite knowledge of calculus and linear algebra, Wavelet theory is an excellent book for courses in mathematics, engineering, and physics at the upper-undergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level." (Mathematical Reviews, 2011) |
1934968390","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":7.16,"ASIN":"1933241586","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":7.16,"ASIN":"1933241578","isPreorder":0}],"shippingId":"1934968390::Oq%2Bp5NRv6Av3v%2Fjn8%2F8AERsmcsqm6OwDoxqR6kVm%2F45zkzthPtYdJv6zhc80UIzX85rDEraYny6qK8GSIlGCLlykyskJYoSX3sXyOtUIvFk%3D,1933241586::e0DiA4K9XMrqwH4LmzXQZZZZcfTberrxtDsjN%2B3wAIldlsvOhTFx%2B2ZuajQo8G0%2BkEnK%2Fvhg9OkZJIdrw%2Ftmg3z3MKCjP1eqlVfoTnQSqng%3D,1933241578::A3hr7NZ0HaPO%2F9pfTaXWlJeg2HqL%2Fu2PFSNEgF4TL5yU7ykvmslrTmhC%2BgYyYIk%2BJShX7g%2BQhatCxETKeucY7Mztb%2FEJj0tpuC1BY7K9Q word problems workbook, I would have expected problems that require more concept-level thinking (what operation is needed here?). The problems provide plenty of repeat practice for basic math functions, but there's not much mix of types of problems in each lesson, so the student isn't really required to do much critical thinking in figuring out what approach to use in solving the problem. Once the student solves the first problem or two, the rest of the problems are pretty predictable. I would prefer to have quicker math functions involved if it meant my daughter had to spend more time critically evaluating the path to the solution.
Will write a second review later; but, just wanted to let parents/readers know that I browsed through the book and it looks great. My daughter did 2 pages of it and the questions in books are already picking her brain!!! It is less boring (better than plain old additions/substractions/multiplication/division exercise rut)! She loves to check on the net to see how many meters are in a kilometer to solve the problems in the book. So far so good; keeping my fingers crossed.
Im am helping a friends daugther study for the AZ Aims test. Not much information on the test is available nor study material. The makers of the test have three released exams, and thats all so, i was looking for a book with similar word type problems to study with. I figured out, the was having issues with certain keywords that changed the meaning of what was being asked. So with this workbook, I know had lots of word problems with similar style as the test. Well after doing a few problems in the book, the child finally started to see the trick question. The child finally passed the exam with ease after having failed it before being tutored. But with out the book it would have been difficult to write so many brand new questions. |
"Combinatorial and Computational Mathematics serves as an introduction to the current state of knowledge in various areas like Formal Languages, Applications of Fuzzy Set Theory, Combinatorial Problems, Fluid Mechanics, ect."--BOOK JACKET |
You are here
The Real Numbers and Real Analysis
Publisher:
Springer
Number of Pages:
553
Price:
84.95
ISBN:
9780387721767
The most distinctive characteristic of this text on real analysis is its three-in-one feature. It was designed specifically for three distinct groups of students. The first audience consists of mathematics majors taking an ordinary introduction to real analysis. The second is prospective high school mathematics teachers getting an introduction to real analysis. The third group is prospective high school teachers taking a second real analysis course, presumably as part of a Master of Arts in Teaching (M.A.T.) degree. The book was motivated by a need for a textbook for the M.A.T. students, but is intended to have enough flexibility to serve the other groups as well. The author provides suggested paths through his book for each potential audience.
The topics treated are more or less standard. Beginning with the construction of real numbers and their properties, the author proceeds through limits and continuity, derivatives and integrals. Transcendental functions are treated in some detail later in a separate chapter. Sequences and series — first of numbers, then of functions — are presented at the end. The treatment of the real numbers in the first two chapters (more than a hundred pages in total) is a good deal more extensive than in comparable texts. It includes, for example, the Peano postulates, axiomatic treatment of the integers, rationals, and real numbers, and construction of the reals via Dedekind cuts.
The approach is detailed and rigorous, and the level of sophistication is in the middle of the spectrum. The author is a believer in "slow and steady", so the proofs he provides are usually written out in full detail. He also prefers to minimize technicalities, so he omits things like limits inferior and superior, and avoids proofs that are too slick. Because sequences and series are not treated until the end, some theorems about continuity or derivatives, for example, have somewhat more tedious proofs than they might otherwise.
This is a talky book. The author includes historical remarks in every chapter and "Reflections" in every section. The purpose of the reflections is to try to give the student a broader perspective on what he or she is learning. In general, these additions are admirable and desirable, but combined with a wordier style throughout, the text occasionally has a flavor of "too much-ness". While one might readily choose this over a terse style like that of Rudin's, there is clearly a tradeoff. Students, in my experience, do not readily read mathematics textbooks, so more is perhaps not better.
I have learned to be wary of historical background material in mathematics texts. I am by no means an expert, but I fear that too many authors uncritically pass on historical errors and misconceptions. While I have no special concerns about the historical notes in this text, there does seem to be a tendency in this book to portray the history of calculus as an onward and upward path without the missteps and dead ends we all know are a real part of mathematical development.
As I find with many other introductory analysis texts, I am troubled again here by the tendency to treat the course as a means of ratifying the theorems of calculus. As Thomas Kőrner says in A Companion to Analysis, "It is surprising how many people think that analysis consists in the difficult proofs of obvious theorems." Ifeel very strongly that students need to understand why they are pushed to this level of rigor, what can go wrong, and why a little doubt is a good thing.
In spite of my quibbles, this is a strong text, especially for students who need more guidance and support. The book gives an instructor plenty of options for planning a course |
ittinger series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept ...Show synopsisThe Bittinger series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed readers to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting readers with quality applications, exercises, and new review and study materials to help students apply and retain their knowledgeDescription:New in new dust jacket. Brand New as listed. ISBN 9780321771889....New in new dust jacket. Brand New as listed. ISBN 9780321771889. Clean! Out of sight Shipping & Customer Service! We process all orders same day! !
Math book aided in my college course, however after 2 weeks of use, the book is falling apart and it's very difficult to keep pages in line. Only used book at a minimum and it's falling apart very easily |
English
Wouldn't it be great if there were a statistics book that made histograms, probability distributions, and chi square analysis more enjoyable than going to the dentist? Head First Statistics brings...
View More
Ever wished you could learn C from a book? Head First C provides a complete learning experience for C and structured imperative programming. With a unique method that goes beyond syntax and how-to...
View More
Exploring Religious Diversity analyzes the philosophical questions raised by the fact that many religions in the world often appear to contradict each other in doctrine and practice.Analyzes the ph...
View More
Offers information ranging from basic point and line notation to the time-saving secrets of similar and congruent angles. Through stories and practical examples from the world around you, this book...
View More
Provides a set of principles for doing educational research for social justice. This book is helpful to various researchers, whether they are just beginning their first project, or whether they are...
View More
This substantial anthology is a comprehensive, authoritative collection of the classical and contemporary readings in the philosophy of religion, providing a survey and analysis of the key issues,...
View More
How can one best work for justice and empowerment in the ever-changing, real-life messy world of primary school classrooms? Written by a full-time teacher and an action-researcher, this book points...
View More
You may be comfortable with conventional two-dimensional geometry, but what about 3D geometry? Do spheres, polygons, and the Pythagorean equation just make you dizzy? With \"Head First 3D Geometry,...
View More
In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaru...
View More |
MATH 0103
Intermediate Algebra
Course info & reviews
Review skills necessary for pre-calculus and college mathematics and statistics. Topics include a review of sets, operations with polynomial and rational expressions, solving equations and inequalities, and an introduction to the coordinate plane and functions. RESTRICTIONS: Requires arithmetic and Algebra I skills. MATH010 does not ea...
The figure below shows the geometric basis of the algebraic process of completing the square. Study... Show more
The figure below shows the geometric basis of the algebraic process of completing the square. Study the figure, where each algebraic expression is
related to the area of the corresponding geometric figure. What is the area of the missing piece that will "complete the square�
Draw a similar figure that shows geometrically how to complete the square on the expression x^2+8x. |
New to This Edition
Math and Drug Calculations Online for Nursing and Health Professions is a complete drug calculations online course that provides students an opportunity for application and practice. Since it is not tied to a specific text, this online course is suitable for all drug calculations classes. It incorporates the ratio and proportion, fractional equation, formula, and dimensional analysis methods and presents a step-by-step approach to the calculation and administration of drug dosages. Animations, voice-overs, and interactive self-assessment activities are used to provide an engaging and interactive course platform for students. This online course consists of three comprehensive modules; Module 1: Math Introduction and Overview, Module 2: Medication Administration, and Module 3: Medication Administration in Specialty Areas. Each module includes practice problems to promote active learning and quizzes that instructors can use to evaluate students' understanding of content presented in the course. A comprehensive test bank of approximately 300 questions is also provided for instructor's to build quizzes and test.
Key Features
Includes the four drug calculation methods (ratio and proportion, fractional equation, formula, and dimensional analysis) to expose the user to key calculation methods so they can apply the method which works best for them.
Modules are organized by topic sections that include an overview, objectives, what you need to know, example problems, practice problems, and one or more quizzes.
Follows abbreviation and dose designation recommendations from the Joint Commission on Accreditation of Healthcare Organizations (JCAHO) and the Institute for Safe Medication Practices (ISMP).
Many of the math practice problems include a tutorial button for each of the four drug calculation methods that provides a step-by-step tutorial to solving the problem in the chosen method.
Animations demonstrate various concepts related to drug calculation and administration, with some animations requiring user participation.
Interactive self-assessment activities are incorporated throughout the course to allow users to apply their knowledge in context.
Voice-overs enhance the step-by-step explanation of medication administration procedures and the drug calculation methods demonstrated throughout the course.
One or more Quizzes are included within each module to evaluate understanding of all the major topics covered in that particular module.
Provides the latest drug administration techniques and devices with detailed explanations of the various ways to administering drugs, including oral, intravenous, intramuscular, subcutaneous and other routes.
Includes the most up-to-date, commonly used drugs so users have exposure to what is being used in the "real world" of clinical practice.
Presents information on infusion pumps (enteral, single, multi-channel, PCA, and insulin) to help users understand their increased use in drug administration.
Related Links
Math and Drug Calculations Online for Nursing and Health Professions is a completely new drug calculations online course |
Pages
Saturday, August 31, 2013
Algebra 2 Interactive Notebook Pages for Unit 1
This year, I have resolved to do a much better job at the interactive notebook in Algebra 2 than last year. Last year, we had 12 students in my entire school who were enrolled in Algebra 2. This year, that number is just under 40. This is both exciting and kinda terrifying. First of all, it means that more students are prepared and willing to take Algebra 2. At the same time, my Algebra 2 students this year are greater in number and much more varied in level. This presents many challenges. But, these are challenges I am excited to attempt to meet.
Our first unit in Algebra 2 is an introduction to functions, function notation, domain and range, intercepts, maximums and minimums, intervals of increasing and decreasing, finding solutions, and transformations. My goal is to create a foundation which I can build off of once we start linear functions. I am also working hard to prove to my students that they are capable of doing Algebra 2 level work. Many of my students have extremely low confidence. We are also learning how to use the graphing calculator. This is the first experience any of my students have had with a graphing calculator, and I am working hard to make it a positive one.
So far, my Algebra 2 students are loving our interactive notebooks. They thank me on an (almost) daily basis for making Algebra 2 visual, fun, and easy. I have some students who are complaining right now that Algebra 2 is too easy. I told them that they just had to wait. Before they knew it, we would be exploring logarithms, exponentials, conic sections, and all kinds of other exciting mathematical relations. After going on and on about how excited I was about everything we were going to be learning and studying this year, one student asked, "Do you like math?" I was a bit taken aback by this question. Are there math teachers who don't like math? Of course, I like math. I love math! I eat, breath, and sleep math.
As usual, I have embedded the files for these foldables at the end of the post. If you have trouble viewing them, please make sure that you have Flash/Shockwave installed. If that does not correct the problem, please send me an e-mail and let me know what documents you are needing. I will be happy to send them to you!
My Algebra 2 Interactive Notebook
Algebra 2 Unit 1 Table of Contents (Thus Far)
I have already blogged about the NAGS foldable I had my Algebra 2 students create here.
NAGS Foldable - Outside
NAGS Foldable - Inside
NAGS Chart
I still haven't found a better way to practice differentiating between function/not a function than this card sort. I blogged about this last year.
Function / Not a Function Card Sort
We also created a Frayer Model for the word "function."
Function / Vertical Line Test Frayer Model
I stole this coordinate plane foldable from Ms. Haley and her wonder Journal Wizard blog! I think this is a big improvement over the coordinate plane foldable I did with my students last year. I created a template for this foldable which I have embedded below.
Parts of the Coordinate Plane Foldable
Parts of the Coordinate Plane Foldable
Parts of the Coordinate Plane Foldable
Our notes over independent and dependent variables were less than exciting. Maybe next year I will come up with a card sort or something. Hmm...
Independent and Dependent Variable Notes
Last year, my students had a TERRIBLE time remembering the difference between domain and range. This summer, at the amazing Common Core Training I received from the Oklahoma Geometry and Algebra Project (OGAP), I was introduced to an amazing resource--Shmoop. They have amazing commentary for each and every high school common core math standard! I learned about the DIXROY acronym from their commentary on F-IF.1.
I was able to re-use the domain/range notation foldable that I created last year for my Algebra 2 students. My students were VERY confused by the different notations. I haven't yet figured out a way to introduce these notations without overwhelming my students. They recovered, eventually.
Domain and Range Notation Foldable - Outside
Domain and Range Notation Foldable - Inside
I downloaded the domain and range cards from this blog post. There are 32 cards which give my students 32 opportunities to practice finding the domain and range!
Domain and Range Foldable
We made a tiny envelope to hold our 32 cards. Let me just say - having the students cut out all 32 cards took WAY too much time. I was about ready to pull my hair out. I think we might of spent half of a fifty minute class period just cutting these cards out. But, we used them a lot, so I think it was worth it.
I LOVED the envelope template that Kathryn (iisanumber.blogspot.com) posted earlier this summer. I downsized her template to the exact size needed to fit the domain and range cards I linked to earlier.
As you can see, this foldable perfectly holds the domain and range practice cards from our handy-dandy envelope!
The foldable is made to perfectly hold our domain and range practice cards that are housed in the envelope.
Students fold over the domain tabs to help them determine the left-most and right-most points on the graph. If the graph goes approaches negative or positive infinity, the students leave the flap open where it reads positive or negative infinity. I wanted my treatment of domain and range to be much more hands-on this year, and I think this foldable does the trick!
After doing many, many cards together, I had students find the domain and range of all 32 cards as homework. They had to write the domain and range in both interval and algebraic notation. (And, the discrete graphs had to have their domain written in set notation.) The next day, I gave them an answer key to use the check their work.
The Domain and Range Foldable in Action
One of the main thing my students need to be able to do on their Algebra 2 EOI is to describe graphs. This foldable is an attempt to introduce my students to the concepts of x-intercepts, y-intercepts, relative maximums, relative minimums, increasing intervals, decreasing intervals, roots, solutions, and zeros.
Describing Characteristics of Graphs Foldable - Outside
Because there is so much information on this one foldable, this was a perfect opportunity to use COLOR WITH A PURPOSE. Each term was marked with a different color. And the corresponding part of the graph was marked with the same color. This is one of my favorite foldables that we have done this year!
I've posted some close-ups of the flaps if you'd like to see what I had my students write.
Close-up of Right Flaps
Close-Up of Left Flaps
Last year, my Algebra 2 students really struggled with the concept of an inverse. So, this year, I decided to start talking about inverses very early in the school year. This will allow us to revisit the concept over and over as we explore different types of function in a much more in depth manner. By the time the EOI rolls around, my students should no longer be scared when they see the word inverse!
This foldable was inspired by @druinok's post from February.
Inverse of a Function Foldable - Outside
I want my students to be able to find the inverse if they are given a set of points, a graph, or an equation. Since we have only just started exploring functions in general, the examples we went through were quite simplistic. We will explore much more complicated inverses as the year progresses!
A lot of my students were terrified when I told them that we would be learning about inverses. By the end of the lesson, they were amazed that inverses were actually quite easy.
Inverse of a Function Foldable - Inside
Inverse of a Function - Important Fact!
I still have to figure out how I want to introduce transformations to my Algebra 2 students. That topic should end our first unit. Hmm...
Wow, wow, wow! That's an excellent collection of foldables. I love what you did with the domain and range foldable. I've had my kids use post-its before to mark off the lowest and highest values, but I love how the flaps show +/- infinity as well. My other favorite is the foldable that goes over all the characteristics of a graph. By the way, thanks for the shoutout on the envelope. Glad it could help!
I just have to write to tell you how amazing you are for sharing all this, and to thank you from the bottom of my heart. You are an excellent resource, and I almost feel as though I'm "stealing" these ideas from you, but the fact of the matter is that I'm teaching five preps (Geometry, Algebra II, PreCalc, AP Calc, and Business Math), and I just don't have enough time to devote to planning each lesson that I would like. I've been dying to try interactive notebooks with my Alg II students, and thanks to your insight, I'll be able to do it, at least part-time, this year. Thank you again and again!
To answer your question-- yes, there are math teacher who don't like math. I am one of them, LOL. I am a middle school special education teacher who is only teaching math this year, which is quite possibly my least favorite subject-- though my kids can relate to the fact that I struggled with math during middle school. :P While many of these foldables are beyond what I will cover with my students, you have given me a lot of great ideas. I will definitely be consulting your blog as a resource!. :)
I love, love, love your blog! I am new to Alg 2 and to interactive notebooks and you have such great ideas. I wish I would have remembered to come to your blog last week as it would have saved me some headaches. Out of curiosity, are you teaching from the Common Core State Standards?
Thank you! This year, I am not teaching directly to the Common Core State Standards. We have one year left of testing over our old standards, so I am teaching primarily to them. However, I attended an amazing Common Core training this summer, and I am trying to match my teaching to the CCSS whenever possible.
I absolutely love your blog! I have implemented an Interactive Notebook in all of my math classes this year and I love it so much. Your ideas, as well as many other bloggers, are so fantastic! I figured you might get a chuckle on some of the following student commentary this week:
-Ugh this class is like kindergarten. -But kindergarten was fun! -Stop complaining or Miss Nelson will take away the markers and crayons, guys! -Miss Nelson, I'm glad you don't make math hard. -Do we need markers AGAIN? -I don't want to take a boring math class next trimester. Can you teach {insert other math class here} next time? -I like coloring. -I don't like coloring. -When are we gonna be done with these foldy things? -Why do we have to glue something EVERY day, Miss Nelson? -Someone stole my notebook!!! -Wait... I found it. -This is weird. -Can you cut this for me, Miss Nelson? -Who wrote my name on my notebook?
Oh, the joys of interactive notebooks and math class. Thank you for providing me with an excellent resource for helping remedial students who are preparing to retake their state Algebra ECA in 2 months! Keep up the amazing work!
Thanks! I LOVE your students' comments. Your students sound a lot like mine! This week, I asked my students to draw a table in their interactive notebooks. I was looking for an x/y chart like the one I had already drawn on the Smart Board. As we started filling out the chart, one of my students became really confused. When I had asked my students to draw a table, she had taken this to mean a literal table... There is definitely never a dull moment when working with high school students!
I love your blog! There are so many great ideas. I am curious about how you go about making the foldables and visuals while lecturing. Do you lecture as you make them or make them first and talk about them second...or finally (haha) lecture then make the visuals to reinforce the concepts?
Hi Jill! I know this isn't exactly the answer you are looking for, but I do a combination of all the different ways you mentioned. Usually, I lecture as my students make them, but I've also done them both of the other ways. Sometimes, I even let my students vote on which way they would prefer me to do it.
If it's a topic that I believe my students might already be familiar with, I will probably lecture as we make the foldables. If it is a completely foreign topic to my students, we will usually make the foldable, talk about the material, and then reference the foldable as we work practice problems. This past week, I introduced my students to transformations of functions. We spent an entire class period exploring transformations with as little lecture as possible. Then, on the second day we created a foldable to summarize what we had discovered on the previous day.
I just want to say THANK YOU for your blog. It's been a few years since I took Algebra I & II in high school and now that I'm in college algebra it seems that I've forgotten much of what I learned. Your simple foldables (which I am a huge fan of) have put a lot of the ideas into an easy to understand format that just might help me get through this math class. THANK YOU again! |
Mathematics is a vital life-skill. Not only is it is widely used in science, engineering and technology, in estimating, modelling and forecasting, it is also used in other professions such as computer graphics, finance, economics, insurance and linguistics. In studying mathematics, students gain a firm foundation in the basic principles and techniques of modern mathematics, and understand how mathematics is applied.
Flinders University believes that mathematics is an essential component of a modern economy. It underpins many essential disciplines and is an important life-skill. As such, we believe that it is vital we contribute to mathematics education and research, both through direct mathematics research and teaching and in its many other guises including biostatistics, epidemiology, and econometrics.
Flinders has a proud history of mathematics and statistics teaching and research, counting amongst its alumni, Professor Terence Tao, UCLA Professor and Australia's only Fields Medal winner, and Rodney Brooks, Professor of Robotics at the Massachusetts Institute of Technology (MIT).
Mathematics at Flinders is growing steadily. The School has recently established the Flinders Mathematical Sciences Laboratory focussing on advances in modern applied and pure mathematics and their nexus. Recent senior appointments in this area have strengthen the discipline and further expansion is expected in the future.
Research and Higher Degrees
Further Information
If you would like more information about the courses or research opportunities in mathematics and statistics please contact Professor Jerzy Filar. Current opportunities for higher degree study are outlined on the Research Higher Degrees page.
Courses in Mathematical Sciences
The mathematics awards at Flinders focus on giving a strong mathematics education covering all areas of mathematics but with a particular focus on modern applied mathematics - mathematics that can be used to solve today's problems. Thus the award is strongly oriented to looking at the practical application of mathematics in industrial, environmental, scientific and social contexts. Second, the award allows for topics taught by other disciplines that represent applications of mathematics such as in medicine (in epidemiology), in business (in econometrics), physics (in mathematical physics) and the environment (such as groundwater modeling).
The number of students taking at least one mathematics topic has trebled in the past four years and in 2013, Flinders approved the introduction of two new Bachelor of Mathematical Sciences awards to join the successful major in mathematics available in the Bachelor of Science. (For students without the necessary prerequisites for the BMathSc, there is a pathway through the Bachelor of Science). In 2014 this will be joined by a new combined degree Bachelor of Mathematical Sciences with Bachelor of Computer Science. |
Book DescriptionEditorial Reviews
From the Back Cover
Get a firm grasp of trigonometry with this simple-to-use guide! It can help you pump up your problem-solving skills, ace your exams, and reduce the time you need to spend studying. Students love Schaum's Outlines! Each and every year, students purchase hundreds of thousands of the best study guides available anywhere. Students know that Schaum's delivers the goodsin faster learning curves, better test scores, and higher grades!
If you don't have a lot of time but want to excel in class, this book helps you:
Brush up before tests
Find answers fast
Study quickly and more effectively
Get the big picture without spending hours poring over lengthy texts
Schaum's Outlines give you the information teachers expect you to know in a handy and succinct formatwithout overwhelming you with unnecessary details. You get a complete overview of the subjectand no distracting minutiae. Plus, you get plenty of practice exercises to test your skill. Compatible with any classroom text, Schaum's lets you study at your own pace and reminds you of all the important facts you need to rememberfast! And Schaum's is so complete it's the perfect tool for preparing for graduate or professional exams!
Inside, you will find:
Hundreds of detailed problems, including step-by-step solutions
Hundreds of additional practice problems, with answers supplied
Clear explanations of trigonometry and the underlying algebra
Understandable coverage of all relevant topics
If you want top grades and excellent understanding of trigonometry, this powerful study tool is the best tutor you can have!
About the Author
McGraw-Hill authors represent the leading experts in their fields and are dedicated to improving the lives, careers, and interests of readers worldwide
I found the book very useful for brushing up one's knowledge of trigonometry.It gets down to the point right away without wasting any time on unnecessary theory.After all, its your problem solving ability that counts and not how well you know the theory.:) What I like most about this book is that it treats the subject matter from a student's point of view (like it teaches you how to get the answer using a calculator STEP by STEP.....it actually tells you which buttons to push).Also, throughout the book ,the application of trigonometry in surveying, construction,astronomy and air-navigation is emphasized. I recommend this book for anyone who's taking computer science/engineering/technology courses in college.
Trigonometry was never a good subject for me--I never "got" it. But when I was taking advanced math and science courses, I needed trig. This book helped me to "get" it, finally, and be able to solve trigonometric problems. It's very clear, up-to-date, and well-written.
I am observing that my test scores on tests on tests involving trigonometery are increasing, thanks to this thin aid. It is thin, yet it is good. That is almost impossible. This book is one of the best created. Trigonometry is hard, and is mentioned and applied almost everywhere. This book is comphrehensive and is both easy and advanced. Everyone should have this. You can have knowledge of math, science, and computers with it.
I used Schaum's for a Summer School Trigonometry course. It doesn't replace a textbook but it covered all the necessary topics effectively and provided a good alternative to derivations and problems found in the text. I relied on it as a backup and a good check on the work being covered in class.
I had to have this book for a trigonometry class, and it sucks. There were so many miss prints in the book, that it was actually a disgrace, and I personally think this book is unfit for learning. There are way too many miss prints in this book, and it is piss poor at explaining concepts and how to work problems. When I went to study for a test on graphing sines, cosines, and tangents, I actually had to look the material up online because I could not understand the book, and almost everyone that's in my class says that they cannot understand the book all that great, so I know it's not just me, and my professor even went so far as to say that the book had a lot of miss prints. If you ever have to buy this for a class, BEWARE! My best suggestion would be to buy an alternate text that's a lot better if this is a requirement for a class.
The preface claims that the book can be used by students who are studying trig for the first time. What a load of bunk. As a "first time" trig student, I couldn't get through the first chapter without getting confused. I moved on to the second chapter, and the information was just as muddled. This book may be more useful as a study guide to be used together with a real trig textbook, but I don't believe it is useful for the beginner. Kahn Academy on Youtube gave more clearer explanations and examples on the subject. Beginners should stay away from this book!
I purchased this book and Schaum's Precalculus for a self-study of precal over the summer, and it is NECESSARY to use the PRECALCULUS book even for a regular trig course since it covers vectors, polar and parametric EQs, and complex roots in radians. As for this book, it covers area of triangles, more identities, and more applications than the precal book, but is still not sufficient for a complete self-study. Taking the actual course, whether it be trigonometry or precalculus is necessary to be prepared for the rigor of calculus, yet for a self-study, it would be necessary to DO ALL THE PROBLEMS in this book, and purchase a separate text with more practice problems. In conclusion, this is an excellent book for review or as a supplement to a full course. |
Shows science, math, and engineering students and professionals how to make the most of this top-of-the-line graphing calculator Describes step by step how to harness the calculator' s 3D graphing capabilities, advanced built-in functions, USB connectivity, and 16 preloaded applications, including calculus and electrical engineering tools The book' s accessible, plain-English explanations are a must for users who find TI' s instruction manual difficult to slog through Since students...
DESCRIPTION The TI-83 Plus and TI-84 Plus series is the de facto standard for graphing calculators used by students in grades 6 through college. With so many features and functions, the TI-83 Plus/TI-84 Plus graphing calculator can be a little intimidating. Using the TI-83 Plus/TI-84 Plus is an easy-to-follow book that starts by giving a hands-on orientation to the calculator so readers will be comfortable with its screens, buttons, and the special vocabulary it uses. Then, it explores key features... |
It is not necessary to purchase the text. You should have some
reference book on algebra, but all basic algebra text books contain
more or less the same material. Alternative suggestions include:
Lang, Herstein, Artin. Any of these should get you through the
course. The edition is irrelevant.
Prerequisites:
Math 322 or equivalent. Math 323 would be desirable but not
absolutely necessary. You have to have a good working knowledge of
group theory, especially group actions, as covered in Chapters 1-4 of
Dummit and Foote.
Course Outline:
This is a standard course in Galois Theory.
Not only does Galois theory itself appear in many diverse
areas, but the principles of symmetry we cover appear in different
guises throughout mathematics. This, or a course like it, is a must
for anyone serious about our subject. Besides, Galois theory is
beautiful and fun!
After briefly reviewing the basics of fields and polynomials, we move
on to
Galois theory. This is the study of the symmetries of algebraic equations. Highlights include Ruler and Compass constructions: the impossibility of doubling the cube or trisecting an angle. Impossibility of solving the quintic by radicals. Cyclotomic fields. Kummer theory.
If time permits, we will explain that the Galois group of a
field is really the same thing as the fundamental group of a
topological space. Or we will do a little algebraic number
theory.
The minimum material covered consists of Chapters 13 and 14 of the
text. This course covers the material in the field theory part of the
algebra syllabus for the Qualifying Exam for the PhD programme at UBC. |
Biology and it's principles touch every person every day - that is a fact. Our ecosystem, our food, our beverages; literally everything we do in a day can be tied into biology.
We all remember learning biology in grade school; it involved a lot of reading, and very little "math" per say. However, as a student moves through his or her sholastic career, biology becomes more involved and complex. Equations, and formulae based solely in mathematics soon begain to emerge, and play a heavy role.
This section of StudyUp is filled with articles and discussions on just that - biology and it's relation to mathematics. Math and biology are 2 of the more important and challenging topics a student will face in the teen years of education, so we hope to give you a head start. Read on, and learn how biology can affect you!
At 10th grade, many students are old enough to understand the biological concepts and that is why their biology syllabus includes important topics such as cell biology among other significant topics important in the biology curriculum. The biology exams at this stage are quite challenging and the candidate must prepare adequately to take the exam that will follow as it is likely to have a major impact in your future career decisions as you try to shape your destiny in life.
By passing or failing your biology exam in 10th grade you are likely to shape the path for any future...
Ninth grade is the threshold where students should get sufficient knowledge in subjects to face their OL and AL level with confidence. For this matter, besides their classrooms books, they have to read additional books to acquire enough knowledge on subjects to pass these crucial examinations and enter the next level. The subject of biology is complex. Students may like it, if they are exposed to well-written books on the subject. Though some 9th grade biology books are available, most of them are not up to the mark.
Finding good books for studying a complex subject such as...
Biology is a difficult and detailed study that contains many levels of learning. It is a growing field that continues to attract curious students who aspire to learn more about the organism life on earth. A variety of streams generate from biology, there are many levels contained within the study and many more contained underneath that particular level. The list goes on, there is such a large amount of information, such a large amount of detail that it is important to grasp each concept as it comes. There is a high order of mental organization required in order to fully...
Students are required and expected to be organized for the higher level of education before getting admitted in to the colleges. The Advanced placement courses or the AP Biology courses prepare the students for the advanced studies ahead. These AP biology courses are held in high regard as quite taxing and posing a real challenge to the students. Though these courses are considered quite a hard nut to crack, yet these AP biology courses should be regarded as value for the hard work.
AP biology courses can also be considered as a testing ground for the future aspirations of the students....
The best college biology you choose should suit your budget, lifestyle and needs. In general, famous large universities provide students with the best biology diplomas, degrees and PhDs that include extensive opportunities for autonomous researches, highly specialized course work, and exposure to special disciplines while colleges provide the facilities to interact with professors, personalized instructions, small group classes etc. To find a good college for biology, you have to focus on some key factors and see whether your selected college has these facilities to provide you with best biology programs.
Professors and lectures with PhDs and diverse faculties of a...
The subject of biology is divided in many segments depending on the particular life in question that is being studied. The main aspects in the biology study are plants, animals and marine life. The best way of studying any subject is to do it practically. By taking this advice into consideration, it would therefore be very advisable to enroll for a course in marine biology in an institution that is near the sea or that has branches or departments in places where the students can be exposed to practical experience in the sea.
Most of the highly ranked marine biology colleges...
Biochemistry and molecular biology are some of the easiest branches of biology that you can tackle and if you were thinking of specializing in both, kudos to you. Forget about what your colleague will tell you; what they think does not matter, it is what you are interested in that matters the most. Biochemistry is the study of the chemical components of the life processes that take place in living organisms. The cell is the basic unit of life and in it takes place some chemical reactions that keep the cells alive and functioning.
Emil Fischer is one of the best...
One of the fundamental advantages of the new age is that there is enough knowledge and reference for biological sciences and other related material. The precision of the new scientific advancements have brought a better understanding to the units that make the molecular realm. Physical and chemical development, though a gradual process, has actively simulated the principles that make the biochemistry of life. They have been so successful that it has brought meaning to phenomena that had earlier been impossible to comprehend. Biochemistry is a broad field of science developed from the field of physiology. Biochemistry puts emphasis on the...
Now you have the knowledge that molecules make up any organ. The other fact at hand is the organs are responsible for the being of a species. Now you must have wondered what special features possessed by these molecules that give them their capability are Why this and why that? This is a baffling subject bound to raise lots of questions. Early scientists and philosophers would answer that life is a mysterious divine force. From whatever point of view life has an endowed characteristic difficult to comprehend. At the same time attempts to have a duplicate are not easy...
Biology cell tests can be very good for helping biology students in gauging their degree of understanding. The tests are available from many educational; programs which have programs in biology study. While some of the tests may simply be used to evaluate the candidates before thy take their real final exam in biology cell exam, others are simply meant to jog your min as a method of improving you general knowledge.
Preparation for any test is usually important, the preparation that a candidate needs to make before sitting for a biology cell test is very important in ensuring that you do...
Learning biology is not an easy task. Students need great commitment and concentration if they want to excel in the subject. The divisions of biology include zoology (study of animals), botany (Study of plants) and microbiology (Study of microorganisms). OL and AL students have to follow basics of these three sectors and then pursue in selected field such as genetics, medicines, health care, biochemistry etc., when they enter a higher stage of biology education.
At high school level, students go through further introductory lessons in biology as well as related scientific awareness. They also gain practical information of the subject...
Biology distance education is very popular among students who are unable to attend a university, or a college to pursue higher education due to their other commitments. Like other distance educational programs, biology distance education also offers courses from certificate up to PhD levels. Many adults follow online distance biology courses from these universities and colleges even while attending regular employment.
The advantages of registering with a biology distance education course or a degree program are manifold. Nevertheless, students must have their utmost commitment to complete the selected courses, as there are no overseers to ask them to complete relevant...
Choice of biology graduate programs is determined by many factors. Some programs are internationally renowned while others are hardly known by quarter of the state's population. Many factors contribute to this scenario. In some universities, the culture of interactivity between students and lecturers is so entrenched that it has become the main distinguishing factor. In other universities, students hardly interact with their lecturers.
Armed with all this information, you can make your choice of the right program. However, you have to understand that there are always new things to be learned. You have to understand your own research interests and...
Any biology student study guide worth its name must start by mentioning that the success of a student in any biological endeavor must be motivated by knowledge and not ignorant assumptions. You must reach out to develop yourself, to arm yourself with knowledge and to position yourself in such a place as will help you be somebody in our society today. If that is the objective of your study, that of curving your niche in an already competitive filed, then the challenge of mastering biology will become an honor instead of a bother.
Such organisms as swine flu virus, MRSA, bird...
In a sense, biology has become an informative field for life existence. Most institutions have incorporated biology in their programs. Whether it is just as a lay man or an expert, it has an interesting medley for learning. Even for an inexperienced observer, biology plays a fundamental dependability. Here is such a case of an impassive individual; though not equipped with these skills he or she has knowledge to maintain a good relationship with life nature. Eating habits and other minor practices in the farm; like when culturing plants, is seen as having at least a general comprehension of biological...
Cancer is one of the most deadly diseases affecting the human race. This disease is not only chronic, but also fatal and life threatening. The vast occurrence of the cancer all over the globe has led to threatening proportions. The most pathetic part is that though this is a very common disease and despite the fact that lots of research has been conducted on this disease, yet the research findings are not yet totally enough to explain the mysterious behavior of the cancer cells. There are loads of other aspects to be studied in cancer too, other than the growth...
What entails a comprehensive and well-structured syllabus of any cell biology course? An ideal cell biology course should be designed to help the student gain a thorough understanding of cell biology, by first introducing the core concepts and then developing on them towards the available body of contemporary knowledge. That means that the course material should be presented based on the levels suitable either the undergraduate, advanced undergraduate or graduate whose intent is to major in any of the life science fields.
For instance, an ideal cell biology course should start on prerequisite basics such as principles of genetics and...
Cell biology is a science discipline that has its focus on the simplest living unit- the cell. This study of cell has its origin from a German scientist. The first time the term 'cell' was hinted in the biological world was by the man called Robert Hooke. He had conducted an experiment which he observed small blocks in a plant specimen. He gave it a reference name as the cell. That was in the 18th century and in the year 1825 there came another scientist called Robert brown. He fronted more inquiries into the first discoveries of Robert Hooke 'the...
The cell biology lab is a setting of all the necessary equipments used in the study of cells within a particular building. The building should be equipped with important instruments used n the research or study of cells and other related subjects. The study of the cell is very important in the course of research involving the discovery of new medicines and information regarding new health related facts.
Cell biology labs are always common in colleges or schools that teach the biology subject, in hospitals or research institutions as well as in investigative institutions. In colleges, the labs are used...
Cell biology is the study of cells, their structure, organelles, the life and death of cells and their life cycle. This subject also explains about the physiological structure of the cells. The cell biology is an academic discipline for students who study biology and the advanced subjects in that field.
Notes for subjects can be very handy and helpful for students many times, the notes are in general the contents of a book which is presented in a very simple form. Many students like to study the notes instead of the books, because they find it easier to understand the concepts...
With the current internet technology, most of the information that took a long time to access can be accessed in a matter of seconds since it is just a mouse click away. In a matter of minutes, you will be able to find out what the expert in the field of cell biology are up to and what are the most recent findings of this group of scientists. In fact, what makes cell biology online more attractive than books is that the information is up to date. You need up to date information if you are going to create a...
Most recognized universities and colleges offer Cell Biology PhDs. The field of cell biology is vast and, therefore, students have to follow extensive research, laboratory tests, classroom lectures, etc., on the course to reach a Cell Biology PhD score.
Students who apply for cell biology PhDs are requested to have substantial knowledge in science. For example, Temple University in the USA requires students of Cell Biology PhDs to have at least chemistry (12 Credits), mathematics (6 credits) Physics (9 credits), and biology (16 credits). Although most universities do not require a master degree in cell biology to follow the...
Cell biology is a vast field; it is the study of structure, organizations of cells in a body, it is known that cells are basic building blocks of a body. And hence there are lot of functions involved in a cell. There is lot to learn about cell biology and it is a field where latest researches are made till date.
Since there is a wide range of content and information in the subject of cell biology, to test a person's knowledge in the field of cell biology can be tricky. There are numerous questions that can be asked in...
Cell biology research refers to the study of cells. The areas of study include their structure, their interactions with their surroundings, their properties, what they contain among other characteristics and issues related to cells.
In biological studies, knowing the contents of cells and how they work is a basic fundamental. Consequently, it is possible to understand and recognize the differences and shared similarities between different types of cells. This is a very important aspect in the field of cell biology.
In cell biology research, it is apparent that in the differences between cells, there are several similarities that allow scientists and...
Cell biology study covers extensive subjects. Students of cell biology are taught first the value of scientific method. They are exposed to the scientific methods used by Edward Jenner called spontaneous generation debate, when he discovered the first vaccine. In the foundation course students are taught the related steps in following the scientific method, the gap between law and theory of science, the difference between pseudoscience, science and non-science and also how to implement actual scientific methods to a research.
After acquiring the basic knowledge, cell biology study leads students to inorganic chemistry that teaches them about isotopes, compounds,...
Cell division refers to the process through which two or more cells are formed from the division of a parent cell. Cell division is only of the many segments that make up the cell cycle. The type of cell of division whereby the resulting eukaryote daughter cells have the ability to divide again is known as mitosis.
Binary fission is the corresponding cell division process that involves multiplication of prokaryotes. Meiosis is another type of cell division whereby the daughter cell is completely transformed and made into a gamete that cannot divide again until after it has been fertilized. In...
Cells are an amazing part of all living creatures. Although it may be difficult to get it, we are all "made" out of the same "material"- the cells. If you are looking for the exact definition of the cell structure, you are going to get it right now. The cell is a structural unit of all organisms in existence on our planet. It does not matter whether we talk about viruses or mould, cats or even humans- we are all a pile of cells!
Of course, creatures do not have the same amount of cells. Several types of living organisms...
The most basic and fundamental structure and functional unit of all living creatures may be called a cell. Cells are usually so small that these units are commonly viewed under a microscope. They are very minute, but are very much essential to the sustenance of all living organisms. Cells are often referred to as the building blocks of all living creatures. Cells are very important to all living creatures and are also important in the study of life. Cells play a major role in biology which is the study of all living creatures. There is also a sub-division of biology...
The structures of all living things are made up of cells, which are described by MBV Roberts in Biology a functional approach as "the tiny microscopic units which collectively carry out the processes that make the organism a living entity." Cellular biology was hence coined to refer to the study of the cell and all that encompass it. By observing the cells of plants and animals under the light and the electronic microscopes, many molecules have been discovered in the cells hence the name molecular biology. Cellular and molecular biology cannot be divorced from one another as they are intrinsically...
Cellular molecular biology has been greatly advanced by regular edition and distribution of journals. These journals focus on disseminating basic knowledge and information regarding various areas of cellular molecular biology. In these journals, the main subject areas that are tackled include biophysics, biotechnology and biochemistry. The journal contains short information and regular writing on new information that may be of interest to the general public and cellular molecular biology professionals and students. The journal also welcomes submissions from scholars, students or practitioners in this field. Often, the most sought after topics for the journal include molecular biotechnology, cytoskeleton, cellular membranes...
When you have completed your high school education and you are contemplating joining college, your interests are the main driving force. In case your interests are in biology, you will start to note that you feel the compulsion to learn more and more things about the courses that you will have to take during your education.
College biology courses vary depending on the college that you want to attend. They also vary depending on the nature of the course that you want to pursue. These days, online biology courses are all over the place. You might also want to give...
If you love water and the animals that it contains, you may want to pursue an education in Marine Biology. Marine Biology studies the relationship of ocean life and its relativity to the environment. These studies are done from oceanography centres, aquariums, from boats, and even coastal areas. You must first possess advanced degrees in research and you must be very responsible.
It is very challenging to become a marine biologist and in order to do so you should have taken lots of science classes while in high school. Classes like biology, physics, chemistry, advanced mathematics,...
Marine biology is a field that is exciting for only those who love the study! Marine biology focuses on the study of the marine ecosystem in general. Hence, you would be looking to study areas within the ocean that contain several different animal and plant lives that contribute to the workings of an entire ocean and its immense eating classification and toxicology. Within marine biology there are a whole wide range of smaller fields that people tend to choose and stick with.
Many mixes imagines are formed when it comes to appropriately issuing the difference between marine biologists and zoologists....
How can one choose the best college from where he or she can pursue excellence as a biological scientist? The ideal colleges for biology have distinct traits that should help any student interested in biology to choose which best suites his or her interests. At the very basic, ideal biology programs should give the student a very broad knowledge of all key biological sciences as the foundation. The early sections of most biology programs expose a student to incredibly diverse faculties of biology.
Once this foundation is cemented, the program should then allow the student to pursue a specific area in...
Computational Biology PhD is one great way of advancing knowledge to aid in progress in life sciences. This is because there is increased dependence on technology, and with technological advancement, research techniques are also keeping pace with the trend such that research based studies also become highly advanced. For instance, it is recorded that there was a collaborative effort among three corporate entities in Portugal in the year 2005 in a bid to promote a PhD Program in Computational Biology.
This program is basically aimed at ensuring that the training of PhDs in this particular field is internationally competitive. Thus, there...
Understanding life on the earth is a fantastic and exciting field for many capable students. A variety of departments are offered within all educational institutions that focus on countless and diverse areas of academic or perhaps even practical application fields. Biology is one of the top departments within the academic field. It is an exciting field that leads to countless streams of occupations possible within the study and continuation of biology. The field is popularly known for its recognition of life on earth and its combination of a variety of other related scientific fields. Let us take a close look...
The Department of Microbiology in most scientific institutions offers students with both major and minor options. Further, some institutions provide for Masters Programs, which in most cases is intended for students who have successfully completed an undergraduate course in the same field. Oftentimes, microbiology is ideal for strong students academically, and those with an interest in pursuing related careers.
Microbiology courses usually involve an extensive study and knowledge of sciences such as biology, chemistry, physics and mathematics. In addition, specific requisite subjects and training are also offered. Most microbiology departments offer a certain number of course units to be completed by...
In most university, the department of molecular biology enjoys a very high status. It is often headed by people who have distinguished themselves from the rest of the club of molecular biologists in terms of professional, academic and research credentials. In other words, for this department to function in the right manner, it has to be professionally run. This is what makes the departments at Harvard University, California University and many others excel in research and policy matters.
This is one of the departments that have a very challenging task of coordinating many laboratory works, classes, tutorials and researches. It...
Essential Cell Biology is a book that was authored by Bruce Roberts, Julian Lewis, Alexander Johnson, Peter Walter, Keith Roberts, Karen Hopkin and Denis Bray. It is 896 pages long and contains illustrations. This second edition is very ideal for people who want to grasp the basic molecular biology concepts.
The book is written in an easy-to-follow manner that makes the molecular biology concepts very easy for an introductory student to understand. Each section is packaged logically, in much the same manner that stories are always written. Questions are posed in every section in order to give the reader some...
In order to prepare for a biology exam, you have to be very properly prepared with all the books that you need. It is very expensive to buy biology books. It is even more expensive to buy all the biology books that you need for your coursework revision. The world is full of free biology books for the poor.
You do not have to be poor to pursue tips on how to get free biology books. Some of the books that organizations offer free are very difficult to find in any bookshop. Affordability factor in bookshops could be the main...
The Gcse biology coursework contains various topics in Biology of which some are discussed in this article.
'An Investigation Into The Aspect Of Human Variation' is one important topic which aims at investigating, analyzing and exploring the aspect of variations in human beings. This topic bases on the idea that there is a direct correlation between the length of the foot of an individual and his or her head span. This topic intends to find out the reason behind this supposed correlation.
Moving on, 'An Investigation Into The Affect Of A Temperature Increase On The Breathing Rate Of A Locust' seeks to...
...
For most biological pros, one of the interesting subjects that still attract world wide speculation is evolution. As much as what is to be learnt in grade 12 biology, there have been numerous ideologies on the changes and new developments in the discoveries. In the broad framework of evolution, it is necessary that one digs up facts and historic evidence before making any proposals or hypotheses. It is now fact in the mind of many biologists that the current existence is as a result of the previous species. The constant but gradual change can only be evidenced in time.
In...
Biology deals with living beings and that is why we call it life science. Scientist often prefer using the term life science instead of biology in an effort to emphasize the focus on the study of life processes.
No one can say when the study of biology exactly began. Primitive man collected food from natural vegetation and hunted wild animals. Through this activity he learnt about life and habits of different animals and recognized various plants and their parts. That was the beginning of biology.
With the invention of microscope in the sixteenth century, a new world of minute animals and...
There are lots of subjects which are studied in high schools. Unfortunately, most of them are not learned to the level needed and students are only taught basic knowledge about the subject. However, it can be claimed that biology is one of the few school subjects which are studied thoroughly and deeply in the high. We are going to follow all the material which is learned throughout the education of high school.
Usually, students start studying the biology subject in the very beginning of their education (1st or 2nd grade). It usually has a different name (but not biology) such as...
Human biology courses are available in many colleges across the world. Taking human biology courses from a variety of colleges is very possible. If you are a new candidate who is enrolling in biology courses for the first time either for the undergraduate course or for the main bachelor's degree course, it would be advisable to take an onsite course as this will help you in getting the real concept of then course much faster. As much as there are several online programs on human biology, it would only be sensible if you enroll for the full time in campus...
It is from the close look of the human cell biology that you can be able to identify every individual according to his/her unique composition...
The human molecular biology is a very common subject that a student needs to be familiar with who is in the field of medicine or health. This subject is considered as a basic study material for scientists and students who would like to specialize in health and in the field of biology.
The human molecular biology tells about the basic information regarding health and disease, this is very helpful for the scientists who are new to this field. Technically this subject can be called as a mixture of molecular genetics and biochemistry put together with the help of latest research...
Biology is the study of different forms of life. Biology is very wide field. It covers areas ranging from the minute issues relating to the manner in which chemical machines work inside our body cells to very broad concepts relating to world ecosystems and the causes and effects of climate change.
Biologists are also interested matters of the human brain. They are also interested in how the brain functions and the effects it has on the way we perceive things. They are also very interested in ironing out all the mysteries of reproduction. Reproduction is the process through which new...
Molecular biology, as all of us are fully aware is the science of the molecular basis of life. The realm of molecular biology is so vast that it encompasses numerous branches and numerous career opportunities. The molecular biology being the study of the basics of the human and all living beings in general, has many wide and far reaching aspects.
The molecular biology and its related scientific research have been applied not only to the medical and genetic field but also to the agricultural and the nutritional field too. This only is the tip of the ice berg. The aspects...
What are the current cell biology journals publishing about? What are the hottest issues being covered by most cell biology journals currently, especially in the filed of research?
In any modern science journal, cell biology will be defined as a distinct branch of science. Further than being a science, cell biology is a branch of biology that exclusively studies cells at two levels namely the molecular level and the microscopic level. In studying cells, the component involved include studying the cells' structure and sub-structures, the cell's physiological properties, the cell's organelles, the cell's interactions with other cells and with their...
Cells and other organs in the body are identical in a way. They are similar in both living and non living matter. Studies to explain the structure and characteristic behavior of individual molecules create the study field for molecular biology. In essence, there are two classifications of compounds; these are either organic or inorganic. Organic compounds are those carbon compounds having complex structures and can be found in both living and nonliving things.
Back in the 18th century it had been thought that organic compounds are special only to living organisms until Frederic Wohler, a German synthesized urea in an experiment....
Marine biology deals with the understanding of the various life forms in the oceanic life and other water bodies. In this study, in-depth understanding is brought to the conservation and maintenance of these oceanic and marine treasures. In the past decade, there has been an increasing demand in study of oceanic life forms. This trend has led to the training and availability of marine biological courses in most world marine conserving counties. Introduction in most colleges and other learning institutions have met significant success and with technological advancements, one can even learn online, and have exams rewarded with a certificate...
There are different types of environments that you can expect to work in when you are a marine biologist. Marine biologists can work at sea on a ship, or a marine biologist can work in a laboratory. Regardless of the environment in which you are working you will be performing the same duties, such as: running tests, performing experiments, recording your findings and putting all of the research information together in reports.
Collecting specimens, conducting experiments on the specimens collected, and putting the information gathered into reports will be some of the training that marine biologist must...
Becoming a marine biologist is very easy these days. There is a wealth of information online. You can even do a marine biology course at the university without having to set foot at the brick and mortar structures of the university; in other words, marine biology online courses are a reality in modern times.
Once you are through with your degree course, you are ready to advance your career in this field. Some people do this course out of sheer fascination. Others do it for the love of marine environments. More importantly, others are motivated by the need to conserve...
Biology is among the most marketable fields of study. Marine biology is an important unit of life sciences, and said to be one of the most admired areas of biology. In this field, besides the study of marine life, students get to learn about oceans and seas, with all the life that is supported therein. Pursuing a career in marine biology requires the individual to study all the sea life forms, ranging from corals to large sea animals. With the increased availability of online courses, it is possible to pursue this career path through a Marine biology online degree. ...
Taking Bachelor of Science in Marine Biology is one of the best opportunities that you can have of preparing yourself for graduate work the marine biology field. During the first two years, students get exposure to physics, chemistry and biology. They also do coursework that is lab-based, dealing with marine biology and oceanography.
Many universities offer a BS in marine biology all over the world. The choice of courses depends on the specific guidelines that are offered in different universities. One also takes elective courses with related fields such as physical sciences, environmental science, chemistry and biology.
People who have...
Studying of gene structures, its functions, proteins and sequencing, including DNA and determination of macromolecular structures are covered in masters' molecular biology. Students of Masters molecular biology have to attend numerous workshops, projects, researches, lectures etc., and undergo training in methodology of research as well as conduct their independent researches to enter a master's territory.
There is now a large niche in the job market for persons who have advanced knowledge in molecular biology. Pathology departments, hospital labs, food related industries, pharmaceutical and biotechnology companies in government as well as private sectors seek graduates in molecular biology to advance their researches...
Biology is an important branch of science which deals with the study of living things. Biology is a vast science and has many disciplines like genetics, molecular biology, immunology, biotechnology, cell biology etc. The research and development in the various fields of biology has lead to the welfare of human race.
Cell biology deals with the study of cell which is the basic structural and functional unit of life. There are mainly two types of cells - Prokaryotic cell which lack a cell nucleus and is found in lower life forms and Eukaryotic cells which have nucleus and is found...
Molecular biology plays a particular focus on living cells. All organisms function through complex exchanges in their internal organs. The bimolecular cells use up chemical compounds in either inanimate form or organic. Though present in living matter, they have various complexities. These molecules can not exist on solely, but in a mass grouping they unite to accomplish great tasks. Taking a simple molecule for instance, you will see that it inhibits its own characteristic similar to life of the host-everything has a general code of conduct and responsibilities.
If you view a general sample of large organisms, they contain proteins or...
Molecular biology studies the methods through which the cell, the basic unit of human and plant life provides for itself and how it utilizes its contents to make sure life continues. These methods molecular biology or more conveniently the molecular biology methods of life maintenance can be summed up into two; the inorganic and the organic constituents of the cell, which participate in the chemical reactions, which then produce the energy that is required for the functioning of the cell. We all know by now that the cell is made up of organic and inorganic compounds and that molecular...
Microbiology coloring book is focused on microorganisms such as bacteria, viruses, protozoan, fungi and other parasites. This is a book in microbiology covering the basics of the broad subject of microbiology, industrial microbiology and medical microbiology. This book is unique because it provides an opportunity for the learner to learn by coloring the drawings and illustrations in a specific sequence. This approach opens up the subject of microorganisms to high school students. College and graduate students equally find the content of this book challenging and very useful.
By studying and working on well illustrated drawings, one can gather a great deal...
The study of microscopic organisms is known as microbiology. Examples of such microscopic organisms include bacteria and viruses. The study of these organisms is vital in helping researchers to understand and consequently know how to manage human and other living species' diseases. Considering the importance of such a field of study, online courses are useful in disseminating skills and knowledge to as many people as possible throughout the world. With online courses, it has become possible to reach more students from every part of the world.
Microbiology online classes are mainly focused on students intending to pursue careers in health services...
Microbiology is basically concerned with the study of cells. However, this is not as simplistic as it appears. Rather, there are numerous fields and categories of cells, ranging from single cells to fully-fledged organisms like humans. As a result, biological research and into cells is critical in finding solutions to medical and biological problems afflicting not only humans but also other organisms.
Consequently, the study of microbiology can be divided into various categories, ranging from animal cells, plant cells to any other type of cells that are studied in the broad field of microbiology. Given the wide scope of...
We have or are studying biology as a part of science. Of the three main branches of science, biology mainly deals with the study of living beings, that is, all plants and animals including humans.
Modern biology textbook makes special effort to deal with the subject in a manner that the student develops a spirit pf enquiry, inculcates scientific temper, strengthens power of observation, learns to perform simple experiments and to draw appropriate conclusions from them. Above all, throughout the biology textbooks a proper effort is made to see that the student develops an appreciation for nature.
Though the language...
Given the importance of molecular biology, the importance and necessity of books dedicated and focusing on the subject cannot be overemphasized. For a proper study and understanding of any subject, texts and references are necessary. Likewise, for any student of the subject, a molecular biology book is necessary to supplement the studies, lectures and laboratory experiments he or she engages in. As opposed to other subjects of study, molecular biology is unique and highly specific. This means that there are not as many books as there are in other fields of study like, say, geography, history or languages.
Among the few...
Molecular Biology is a branch of biology that deals with the nature of biological phenomena at the molecular level. It is a study of DNA and RNA proteins and other macromolecules involved in genetic information and cell function. It is also a study of life processes like respiration, reproduction and excretion. With the increase of techniques in the study of Molecular Biology, its demand is increasing. Various scientists discovered the different types of molecules, thus inventing genetic engineering. This enabled higher studies of plants and animals by molecular biologists. This also helped in adjusting animal and...
Molecular biology cell is the study of the constituents of the cells that are the basic unit of living organisms. These contents of cells; organic and inorganic are held together by chemical bonds and that is why biochemistry and molecular biology can be weaved into each other with none being different from the other. This article will take a brief look into molecular biology and give you a re rough ides of what you should expect.
Organic compounds in the cell are fats, protein carbohydrates and nucleic acids, the inorganic contents are water acids and bases. In a cell's molecular biology,...
Molecular business courses are the more detailed course in cell biology. The courses are available in the many colleges across the globe that offers courses in cell biology. The costs of taking the courses are variable depending on several factors. The level of enrolment plays a major role in determining the cost of the course as well as the duration of time that the course will take.
The location of the candidate should also be taken into consideration when thinking of taking molecular biology courses. There are several free courses on molecular biology from some of the molecular biology colleges....
With the technological advancement including the biological fields, the need for laboratory professionals has risen. This is because of the need for personnel to examine forensic evidence and such other scientific work. Therefore, for those with a passion in life sciences, molecular biology employment as laboratory technicians is one of the most sought after jobs.
To qualify as a laboratory technician, one needs to begin with acquiring the appropriate college education, specifically a degree in life sciences, medicine or technology. Due to an increased advancement in technology, one can be assured of expansion in job opportunities and career advancement in...
Molecular biology deals with formation, functions and structure of all macromolecules that are very essential for life to exist. Elements such as proteins and nucleic acids are often studied in molecular biology. Focus is often put on the role that they play when it comes to transmission of genetic information and cell replication.
This branch of natural sciences explores ways in which DNA is manipulated in order to be mutated or sequenced. If DNA is mutated, it has to be inserted into an organism's genome so that the effects of mutation can be properly studied and understood.
The chemical structures...
Are you interested in pursuing a molecular biology graduate program? If so, you need to know what you are to expect in modern graduate programs of molecular biology. Current programs vary from those in the traditional approaches that were mainly offline in their knowledge dissemination and access.
We have finally evolved from the times when molecular biology graduate students had to read numerous volumes of the highly acclaimed scientific journals printed and stocked in school libraries. Today, students can fully utilize online bibliographic databases that are storehouses of important information on molecular biology. Online resources are now the main avenue exploited...
The "chemistry of life" is often how a biochemist is described, and this is a very good description. A biochemist solves problems by working with chemicals like DNA and the function of enzymes that make an organism's life processes. Molecular biology and genetic engineering are central fields of biochemists. In fields such as industry, agriculture, industry, the demand for biochemists is growing since biotechnology is becoming more important in today's environment.
It may prove to be very challenging to become a biochemist; however, here are a few steps that will assist you in the process:
1. While still...
Molecular biology is a very broad discipline with many opportunities in terms of jobs. Molecular biology deals with different ways in which interactions take place between cells. This is a field with very many applications in the job market. Because of this nature of the course, molecular biologists can apply their knowledge in very many areas.
Websites of different companies
The simplest way to get information about job openings is to visit different company websites. Here, we are talking of biotechnology firms, universities and pharmaceutical firms. Many of these companies have no problem with submissions made through websites.
Specialty job websites
These are websites...
Molecular biology is the branch of biology which is the study of the molecular levels. The human body is a complex system with a myriad of different chemical and physiological processes going on inside. Molecular biology follows a line of investigation of the different inter actions between the different systems of a cell. Molecular biology also tries to interpret the relation between the DNA and RNA and also the process by which the synthesis of protein is carried out.
A molecular biology lab is a place where all the different experiments related to the molecular biology field are conducted. The basic...
The most fundamental purpose of a molecular biology laboratory is to disseminate knowledge and understanding of molecular biological processes, with the intention of applying the acquired knowledge to solve specific problems with regard to diseases and human health. This knowledge and understanding is intended to be disseminated to people from different backgrounds, affiliations and beliefs from all over the world, as long as they have the ability to grasp and follow through what is going on in the laboratories.
Indeed, the origin of contemporary molecular biology is a molecular biology laboratory, and several skills were originated in the laboratory, especially the...
Molecular biology of gene is a highly specialized field of biology, with medical professionals and Science scholars carrying out extensive research into genes. In Germany, for example, researchers have delved into the in-depth study of the genes of fruit flies.
The fruit fly is one of the most preferred organisms in biological research, and in the study of molecular biology of gene, it would be the natural choice for many researchers. This is attributed to the genetic comparability of the fly in several ways to that of other mammals, including human beings. In addition to the similarities, fruit flies are...
For students studying cell biology, it is important to know the major segments in the line cell study, by talking about cell biology, the term could cover a wide area of the cell biology study curriculum but when you narrow down the study to the molecular study of the cell, then the subject matter is concentrated on the particular study of the molecular composition of the cell. This area being a very wide and complex one is covered in detail and there are several reading materials on this subject ranging from books, journals and magazines to the internet books among...
Of the most resourceful biology books published in the last decade, the hard-covered fourth and fifth edition of Molecular Biology of the Cell might arguably rank number one. The co-founder of Google, Sergey Brin, recently said that, Molecular Biology of the Cell Alberts book is technical manual for the study of biology life. Every living organism able to read should own a copy of this manual.
We are living in an era of information explosion. We are drowning into the seas of information published online and offline. A good textbook in an area such as biology should be a guide...
Molecular biology is the subdivision of natural science which is the study of the molecular levels. The human body is a compound structure with a numerous of diverse chemical and physiological course of action going on within. Molecular biology pursues a procession of examination of the diverse inter actions connecting the diverse systems of a cell. Molecular biology also attempts to deduce the relation connecting the DNA and RNA and also the course of action by which the production of protein is carried out.
Molecular biology of the gene is the study of the gene and the research associated with this...
Molecular biology is a very interesting course for those who pursue it at different brick and mortar colleges and universities. However, some people say that molecular biology online courses are rather different, in that they are more challenging and require higher investment in time and resources.
The main reason why they are perceived to be challenging is that people who pursue them are already engaged on other fulltime or part-time activities that take a sizeable proportion of their time. Generally, doing the course online is a very good idea since accommodation and travel costs are eliminated. You only need to...
Postgraduate studies in molecular biology can be in the form of a PhD. In case you want to understand more about this discipline, studying at this level will open new windows of opportunities. Many universities all over the world are offering molecular biology PhD courses. Some courses come in the form of scholarship while others are not.
In some cases, governments fund researches whose findings will be very significant in changing a particular field such as medicine, microbiology or cell biology. At this level, the research topic that one covers is so specialized that it identifies only one problem and...
Regardless of your level of higher learning; undergraduate postgraduate masters or PhD, a molecular biology program calls for specialization. If you thought you were going to study molecular biology as a whole, think again. Molecular biology is the study of the structure and component of the molecules that make up the cells which are the basic units of life. It had established itself as an area of biology and is interrelated to biochemistry.
Molecular biology programs are offered all over...
The current molecular research has gained greatly from the internet. For a long time, research on molecular biology trailed behind due to lack of resources whereby each researcher held his or her findings without being able to publish the same as widely as would benefit the global body of knowledge. Researchers used to spend demanding sessions in their laboratories yet what they found out was limited in its distribution.
This scenario led to several negative impacts to the field such as:
* Re-invention of the wheel where most researchers were spending time to study the same area of molecular biology that...
The 20th century seemed to be a new era as far as biology is concerned. The reason is that scientists started working on the development of an extremely new subject- the molecular biology! However, don't let the subject's "youth" make you think that it has not been developed enough! Just on the contrary- doubtfully, there is a subject which has been developed more for the last 50 years.
But the truth is that most of the people do not care about the discovery and development of the molecular biology but how all these achievements make their life easier and what...
Biology is complex. There are thousands upon thousands theses, methods, techniques, agents, organelles etc., involved with the subject - biology. Students and people work in related fields find it difficult to grasp all these factors unless they have some help from the latest technologies. Molecular biology software is very important in this respect for students and others who are working in the related sectors.
Molecular biology software is mostly written by chemists, biologists, and software experts who are well versed in the field of molecular biology. They receive funds from universities, organizations, individuals for their efforts, or they...
Information on molecular biology techniques is always a very detailed matter. Professionals in this field are the best-equipped people to discuss it exhaustively. Perhaps the most commonly used technique is one that involves use of botanical protocols.
Another commonly used technique is protoplast isolation protocol. In this technique, metabolically competent protoplasts are isolated very easily. Flow cytometry is another commonly used technique. It is used in almost every DNA study. Many people know about DNA cloning but they know not that this is one of the most important techniques used in molecular biology.
Agarose DNA Gel Electrophoresis is another commonly...
A growing academic field, that students seem to being pursuing more and more, is the competitive and fast growing field of molecular biology. The department of biology is an exciting field for many aspiring students that look forward to pursuing academics within the fields of medicine, biotechnology, cell biology, environmental science, and its next major stream; molecular biology. The field is extremely popular and known for its level of difficulty and hard work required. Students must work hard within any streams that are related to the department of biology. Let us take a look at what molecular biology focuses on...
Bruce Alberts is one of the six authors of a book entitled "Molecular Biology of the Cell". The most current edition of this book is the fourth edition. The other authors include Alexander Johnson, Martin Raff, Peter Walter, Keith Roberts and Julian Lewis. Previous editions of this book have received the same acclaim as the latest edition.
The book has 1616 pages and is therefore very voluminous. Currently, this hardback book is not available in the North American market. It contains all the fundamental concepts that one would want to find in a molecular cell biology book. This is a field...
Biology is such a vast subject that no one book can cover all its concepts. Even if we consider a particular discipline of biology like molecular cell biology, no one book can have all its principles and concepts described. Different books will have different matter on the same subject. Since biology is the study of life and cell is the basic unit of life, so the study of biology revolves around the study of cells. Molecular cell biology is the studying of biology at molecular level and it also involves the concepts of genetics, chemistry and biochemistry.
A book on...
Essentially no two individual are alike or rather all organisms are unique in their own way. These are the physical variations brought by the characteristic molecules that make each species. For example, take a look at the surrounding. All animals, even humans, inhibit their own distinctive characteristics: hair color, eyes skin pigmentation among many others. Even in similar species there are various differences in their structure brought about either by genetics or their background. It is amazing and quite interesting that a man without a finger does not necessarily bore a child without one. The molecular cell biology is a...
Plant biology is at the core of human existence in terms of its relevance. There cannot be life without plants. Plants are interesting to learn about, attend to and use. Plants are as fascinating to children as they are to a post-doctoral molecular plant biologist.
In order to understand what molecular plant biology is all about, you should have a basic understanding of plant biology. You should understand everything about introductory parts. You should understand things like different types of plant, scientific classifications of plant, as well as different uses of plants that are found in your locality. In other words,...
Since the arrival of the internet, there have emerged positive developments in virtually all aspects of life, including the area of textbooks. Today, it is possible to buy and read books online, including online biology books. One advantage of online books is that they are relatively cheaper, in addition to the convenience they offer.
Buying biology books online saves a lot of time that would have otherwise been used to search for the book in stores, considering how some biology books can be rare and difficult to find. To be able to get the best online biology books, below are some...
Online biology courses are offered in several universities around the world. These courses are specifically tailored for students who are not able to attend courses physically. In one University in the United States for example, the online biology course offers lessons on basic biology, particularly focusing on issues of reproduction, growth, theories among other biological fundamentals. At this level of study, there is no need for practical or laboratory work. Therefore, it is possible and common for students to enroll and take the entire course online from the beginning to completion.
Other biology courses available online include human anatomy and physiology....
With the rapid increase in technological innovations, many educational institutions and colleges have decided to use technology to improve their student enrolment by being able to reach as many students as possible regardless of their residence in relation to the location of the colleges. Biology education has not been left behind in any way in this move and several biology education programs are being offered online.
Even the regular in campus colleges in biology have come up with long distance learning programs to cater for the students who would want o enroll in their programs but are hampered by the...
These days internet is not just a source of entertainment but also is an unlimited source of education. The concept of online education is already very popular. Be it any field like information technology, management, science, arts etc, and internet provides opportunities for independent exploration and enhance their knowledge.
Although internet provides vast theoretical information about a subject but it also does not lag behind in offering practical knowledge. Many biology subjects like zoology, botany, life science, genetics, molecular biology, cell biology requires practical approach to understand the theory properly. There are many biology labs available online which are web based...
Taking a test or a quiz online is very common these days; there are lots of online activities that are going around in the internet world. There are websites that offer fun quizzes, polls, exams in the internet. And hence a biology quiz is not an exception. There are websites that offer biology quizzes for self-check or even as a test.
Most of the online quizzes are of the choose the correct answer type questions, a question is presented and at the bottom three or four choices is given, the person who takes the quiz should choose the correct answer...
Now that everything is moving to the computer, including textbooks, you can now study entire books online. Because of the high cost of publishing textbooks for schools combined with the urgent need for books, some students are obtaining their reading materials from the internet. The online books will offer cheaper pricing since they do not have to be printed. Something to remember when obtaining textbooks online are: if the required textbook normally comes with a workbook you may not be permitted to print out the pages to highlight some of the important information. When you...
The fact that most students are always short of money is almost universal. Yet, to master biology and master it well, a student needs access to biology textbooks more than is availed by the school library. Once given a reading list, a biology student has one prominent question in his or her mind; how am I going to buy all these textbooks?
If that is your dilemma, or the dilemma of your child, then this article is just for you. The following are tips on how you can access all your quality biology textbooks, cheap, easy and fast without worrying about...
Some of the greatest schools in America today have embraced the nontraditional approach of teaching biology. Some years ago, potential biologists had to pursue degree programs in their area of interest for that was the only avenue available. Today however, prominent schools have started offering college biology courses that last for shorter periods and are research oriented.
Even more appealing is that these biology courses are available online, such that students of biology can pursue their passions while in the comfort of their homes, wherever that be in the country or even abroad, and during their most convenient study time. Today,...
During high school days, one needs to have all the motivation that can be gained in order for success to come about. Biology is a very challenging subject especially when it comes to revising for the examination. For this reason, you need to have all the necessary books in order to revise using different approaches. It is all about versatility. Online high school biology books and tips are becoming more and more readily available.
As time goes, high school education is going to hit the online platform. Today, degree programs are being offered online. The same thing can happen with...
The technological advancement has seen a sharp rise in the number of the elite in the world. Ever since the first electronic mail was sent, it created a series of reactions that led to the current internet. The internet has been very resourceful in bringing a school close to home. This is speaking in terms of online schoolwork. In the same reverence, biology and its co-related sup-fields have not been left behind. The establishment of online microbiological courses has taken stage in most academic institutions. You can imagine the operations of a fully fledged university and its undertakings; in a...
Biology is divided into several branches. This division can be made in at least three different ways. The principal sub-divisions of PhD biological sciences in terms of major kinds of organisms are:
Botany - the study of plants
Zoology - the study of animals
Human Biology - it is the study of man as a living organism and his relationship with other living organisms.
Principal sub-division of PhD biological sciences in terms of special groups of organisms are:
Bacteriology - the study of bacteria
Virology - the study of viruses
Mycology - the study of fungi
Entomology - the study of insects
Ichthyology - the study...
Advancement in a filed of study is the only way that you will understand it thoroughly and biology is one key area that has many fields of specialization. By the time you are progressing into your PHD, you must have gone through some various levels of specialization already; you are not new to specialization. One thing about PHD biology programs is that they are so specialized that you may be spoilt for choice. Molecular biology and biochemistry are split into many other fields of specialization; choosing becomes a tiring task. Despite the many areas that you may be interested in...
Graduate microbiology programs in America have become very popular of recent. For those interested in microbiology, the field remains the most appealing branch of biology. Various colleges and universities are now offering extensive study programs for those who have their career goals pegged on specific interests of microbiology. These range in the mode of study adopted such that we have full-time, part-time and vocational graduate programs for both MSc and PhD.
The PhD in Microbiology programs in particular are notably very flexible and research oriented. This is because they are tailored to cater for those who are already engaged elsewhere...
Some people may decide to take a certain area and then explore the particular area in totality, as much as many people simply go to college and then stop once they secure themselves a job and can earn themselves an in come. Some people on the other hand are driven by a passion for a certain subject which usually encourages them to continue scaling grater heights in a particular area even in the face of all odds.
Many people are contented with getting the degree in molecular biology or any other disciplines for that matter. The ambitions of any person in...
Biology simply means learning about life. Plant cell biology is the study of the life cycle of plant cells. This is very important since plants are an integral part of all lives on earth. They, unlike most other lives, produce their food for themselves as well as give others sustenance to hold onto their dear lives.
Plant cells are of many types. The typical or simple plant cells are called in plant cell biology as parenchyma cells. It is a cell type composed of walls that are very thin and have less or more diameter wise identical planes. When matured...
This book is based on strong passion for research skills and professionalism of research. It believed without skepticism that life in science is united; this can be brought about by the close resemblance of botany and zoology or other particular sciences of life.
This is a principle advance to the basic biology in explaining already established phenomena and clearing the confusion or misrepresentation of facts among them. It contains the basic introduction in a summary form for make easy understanding and therefore a worthy source for first hand information. Elaborate with pictures, it is aimed at giving supporting facts...
One of the amazing facts about organisms is that they are not spontaneous. They pride themselves in possessing a multitude of behaviors. The lack of simplicity in their life makes a good source of awe through their sensitive nature. The way of life of molecular association to foster a functioning organ, is just one but the least of the spectacular admiration for the body of any species. The book Protocol molecular biology gives a responsive approach to molecular biology
Intricate structures formed are unlike any other that we can compare to. By this same comparison, some similarities in living and non-living...
School biology books are foundation stones for learning the subject reaching to a PhD level. Online bookstores are great avenues to find good books written on biology. They have description of authors, summary of contents etc., to get considerable knowledge of biology book before buying.
The Art of Teaching Science: Inquiry and Innovation in Middle School & High School is an excellent book by Jack Hassard for biology teachers. They can gain extensive knowledge in how to teach and learn biology using a wide range of pedagogical teaching methods. These methods can be used in classroom level maximizing students' attentions, as...
Stem cell biology is a very interesting discipline. Many medical discoveries have been made as a result of stem cell biology. One important study relates to the identification of ZEBI, a special tumor invasion promoter. This element negatively regulates miRNA clusters that always target important factors of stem cells.
In a recent research study, ZEB1 was identified. This is now a most talked about tumor invasion promoter. It is also a regulator in the negative of miRNA clusters targeting factors of the stem cell. These findings provide new insight into the transcription factors network and miRNAs regulating cells of the...
Studying any subject requires lot of assistance and instruction. The assistance can be in the form of instructions or the classes conducted by the teachers and the instructors or may be in the form of some aids or study tools. The advent of online education has opened up new vistas. In addition to adding to the facility and convenience of home based education and training, it has also revolutionized the way the subjects are being given instruction and training in.
The exams are also taken online and the study material is also provided online. The matter of exams brings up...
Before you can start to study marine biology, you have to understand the meaning of marine biology. Well, marine biology has a very long history. By definition, it is the study of different forms of life in oceans and all other marine environments, including wetlands and estuaries.
The study entails a very critical understanding of all animal and plant forms that are dependent on marine environment for their survival. The disciplines that make up marine biology include biological oceanography, chemistry, astronomy, geology, ecology, molecular biology, meteorology, zoology and physical oceanography.
Marine science is considered by many people to be a...
Over the ages, science studies that seek to understand the nature and the living thing that surround us have been on the rampant. The vast and dynamic field of life is by far the only known subject matter that is varied. In the current world, there are several subdivisions which attempt to put an understanding to this. The purpose is to get a more detailed field of research and develop it by expanding it much further into smaller sub-topics.
Biology in a similar sense is broad in its nature. It focuses on the matters of living organisms. And just as... |
More About
This Textbook
Editorial Reviews
Booknews
A textbook for teachers as well as education students, with problem solving as its focus. Problems are presented in a variety of contexts and for a variety of purposes. Many features of the text are the result of recommendations contained in The Curriculum and Evaluation Standards for School Mathematics (1989), issued by the National Council of Teachers of Mathematics |
ELCT 57
Tech Math Elec I
Course info & reviews
This course is designed to provide a basis for a clear mathematical understanding of the principles of DC electricity and electronics, and their analysis. Covered are algebra, equations, power of 10, units and dimensions, special products and factoring, algebraic fractions, fractional equations, graphs, simultaneous equations, determin... |
books.google.co.jp - This is an undergraduate textbook suitable for linear algebra courses. This is the only textbook that develops the linear algebra hand-in-hand with the geometry of linear (or affine) spaces in such a way that the understanding of each reinforces the other. The text is divided into two parts: Part I is... Algebra
Linear Algebra: A Geometric Approach
This is an undergraduate textbook suitable for linear algebra courses. This is the only textbook that develops the linear algebra hand-in-hand with the geometry of linear (or affine) spaces in such a way that the understanding of each reinforces the other.
The text is divided into two parts: Part I is on linear algebra and affine geometry, finishing with a chapter on transformation groups; Part II is on quadratic forms and their geometry (Euclidean geometry), including a chapter on finite subgroups of 0 (2).
Each of the 23 chapters concludes with a generous helping of exercises, and a selection of these have solutions at the end of the book. The chapters also contain many examples, both numerical worked examples (mostly in 2 and 3 dimensions), as well as examples which take some of the ideas further. Many of the chapters contain "complements" which develop more special topics, and which can be omitted on a first reading. The structure of the book is designed to allow as much flexibility as possible in designing a course, either by omitting whole chapters or by omitting the "complements" or specific examples. |
This course is designed to strengthen basic math skills. Topics include properties, rounding, estimating, comparing, converting, and computing whole numbers, fractions, and decimals. Upon completion, students should be able to perform basic computations and solve relevant mathematical problems. There is a $7.50 lab fee for this course.
MAT 060 ESSENTIAL MATHEMATICS 3 lecture 2 lab 4 credit
Prerequisite: MAT 050 or appropriate placement test score
Corequisite: None
This course is a comprehensive study of mathematical skills which should provide a strong mathematical foundation to pursue further study. Topics include principles and applications of decimals, fractions, percents, ratio and proportion, order of operations, geometry, measurement, and elements of algebra and statistics. Upon completion, students should be able to perform basic computations and solve relevant, multi-step mathematical problems using technology where appropriate. There is a $7.50 lab fee for this course.
MAT 070 INTRODUCTORY ALGEBRA 3 lecture 2 lab 4 credit
Prerequisite: MAT 060 or appropriate placement test score
Corequisite: ENG 085 or RED 080
This course establishes a foundation in algebraic concepts and problem solving. Topics include signed numbers, exponents, order of operations, simplifying expressions, solving linear equations and inequalities, graphing, formulas, polynomials, factoring, and elements of geometry. Upon completion, students should be able to apply the above concepts in problem solving using appropriate technology. This course is also available through the Virtual Learning Community (VLC). There is a $7.50 lab fee for this course.
MAT 080 INTERMEDIATE ALGEBRA 3 lecture 2 lab 4 credit
Prerequisite: MAT 070 or appropriate placement test score
Corequisite: ENG 085 or RED 080
This course continues the study of algebraic concepts with emphasis on applications. Topics include factoring; rational expressions; rational exponents; rational, radical, and quadratic equations; systems of equations; inequalities; graphing; functions; variations; complex numbers; and elements of geometry. Upon completion, students should be able to apply the above concepts in problem solving using appropriate technology. There is a $7.50 lab fee for this course.
MAT 090 ACCELERATED ALGEBRA 3 lecture 2 lab 4 credit
Prerequisite: MAT 060 or appropriate placement test score
Corequisite: ENG 085 or RED 080
This course covers algebraic concepts with emphasis on applications. Topics include those covered in MAT 070 and MAT 080. Upon completion, students should be able to apply algebraic concepts in problem solving using appropriate technology. There is a $7.50 lab fee for this course.
These courses are offered through the Math and Physics Department.
MAT 095 ALGEBRAIC CONCEPTS 3 lecture 3 credit
Prerequisites: None
Corequisites: None
This course covers algebraic concepts with an emphasis on applications. Topics include linear, quadratic, absolute value, rational and radical equations, sets, real and complex numbers, exponents, graphing, formulas, polynomials, systems of equations, inequalities, and functions. Upon completion, students should be able to apply the above topics in problem solving using appropriate technology. This course is designed for students attending East Carolina University and is only offered on the ECU campus.
This course is a comprehensive review of arithmetic with basic algebra designed to meet the needs of certificate and diploma programs. Topics include arithmetic and geometric skills used in measurement, ratio and proportion, exponents and roots, applications of percent, linear equations, formulas, and statistics. Upon completion, students should be able to solve practical problems in their specific areas of study. This course is only offered for diploma-level students. There is a $7.50 lab fee for this course.
This course provides an activity-based approach to utilizing, interpreting, and communicating data in a variety of measurement systems. Topics include accuracy, precision, conversion, and estimation within metric, apothecary, and avoirdupois systems; ratio and proportion; measures of central tendency and dispersion; and charting of data. Upon completion, students should be able to apply proper techniques to gathering, recording, manipulating, analyzing, and communicating data. There is a $7.50 lab fee for this course.
This course develops the ability to utilize mathematical skills and technology to solve problems at a level found in non-mathematics-intensive programs. Topics include applications to percent, ratio and proportion, formulas, statistics, function notation, linear functions, probability, sampling techniques, scatter plots, and modeling. Upon completion, students should be able to solve practical problems, reason and communicate with mathematics, and work confidently, collaboratively, and independently. This course is also available through the Virtual Learning Community (VLC). There is a $7.50 lab fee for this course.
This course introduces the concepts of plane trigonometry and geometry with emphasis on applications to problem solving. Topics include the basic definitions and properties of plane and solid geometry, area and volume, right triangle trigonometry, and oblique triangles. Upon completion, students should be able to solve applied problems both independently and collaboratively using technology. There is a $7.50 lab fee for this course.
This course provides an integrated approach to technology and the skills required to manipulate, display, and interpret mathematical functions and formulas used in problem solving. Topics include simplification, evaluation, and solving of algebraic and radical functions; complex numbers; right triangle trigonometry; systems of equations; and the use of technology. Upon completion, students should be able to demonstrate an understanding of the use of mathematics and technology to solve problems and analyze and communicate results. There is a $7.50 lab fee for this course.
MAT 122 ALGEBRA/TRIGONOMETRY II 2 lecture 2 lab 3 credit
Prerequisite: MAT 121, MAT 161, MAT 171, or MAT 175
Corequisite: None
This course extends the concepts covered in MAT 121 to include additional topics in algebra, function analysis, and trigonometry. Topics include exponential and logarithmic functions, translation and scaling of functions, Sine Law, Cosine Law, vectors, and statistics. Upon completion, students should be able to demonstrate an understanding of the use of technology to solve problems and to analyze and communicate results. There is a $7.50 lab fee for this course.
This course provides an introduction in a non-technical setting to selected topics in mathematics. Topics may include, but are not limited to, sets, logic, probability, statistics, matrices, mathematical systems, geometry, topology, mathematics of finance, and modeling. Upon completion, students should be able to understand a variety of mathematical applications, think logically, and be able to work collaboratively and independently. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
This course is an introduction to descriptive and inferential statistics. Topics include sampling, distributions, plotting data, central tendency, dispersion, Central Limits Theorem, confidence intervals, hypothesis testing, correlations, regressions, and multinomial experiments. Upon completion, students should be able to describe data and test inferences about populations using sample data. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
Enrollment in MAT 161 more than three times requires the written permission of the Math & Physics Department chair.
This course provides an integrated technological approach to algebraic topics used in problem solving. Emphasis is placed on equations and inequalities; polynomial, rational, exponential and logarithmic functions; and graphing and data analysis/modeling. Upon completion, students should be able to choose an appropriate model to fit a data set and use the model for analysis and prediction. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural science/mathematics.
MAT 162 COLLEGE TRIGONOMETRY 3 lecture 3 credit
Prerequisite: MAT 161
Corequisite: None
This course provides an integrated technological approach to trigonometry and its applications. Topics include trigonometric ratios, right triangles, oblique triangles, trigonometric functions, graphing, vectors, and complex numbers. Upon completion, students should be able to apply the above principles of trigonometry to problem solving and communication. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural science/mathematics.
MAT 175 PRECALCULUS 4 lecture 4 credit
Prereq: MAT 161
Corequisite: MAT 175A
This course provides an intense study of the topics which are fundamental to the study of calculus. Emphasis is placed on functions and their graphs with special attention to polynomial, rational, exponential, and logarithmic and trigonometric functions, and analytic trigonometry. Upon completion, students should be able to solve practical problems and use appropriate models for analysis and prediction. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
MAT 175A PRECALCULUS LAB 2 Lab 1 credit
Prerequisites: MAT 161
Corequisites: MAT 175
This course is a laboratory for MAT 175. Emphasis is placed on experiences that enhance the materials presented in the class. Upon completion, students should be able to solve problems, apply critical thinking, work in teams, and communicate effectively. This course has been approved to satisfy the Comprehensive Articulation Agreement pre-major and/or elective course requirement. There is a $7.50 lab fee for this course.
MAT 223 APPLIED CALCULUS 2 lecture 2 lab 3 credit
Prerequisite: MAT 122
Corequisite: None
This course provides an introduction to the calculus concepts of differentiation and integration by way of application and is designed for engineering technology students. Topics include limits, slope, derivatives, related rates, areas, integrals, and applications. Upon completion, students should be able to demonstrate an understanding of the use of calculus and technology to solve problems and to analyze and communicate results. There is a $7.50 lab fee for this course.
MAT 263 BRIEF CALCULUS 3 lecture 3 credit
Prerequisite: MAT 161, MAT 171, or MAT 175
Corequisite: None
This course introduces concepts of differentiation and integration and their applications to solving problems; the course is designed for students needing one semester of calculus. Topics include functions, graphing, differentiation, and integration with emphasis on applications drawn from business, economics, and biological and behavioral sciences. Upon completion, students should be able to demonstrate an understanding of the use of basic calculus and technology to solve problems and to analyze and communicate results. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
MAT 271 CALCULUS I 3 lecture 2 lab 4 credit
Prerequisite MAT 172 or MAT 175 with a grade of C or better
Corequisites: None
This course covers in depth the differential calculus portion of a three-course calculus sequence. Topics include limits, continuity, derivatives, and integrals of algebraic and transcendental functions of one variable, with applications. Upon completion, students should be able to apply differentiation and integration techniques to algebraic and transcendental functions. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics. There is a $7.50 lab fee for this course.
MAT 272 CALCULUS II 3 lecture 2 lab 4 credit
Prerequisites: MAT 271
Corequisites: None
This course provides a rigorous treatment of integration and is the second calculus course in a three-course sequence. Topics include applications of definite integrals, techniques of integration, indeterminate forms, improper integrals, infinite series, conic sections, parametric equations, polar coordinates, and differential equations. Upon completion, students should be able to use integration and approximation techniques to solve application problems. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics. There is a $7.50 lab fee for this course.
MAT 273 CALCULUS III 3 lecture 2 lab 4 credit
Prerequisites: MAT 272
Corequisites: None
This course covers the calculus of several variables and is third calculus course in a three-course sequence. Topics include functions of several variables, partial derivatives, multiple integrals, solid analytical geometry, vector-valued functions, and line and surface integrals. Upon completion, students should be able to solve problems involving vectors and functions of several variables. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics. There is a $7.50 lab fee for this course.
MAT 280 LINEAR ALGEBRA 3 lecture 3 credit
Prerequisites: MAT 271
Corequisites: None
This course provides a study of linear algebra topics with emphasis on the development of both abstract concepts and applications. Topics include vectors, systems of equations, matrices, determinants, vector spaces, linear transformations in two or three dimensions, eigenvectors, eigenvalues, diagonalization and orthogonality. Upon completion, students should be able to demonstrate both an understanding of the theoretical concepts and appropriate use of linear algebra models to solve application problems. This course has been approved to satisfy the Comprehensive Articulation Agreement pre-major and/or elective course requirement.
MAT 285 DIFFERENTIAL EQUATIONS 3 lecture 3 credit
Prerequisites: MAT 272
Corequisites: None
This course provides an introduction to ordinary differential equations with an emphasis on applications. Topics include first-order, linear higher-order, and systems of differential equations; numerical methods; series solutions; eigen values and eigen vectors; Laplace transforms; and Fourier series. Upon completion, students should be able to use differential equations to model physical phenomena, solve the equations, and use the solutions to analyze the phenomena. This course has been approved to satisfy the Comprehensive Articulation Agreement pre-major and/or elective course requirement. |
RICHARD J. FATEMAN fateman@cs.Berkeley.EDU Computer Science Division, University of California, Berkeley, CA 94720, USA Received: 16 November 1990 Revised: 16 September 1991
A Review of Mathematica
The Mathematica computer system is reviewed from the perspective of its contributions to symbolic and algebraic computation, as well as its stated goals. Design and implementation issues are discussed.
1 Introduction
The Mathematica1 computer program is a general system for doing mathematical computation described in Wolfram 1988, 1991. It includes a command language, a programming language, and a calculation environment that is oriented toward symbolic as well as numeric mathematics. The back cover of the manual Wolfram 1991 provides excerpts from rave notices like The importance of Mathematica cannot be overlooked ... it so fundamentally alters the mechanics of mathematics." |The New York Times. Fortune says ... it will do, instantaneously, virtually all of applied mathematics ... " Taubes 1988. Hype aside, the program is without question interesting to mathematicians, computer scientists, and engineers because of its combination of a number of technologies that have arisen in initially separate contexts|numerical and symbolic mathematics, graphics, and modern user interfaces. The exploitation of the PostScript language for plotting contributed to its natural t into the package of programs initially released for the NeXT workstation. Not all commentary on Mathematica has been uncritical. For example, reviews in Science Foster & Bau 1989 and Notices of the AMS Herman 1988, Simon 1990 compare Mathematica's features, reliability and e ciency to similar programs for algebraic and or numerical interactive manipulation. Additional commentary on the program includes Hoenig 1990, Vogel 1989. A perceptive book review of the reference manual rst edition has also appeared McCurley 1990. Electronic-mail messages on various semi-public bulletin boards in particular netnews: sci.math.symbolic have discussed features and bugs of Mathematica as well as similar programs. There is also an active mailing list speci cally for Mathematica users mathgroup@yoda.ncsa.uiuc.edu. Such forums provide opportunities for valuable exchanges but, especially as subscribers" become more numerous, the continuing unedited message streams overwhelm the large picture. Therefore there appears to be value in a more widely available and more detailed commentary on Mathematica, speci cally from the perspective of its context and contribution to technology.
This work has been supported in part by the following: the National Science Foundation under grant numbers CCR-8812843 and CDS-8922788, through the Center for Pure and Applied Mathematics and the Electronics Research Laboratory ERL at the University of California at Berkeley; the Defense Advanced Research Projects Agency DoD ARPA order 4871, monitored by Space & Naval Warfare Systems Command under contract N00039-84-C-0089, through ERL; and grants from the IBM Corporation, the State of California MICRO program, and Sun Microsystems. 1 Mathematica is a trademark of Wolfram Research Inc. WRI.
1
In order not to keep the reader in suspense, my major conclusion is that Mathematica has many aws. Some of them are substantial and are unlikely to be repaired because they re ect decisions rather than oversights. The gaps between claims and actuality are substantial. These gaps are not all inherent in the nature of mathematical algorithms or representations since competing commercial programs often provide correct answers where Mathematica fails. One way of improving the state of the art in automating mathematics is to examine current programs critically; our purpose in this review is, in part, to direct attention to shortcomings and suggest improvements.
2 Preliminaries
This review occasionally assumes the reader has a more-than-casual familiarity with Mathematica, and is certainly no substitute for a primer on the subject. Careful study of the major reference Wolfram 1988, 1991 may be an adequate substitute for experience in using the program. Four themes permeate this review corresponding to areas of technology to which Mathematica potentially could make a contribution: the eld of symbolic and algebraic computation, the eld of numerical computing, programming language and human-computer interface design, and the organization of mathematical information. These should be kept in mind as the discussion proceeds, but rst a brief historical perspective seems in order.
3 Prior Art in Mathematica
The idea, of using computers for symbolic rather than numerical arithmetical computation, actually predates the electro-mechanical computer. Ada, Countess of Lovelace and patron of Charles Babbage, inventor of the Analytical Engine", suggested such usage in 1844 see Knuth 1969. It took over a century for the rst actual symbolic computation to be cited in the literature. see van Hulzen & Calmet 1983 or Barton and Fitch 1972 for a survey. The more recent tradition of the eld of Symbolic and Algebraic Manipulation SAM by computer has had a small but loyal following at least since the early 1960's. Members of the Association for Computing Machinery ACM joined together to form a Special Interest Group SIGSAM in 1965. Chronologically, Mathematica probably should be considered as about third-generation" among algebra systems, placing it among the more-or-less contemporary general-purpose systems such as Derive The Soft Warehouse 1991, Maple V Symbolic Computation Group 1990, and AXIOM, previously referred to as Scratchpad II Computer Algebra Group 1988. These are not necessarily better than the second-generation hold-overs | in particular, Macsyma Moses 1979, Mathlab Group 1983, Pavelle 1985, Fateman 1989, provides answers when Mathematica sometimes does not; another secondgeneration system with a wide following is Reduce 3 Hearn 1984. These recent systems were built upon research results plus practical experience in using second-generation systems of the late 1960's, including ALTRAN, CAMAL, PL I-FORMAC, Mathlab, Scratchpad I, Symbolic Mathematical Laboratory, SAC-1, SIN, and others. An early comparison 2
of some of these systems is still worth reading for background Barton and Fitch 1972. Stepping back further in time, the rst generation systems of the early-to-mid 1960's included ALPAK, FORMAC, Formula Algol, PM, SAINT, SNOBOL and LISP. There are also numerous other special purpose" systems such as Schoonschip, Sheep, Trigman and Cayley referenced in Buchberger 1983. Although there are ample historical precedents for Mathematica's symbolic facilities, and a clear intellectual debt, there is virtually no acknowledgment of prior software or algorithms in Wolfram's reference Wolfram 1988, 1991. Indeed, the rst edition p. xvvii indicates in passing that Mathematica represents a synthesis of several di erent kinds of software" including some 16 systems. These are not mentioned in the second edition at all. Algorithm documentation receives similar treatment. If you want to know about the Risch algorithm" mentioned on page 528 of the rst edition, expurgated from all but the index for the second edition or how factoring is done, you won't nd any information or references. Although space limitations prevent us from providing details and full references, it is clear that recent developments in other areas have either inspired or paralleled the facilities in Mathematica. These include user interfaces such as the Macintosh environment or other window systems; alternative mathematics display and manipulation systems such as Mathscribe Soi er & Smith 1986, CaminoReal Arnon et al. 1988, Theorist Bonadio 1990, Milo and FrameMaker Avitzur 1988; operating systems and languages with interprocess communication, display technology PostScript in particular; and programming language ideas object-oriented programming, pattern matching, functional programming. An important point here is that Mathematica arrived riding the crest of a wave: a mass market newly-formed from the appearance of vastly improved low-cost computing hardware. That mass market was a large part of what Mathematica's precursors Macsyma, Reduce, Maple, etc. lacked. Mathematica became news partly because it was new, and not because it was that much better than its predecessors. In summary, Mathematica is more evolutionary than revolutionary. Not only does it depend heavily on unacknowleged prior art and technology, it is in many respects not so advanced as older systems.
4 Examination of the Objectives
Stephen Wolfram, the principal designer of the system and author of the user documentation Wolfram 1988, 1991 for the system, speci es several objectives for Mathematica which are summarized or paraphrased below. This review addresses each of the objectives. Occasionally there will be a comparison to other systems; however, the claims of Mathematica are made in relation to mathematics, and rarely with reference to other computer programs. It seems appropriate to try to review the system in that light. Objectives of Mathematica To provide a system for doing interactive symbolic mathematical calculations interactive Mathematica; x5 To provide a repository for mathematical exposition and education Notebooks; x6 3
To provide a programming language which uni es ideas from procedural programming, functional programming, rule-based programming, object-oriented programming and constraint-based programming; xx7, 8 To provide facilities for exact symbolic computation and arbitrary-precision numerical computation; xx9, 10 To provide a repository for information on the simpli cation and manipulation of mathematical functions, polyhedral objects, etc. Libraries; x11 To provide high-quality plotting from algebraic or discrete computations in a format that can be further manipulated PostScript; x12 To provide built-in functionality data types for algebraic manipulation of formulas comprising polynomials, rational functions, the usual functions of elementary calculus, advanced functions of physics special functions", functions of number theory, combinatorics, as well as composite data structures lists, matrices, literal functions and arguments, etc. and debugging; xx13, 14 In the nal sections we discuss some general issues about the relationship of Mathematica to mathematics.
5 The Interactive System
Mathematica is intended to be used primarily as an interactive program supporting the day-to-day computational needs of a mathematician or scientist. For the most part it ts into a traditional model typical of the last 25 years of time-sharing, or the last 10 years of standard data entry into personal computers. The user types a line or more, and after some computation a display is produced. Given the possibilities that have developed recently, it is somewhat surprising that Mathematica hasn't progressed much beyond line-at-a-time input; for an example of what more could be done, see Milo Avitzur 1988 or Theorist Bonadio 1990, or MathScribe Soi er & Smith 1986 or even the considerably older research system DREAMS Foster 1984. Each of these systems allows some use of pointing devices for selection of mathematical expressions. By contrast, experimental input systems based on tablets and character recognition have just recently resurfaced as palm-top" computer systems. The intuitive attractiveness of hand-writing of mathematics for raw input|rather than keying in new text and selecting alreadydisplayed material | has never been exploited successfully in past experimental systems. Experience indicates that a combination of typing and pointing seems to work better. In editing of commands, Mathematica allows selections only of linear text-strings. In its parser as well as its display of equations limited to xed-width character-grid typewriterfont 2-D" expressions Mathematica seems more like Macsyma circa 1968 than a system taking advantage of bit-mapped workstations. Because of the interactive nature of most uses of Mathematica, the intuitiveness of the system and the user-visible programming language is very important. The two components are discussed in separate sections on the display in the next section and a more extensive subsequent section on the programming language. An advertised novelty in Mathematica is its separation of a front-end interactive" component from a back-end computational" component. In practice both parts of the program often run side-by-side in the same computer, but in principle, they could be running on distinct machines. Mathematica was not the rst system to try this separation since there were prior experiments with the Maple system, and indeed the current commercial version Maple V has such a separation. Even so, the separation in Mathematica has 4
still-unrealized potential. Feedback from mouse-input on plots has been demonstrated on some platforms, but only in recent versions of the program, or on advanced workstations e.g. Silicon Graphics' systems. Overall, the notion of say a Macintosh front-end attached to a supercomputer backend sounds better than it really is in practice. Supercomputers generally do not run symbolic mathematics programs particularly well. Running both front- and back-end parts of Mathematica on a fast remote computer works well, assuming there is a frontfront-end" such as an X11 window system to display the graphics. Since such a window system is probably in use anyway, and it can be totally ignorant of and unlicensed for Mathematica, it may be preferable. Mathematica is potentially appropriate for use as an interactive front-end to other programs written in C or other languages. Through a foreign function" interface, it is possible to link to other systems. It appears to be non-trivial to get this to work, however.
6 Notebooks and the Display
Mathematica runs on a variety of machines, and the quality of the user interface varies across a spectrum from the universal but workable ASCII-terminal mode to specially-tuned versions for the Apple Macintosh and NeXT lines of computers. The most sophisticated interface model provided is called a Notebook. The user types commands as text into an outline processor. That is, there is an option of suppressing details of display of material at lower levels. The computer generates additional text and displays into the outline. The graphics sections can be re-displayed as PostScript and edited. Since the Notebooks can be exchanged between computers, there is a simple technique for reproducing results and building up a library, at least if the Notebooks are properly constructed so as not to con ict in the user's space of objects. Although this is undoubtedly a useful approach, in some respects it is not as advanced for modeling of mathematical problem solving as some other programs such as Milo Avitzur 1988 or Theorist Bonadio 1990. These programs o er facilities missing in Mathematica: they allow the text" parts of the mathematics to be data objects in a system where algebraic expressions can be selected, manipulated, linked to other expressions, etc. They provide interactive type-setting of mathematical expressions. In addition, Theorist supports animation and rotation of surfaces. Although Mathematica provides these latter facilities on a subset of its platforms, it lacks the pencil-and-paper quality of interaction that these other products o er. Although it is possible to import descriptive material from other programs into a Notebook, including digitized pictures or typeset formulas, the linkage is rather roundabout. Perhaps in the future it will be possible to run a TeXForm version of an equation through TEX and put it directly into the notebook. This would certainly create a better environment for Notebooks as an alternative mode of publication of mathematics. In fact, there are nine books cited by Wolfram Wolfram 1991 page xviii which describe the use of Mathematica in scienti c or educational contexts, and some of them are clearly dependent upon Notebooks as an interface. The design of Notebooks seems more supportive of presentation of information than interaction, and this may be just ne. One colleague nds the Notebooks to be the best new feature of Mathematica compared to other symbolic mathematics systems. On the other hand, another serious Mathematica user indicated to me that he found the Notebook front-end to be a hindrance. It may very well be a matter of previous experience and developed preferences. Incidentally, a Notebook will likely contain only a fragmentary record of computations so it departs somewhat from the tradition of the laboratory notebook" containing all raw data, experimental results etc. It is more like a showcase. 5
7 Programming Language
7.1 Overview The challenge of making computers truly useful and perhaps making programmers obsolete is often couched in terms that make it sound like a programming language issue: All one would need is to create the right syntax and semantics to banish all the problems of applications programming. That solution is not here yet, but the approaches to the challenge through the years can be characterized as mixtures from three streams: The elaborate all-inclusive language like PL I, Ada, Common Lisp The extensible language C, Algol-68, Common Lisp The language oriented to a speci c application PostScript, JCL. Mathematica falls mostly in the rst camp in that it has adapted in some way nearly every construction appearing in some general language, and it certainly is elaborate. But it includes some extension techniques, and has certainly packaged together some applicationspeci c subroutines. The evidence to date is that, super cially at least, Mathematica is actually fairly comforting to an experienced programmer|most of the familiar tools, as well as some others, are there. There are generally several ways of writing equivalent" programs using di erent programming paradigms. More so than in other languages, different paradigms may di er by orders of magnitude in resource consumption. Since no particular programming style is imposed on the programmer, some programmers will use the numerous syntactic shortcuts to produce write-only" programs so called because they become incomprehensible shortly after being written. Indeed, the fact that there are so many variations possible is especially discomforting for a programmer concerned about e ciency. Because there is such limited information available on the internal algorithms and data structures in Mathematica, there is sometimes no alternative to trying various versions of an algorithm and timing them. The fact that details might change is not a good excuse|such information could be part of release notes, for example. Mathematica does not have an extensible syntax. It uses all the non-alphabetic symbols of the ASCII character set, and quite a few multi-character symbols. Mathematica's designers did not choose the more conventional wisdom that it may be advisable to leave some characters for the users" for possible syntax extension. Techniques for such extension are fairly easy to adopt using the parser model used by both Macsyma and Reduce see Pratt 1973, and is advocated in Common Lisp, as well. Mathematica builds a complete syntactic box, for good or ill. You can't tinker with it unless you write a new front-end processor. From a mathematician's point of view, J. R. Kudera Kudera 1988 comments ... computer mathematics languages are ghastly to use" and that Problems that lend themselves to this kind of computation user-written programs simply do not occur often enough to allow users to develop pro ciency." Thus intuitiveness is important: when a system deviates substantially from common mathematical notation and semantics as well as from conventional programming, it becomes positively hazardous. Nevertheless, programming seems inevitable since the system constructors simply cannot anticipate all needs. In the next several sections we look in more detail at various aspects of Mathematica's approach to the support of programming. 6
7.2 Object-Oriented Programming The programming language specialist may observe that Mathematica's version of this popular supports neither hierarchies nor inheritance. This omission considerably weakens the faithfulness to the notion of object-orientedness for those who care. Type-based dispatch of operations is nicely integrated linguistically with the pattern matcher, however. What the programmer may think of as a function de nition is, in some senses, equivalent to a pattern-match and replacement rule associated with an object, usually the main operator or Head of the function, but alternatively one of its operands. Thus instead of re-programming some central simpli cation routine to handle a new user-de ned symbol f ... ,and its arguments it is possible to associate, in some piecewise fashion, simpli cation rules with f itself. For example f x_Integer := 0 ;x 0 de nes simpli cation of f at negative integer values. The approach of using some kind of local" control based on the operator e.g. f for simpli cation is actually fairly common and appears in one of the rst algebraic simpli cation programs, Korsvold 1965. 7.3 Contexts and Information Hiding Any modern programming language intended for building large systems must provide information-hiding capabilities. Mathematica uses its notion of Contexts for this purpose. The major construction for modular system building appears to resemble packages in Common Lisp, and even uses the delimiters BeginPackage and EndPackage. In Mathematica it is possible to delimit sections of code by Begin and End brackets, identifying a context in which public names those exported to external or Global contexts and private names those local to this context are separated. Unfortunately, this is less e ective than one might wish, because entering and exiting a Context by setting the $ContextPath does not have the e ect one might expect. For example, one might wish to assert the rule that logs of products should be re-written as sums of logs.
Log x_*y_ := Log x + Log y
But just saying this in the Global context is dangerous. In particular, system programs that rely on a certain behavior from Log may be damaged. Therefore one might wish to contain it in a context, for example:
Begin "logsimp`" Log x_*y_ := Log x End + Log y
However, this rule is placed on the Global Log symbol, rather than the logsimp`Log symbol, and the system does not distinguish between a rule that rewrites Log x_*y_ and one which re-writes Log logsimp`x_*logsimp`y_ . Thus this packaging does not limit the e ect of the Log rule, and one must presumably explicitly delete and re-assert such rules when needed or explicitly apply them when appropriate. Merely asserting them once for all time leads to generally unforeseeable consequences. We found, for example, such a rule had the e ect of breaking the Integrate command. This appears to limit severely the utility of rule-based programming as a technique for adding general information to the system. The programmer has three choices: Avoid the use of any of the same function symbols as the system; Use patterns only as a front-end to the Mathematica system; 7
and or Use patterns only as a back-end to the real Mathematica system. This front or back-end usage might include converting all Logs to logs for special simpli cation, and then converting back to Logs. Perhaps another example will illustrate the di culty of this approach: If you teach the system that x + 1 x, then the system does not use this information to compute maxx; x +1. How can one x this short of reprogramming the system function Max as well as all the proprietary system functions that use some internal version of comparison? By the way, this obvious inequality is not true for all possible x representable in Mathematica; Consider x = 1. A subtle point, perhaps not intended to be noticed by the casual fan of Mathematica, is that the rule on the back cover of Wolfram 1988 de nes simpli cation for log, not Log, thereby not interfering with built-in rules. This trick of using lower-case names does not work very well if one wishes to alter, by rules, any built-in functions, or functions like Factorial or Plus, which do not have lower-case equivalents in their usual representation as ! and + respectively. It is possible to group rules and apply them together, as illustrated by programs in the on-line library as well as in Maeder 1988. Such grouping of rules as in the trigsimpli cation routines eliminates cross-talk" by limiting scope. Unfortunately, such tricks weaken the possible synergy of rules. If the idea behind rules is to have them take e ect when appropriate, without speci c attention by the programmer, the necessity for grouping vitiates the concept. One experienced Mathematica hand advised me not to modify any built-in functions. This simple piece of advice carries with it implicitly the idea that if you don't like what Mathematica does with the Log function, you are free to program anything and everything you want about the log function. But then you'll have to change Integrate which returns answers in terms of Log to say integrate which returns answers in terms of log. If you choose to di erentiate the result, you have a choice of changing log to Log temporarily, or propagating your changes throughout your program: de ning rules for the di erentiation and numerical evaluation etc. of log in e ect writing a shadow" Mathematica. This is made rather di cult because the system's internal functioning is hidden for proprietary reasons. Even if you are willing to pay the substantial penalty in performance, you cannot tell how much functionality must be recreated. An attempt to make extensive use of Mathematica rules in de ning a system using abstract data types is reported by Buchberger 1991 who found it resulted in frustratingly slow computation. Two other shortcomings in Contexts are worth noting: Mentioning a name before reading in the package de ning it shields the name in the package from the global environment, e ectively disabling the package. Debugging, never easy in Mathematica, becomes even harder when packages and contexts are involved. Also, printing out a program de ned in another context see, for example, the data associated with the Bessel functions in version 1.2 involves the repeated display of fully-quali ed and rather lengthy context names. 7.4 Spaces mean Multiplication This is perhaps a minor point, but annoying in its own way. In the Mathematica programming language, one can use spaces or even adjacency to signal multiplication. This idea, used by Wolfram in his earlier SMP system, is initially appealing|that one can simply write 2 x or even 2x instead of 2*x. It looks like the traditional mathematical convention. But is it? Consider that it implies that sin x or even sinx is a product, equal to x * sin. The Scratchpad II system Computer Algebra Group 1988, following 8
another mathematical convention, interprets sin x, sinx and sin.x as function applications. Mathematica requires the syntactic construction sin x or sin@x or x sin for function application, and just to make sure you use the built-in function, you must capitalize the rst letter: Sin x . This departure from conventional notation may be of special concern to a teacher whose students may be struggling with notation in the rst place. There is a weak argument that most potential users have not had experience typing" conventional mathematics in any notation, and therefore requiring square-brackets may not be objectionable. What else goes wrong, though? A few things. For example, a++ * ++b, which to those familiar with the C programming language appears to be the computation of a*b+1 leaving a set to a+1, as well as b to b+1, cannot be written in Mathematica as a++ ++b, since that space does not stand for multiplication. Rather it stands for nothing: Mathematica parses this expression as the necessarily meaningless a++++ * b. Even the author of the output-printing program was confused on this, since echoing back the rst expression by typing Hold a++ * ++b displays the non-equivalent Hold a++ ++b . This particular inconsistency of the display with the internal form was reported as a bug some time ago, but its existence suggests a design error in that the parser and display should not have access to inconsistent precedences. In a well-designed system the two closely related subsystems would use the same precedence data, stored in one place. The precedence of the space as multiplication is not adhered to. As another example, 3! ++a results in an error: Factorial is write protected. As a matter of clarity, I suspect the physicist who is used to writing 1=2 may be lured into writing 1 2Pi which is actually the rather di erent =2. Finally, the accidental omission of a semicolon or comma will not be caught by a syntax check. The forms f a b , f a,b and even f a;b are all unexceptional syntactically. My conclusion is that insisting on the use of an asterisk is preferable, because leaving it out brings up too many problems. One reader suggests as an alternative inserting parentheses when there is any doubt. This advice is hardly likely to be followed by those who need it. 7.5 Other Syntactic oddities The version 1.2 valid input form a,b is not documented. It is not a List, but Sequence a,b . Sequences are most naturally produced by pattern matches involving a collection of arguments. If f x__ is matched to f a,b,c then x is Sequence a,b,c A Sequence has the remarkable property that f 1,2,Sequence a,b means f 1,2,a,b . One consequence is that f@a,b is mapped to the same form as f a,b , but the perhaps easily mistyped fa,b is somewhat unexpectedly mapped into a*b*f. Explanation: fa,b = f*a,b = Times f, Sequence a,b = Times f,a,b . If you type f0,x you get 0. A simple x for this problem is to be less clever and forbid the alternative input-syntax of a,b for Sequence a,b . Version 2.0 has adopted this x. You still should beware that f0 is 0, regardless of the de nition of f. Note that the Head function does not work on a Sequence since Head Sequence a,b is equivalent to Head a,b , which is meaningless. As another example, if you want to rede ne factorial for certain values, you might think to do
Unprotect Factorial ; big!=bigfact
but that won't work. Here you need the space: 9
big! =bigfact
because the language includes a not-equals operator !=. Finally, the expression
4 . 4- 5
which means substitute 5 for 4 in the expression 4," returns 5. It has a rather di erent meaning from the expression without the space in it;
4 .4- 5
which returns the rule 10.- 4 because 4 .4 is 10. What's the point in all this? Simply that it is potentially quite confusing to see a modern programming language design in which the meaning of white space" is not only signi cant but has puzzlingly di erent meanings depending on context. 7.6 Procedures, Functions, Patterns, E ciency There are attractive aspects to the method used to de ne procedures. Were it not for the di culties caused by errors in semantics, and the almost inevitable ine ciency which results from the reliance on matching, it would be even more attractive. In a nutshell, the approach is to Incorporate control structures from all of C, Lisp, APL and functional programming. Unify the three notions: a mathematical function f x; a pattern f x_ ; and a procedure invocation. Thus f 3 is implemented" by a pattern match of f 3 which results in a temporary binding of x to 3. The evaluation of the right-side of a rule provides the semantics for the function. Mathematica has other ways of viewing function objects meant as programs, based on the lambda calculus and Lisp. You can express x; y :x + y as Function Plus Slot 1 , Slot 2 or, in the runic syntax 1+2&. Unfortunately many di culties with details crop up. 7.6.1 Rules and Patterns Basically, any algebraic system that tries to implement mathematics by transformations on mostly uninterpreted trees, is going to fall into pits. Consider the plausible rule x_ - x_ := 0. This might be considered universally true, regardless of the pattern which x matches. Yet it is not true where each apparently identical syntactic expression can really be di erent. For instance, it would be an error to simplify Infinity - Infinity to 0. Indeed, Mathematica returns Indeterminate. And yet to Mathematica, the expressions f Infinity -f Infinity , f RealInterval 0,1 -f RealInterval 0,1 and Random Integer,n -Random Integer,n are each 0 although the latter provokes an error message. Similarly, two occurrences of Ox2 are not semantically identical because each may imply a di erent asymptotic constant. In particular, Ox2 , Ox2 = Ox2 would seem most plausible. In fact, the use of the equality for that expression is an unfortunate convention; a more formalistic approach would use a notation perhaps reminiscent of set inclusion Graham et al. 1989. The problem here is fairly deep, and quite important. How could one modify the rule that x_ - x_ :=0 to make it correct? Perhaps by saying that x must satisfy some 10
predicate? What should that predicate be? Mathematica supports examination of the Head of the expression tree that is denoted by x, and this will sometimes work. If x is an integer, one might agree the rule applies. But if x denotes say an expression headed by Plus, one must examine in principle all components to see if any of them are among the dangerous" kind indicated above. Mathematica has no handle on this problem, and the kind of handle that is necessary is probably based on global information regarding the type of an expression. This problem is eliminated by systems which assign types to expressions, such as Scratchpad II see Computer Algebra Group 1988 or Newspeak Foderaro 1983. Mathematica encourages the user to de ne functions by writing collections of transformation rules, each of which has a pattern, a replacement, and optionally, some conditions. This mechanism has a long history as a model of computation, and each formalism in the past has had to de ne the order in which rules are applied. This order has a strong bearing on the real semantics of a rule set, since di erent orderings can change the answers or even cause an in nite loop. How are Mathematica's rules ordered? The manual explains that the most explicit rules are used in preference to the most general. Yet it is clear that except for very simple cases, Mathematica has not got the slightest clue as to which rules are more speci c. Maeder 1988 p. 59 points out ...Mathematica cannot always nd out which of the rules is more special than the other and it might fail to reorder them accordingly." If the rules are not ordered correctly, it may be fairly painful to write patterns to make sure that the pre-conditions for matching do not overlap. Furthermore, if you cover up" a built-in operator with rules, and then you wish to refer to the built-in routine, you have no direct access to the system's prior de nition. In version 2.0, WRI apparently gave up on getting the ordering right and the user is given a chance to rearrange the rules by setting for example DownValues explicitly see p. 266 of the 2nd edition of the manual. It is implausible that users would nd this a convenient way of correcting" a rule set. Mathematica's patterns provide the mechanism for the left-hand sides" of rules. The notation for 1-, 2- and 3-blank match variables" with optional attached predicates is compact and relatively easy to read. The handling of defaults appears more general than other computer algebra systems; the handling of commutative operators is probably no messier than necessary. One can hope that the implementation is no costlier than necessary. The integration of the pattern matcher into the language extension facilities is super cially neat. The idea of so-called up-rules" to avoid clogging common operators especially Plus and Times with rules, is a clever heuristic. This allows, for example, rules that pertain to the addition or multiplication of special functions to reside with the special functions, not with the common operators. This means that in principle the simpli cation of sums is not slowed down unnecessarily by having to look at inapplicable rules, even if they are nominally rules a ecting sums. These problems have been addressed, for the most part less e ectively, by numerous earlier programs. A sample would include Cooperman 1986, Fenichel 1966, Greif 1985, Hearn 1976, Jenks 1976, Mathlab Group 1983, McIsaac 1985. Whether the melding of rules and conditions that have side-e ects can actually be put together in an e cient full-evaluation mechanism appears to be an open question. Mathematica doesn't quite match its speci cations in this regard see x8.3. Yet even viewed as transformations of trees, there are tricky issues in pattern matching. Consider, for example, the transformations to contract certain expressions involving factorials: Simplify nn , 1! to n!. A rather economical and readable version of this transformation given that one must rst understand that n_ is a match variable that matches anything, and is referred to as n on the right-hand side is 11
Placing this rule on Factorial rather than Times is a good idea: since this would only a ect the speed of simpli cation of products involving Factorial. A version that is more generally applicable, and works on h , 2h , 3! also, uses a condition. Indeed, there is a rule equivalent to this latter piece of code in Mathematica's combinatorial simpli cation library, along with some slightly more ambitious rules:
Factorial : n_! m_! := Product i, i, m+1, n ; n - m 0 && IntegerQ n-m Factorial : n_! m_! := 1 Product i, i, n+1, m ; m - n 0 && IntegerQ m-n Factorial : k_! k1_ := k1! ; k1 - k == 1 *equivalent to our rule * Factorial : n_ * m_! := n! ; n==m+1 Factorial : n_ * n_ -1! := n!
n_ * n_ -1! := n!
Consider now the problem of reducing n=n! to 1=n , 1!. It would seem that a rule would do the trick. Unfortunately, one gets the error message:
n_ TagSetDelayed::notag: Tag Factorial not found in -----. n_! Factorial : n_ n_! := 1 n-1!
This means that the rule cannot be placed on Factorial because it is not among the top two levels of Heads in the left-hand-side. Indeed, only Times and Power qualify, and neither one of those is an attractive choice, being too common. The expression looks like Times n_,Power Factorial n_ ,-1 . The point of this illustration is that the up-rule idea only delays by one level the potential exponential growth of rule-sets to implement complex simpli cations. Delaying the problem by one level is a heuristic that may turn out to be window-dressing if programmers make heavy use of the facility. How much degradation would be caused by implementing it for deeper nesting? What would be the bene t? Consider reducing to zero the expression m!2m + 12 , m + 1!2: The rules do not work, and one presumably has to try for a new rule involving powers, perhaps of the form where the details of the predicate to be applied, as well as the functions f1 and f2, are, for the moment, unimportant. Properly formulated, this rule covers one of the earlier rules. Again what is signi cant here is that the up-rule" technique won't work two levels down: this rule must either be placed on Times, or on Power, and neither option is attractive. Thus, while the Mathematica pattern matcher appears to be handy for patching the simpli cation rules, it is probably unwise to expect this technique to provide simple and e cient implementation of all new code. There is continuing active research into understanding the limitations and techniques for term-rewriting systems. The ad hoc melding of rules as supported in Mathematica, regardless of the sophistication of pattern matching, is certainly not going to guarantee important properties including termination. The double issue of J. Symb. Comp. 3, 1 and 2 1987 discusses rewriting techniques and 12
n_^e_ * m_!^f_ := n^f1 e,f *m!^f2 e,f ; prede,f,n,m
applications. Yet the evidence of the past several decades casts strong doubt on the idea that an e cient version of mathematical knowledge can be imparted to a symbolic system primarily by rule-transformations on trees. The more fundamental approach of transforming expressions into canonical forms when possible has been shown to be quite e ective. For factorials this is probably best done by a calculus of di erence and shift operators. 7.6.2 E ciency There are a number of factors contributing to ine ciency in any computer algebra system. Perhaps the major factor is that most systems, including Mathematica, are interpreter based. Consequently, a simple program that can just as easily be expressed in a compiled language e.g. C, runs orders of magnitude slower than if programmed in C. see, for example, Buchberger 1991, where basic list structure operations are shown to execute 3000 8000 times slower. One trick is to write such programs in C, and call them from the interpreter. There are other factors: Generality. Even with a compiler introduced in version 2.0 it is unreasonable to expect compilation, considering Mathematica's elaborate semantics, to reach the same kinds of speeds as compiled C or Fortran. Consequently, although it may be convenient to express a simple do" loop in Mathematica, it will be slow in execution. The version 2.0 compiler is useful only for speeding up the numerical machine-precision real evaluation of expressions in plotting, numerical integration, or similar functions. Evaluation Model. Mathematica claims to implement in nite evaluation all the time, even for local variables. This means that when a variable is evaluated, a bit-vector is checked see x8.4 and possibly the entire structure may have to be traversed even if nothing happens. Data structure operations that would normally be expected to be constant-time depend on the size of the expression. For example, Part stuff,1 or stuff 1 evaluates all of stuff even though all that is needed is the rst sub-part. This can be very expensive, as illustrated in the next section. Data Structures. Mathematica implements List objects as arrays. Consequently, extracting an item from a list based on its index has a cost that is independent of the index. Yet adding an element to the front or back of a list Prepend or Append depends on the length of the list, since the old array must be copied to a new location. Counter to the intuition one might have from other uses of the word List, adding to the front or the back appears to be equally expensive. Furthermore, picking out the ith element of a list requires that all the elements be re-evaluated, so that indexing is itself not constant time, but a function of the length of the list and the complexity of the elements in it. Consider, for example, a table of 106 zeros set up by h=Table 0, 10^6 . The statement h 5 =h 5 +1 takes 4.3 seconds on a Sun SPARC 1+, because all the elements of h are re-evaluated. Computing a histogram this way would be painfully slow. see also x8.4. It is formally possible to concoct an abstraction of a linked list by function calls" in a Prolog-like form. That is,
list a_,b__ :=cons list a_ :=cons list := nil car cons a_,b_ := cdr cons a_,b_ := a,list b a,nil a b *etc. *
13
but this is quite ine cient, and few of the built-in features would work on this data. You do not have much choice about the substantial overhead caused by the generality of the mechanism for function calls, each of which constitutes a pattern match. Mathematica's Function notation may perhaps be a faster choice when anonymous functions are adequate to some purpose. Size. Within xed memory resources, the more space taken by system code the less is available for user programs or data. Mathematica has gotten substantially larger from its rst version to version 2.0 and, if for no other reason, it may be slower as a consequence of paging or other memory-based activities. On the other hand, there are several sources of e ciency in a high-level language. The built-in algorithms and appropriately chosen data-structures may provide an advantage over naive programs. For example, an arbitrarily clever test for primality, perhaps even coded in assembly language, could be included as a built-in command. High-level programming constructs provided to anticipate common sequences of lowlevel operations can eliminate repetitive interpretation. For example, a single command to map the operation of addition over a list would be faster than an iterative program indexing through the same list. Some expressions known to be evaluable to numbers can be compiled in a conventional fashion especially in contexts such as plotting or numerical root- nding. Results can be cached" for re-use, making it possible to reduce the complexity of some styles of heavily-recursive programming. Mathematica uses each of these tactics. Other computer algebra programs make di erent cuts through the morass of decisions, and it is certainly possible to nd signi cantly faster, as well as signi cantly slower programs for comparable computations. An easily accessible paper Simon 1990 compares four systems, including Mathematica, and may be of interest as a view of their e ectiveness in solving problems on some concrete examples.
8 Major Semantic Problems
8.1 Canonicality Wolfram 1988, p. 212-213 1990, p. 269-270 seems to argue that, because it is theoretically impossible to have a program that reduces all expressions to canonical form, that the set of transformations provided basically, Expand, Factor, Simplify, are su cient. Even within the class of expressions with decidable zero-equivalence procedures, the user is left somewhat at loose ends trying to gure out which combination of commands might be e ective in mapping the di erence of two possibly-equal expressions to zero, and thereby directing the result of an If expression the right way. The description of Simplify in Mathematica Simplify expr performs a sequence of algebraic transformations on expr, and returns the simplest form it nds." fails to inspire con dence. Especially in the realm of rational functions since in this case the answers are easily computed, the failure to use canonical simpli cations can be a source of wrong answers: programs which depend on zero-equivalence being decided correctly can be misdirected. Any routine that does divisions can fall prey to this problem. While commands such as Factor and Simplify provide a nominal zero-equivalence decision-making capability, they must be called explicitly, and they do not, in general, take advantage of the possibility of the far more compact 14
representation and manipulation that limited domains a ord. Particular problems involving complicated expressions and their negatives inside radicals are repeatedly cited as di culties in public bug reports" in the network newsgroup sci.math.symbolic. This is discussed further in section 15.1. 8.2 Dummy arguments in patterns are global As we have discussed earlier, Mathematica combines function de nitions and pattern matching with an interesting technique. A simple program de nition f x_ := ... looks like a pattern-match and replacement for the expression f ... . This simpli es the definition of a function over distinct structural cases or types of arguments by allowing more elaborate dummy argument" speci cations. Thus one can de ne f on a special case expression where x_ would match Sin ... as f Sin y_ := .... One can de ne f for integer arguments by f x_Integer :=..., etc. Among algebraic manipulation systems perhaps Scratchpad I Jenks 1984 is a precursor, although other programming languages including ML, Prolog, and SNOBOL have similar notions. Another place the merging of the concept of matching and parameter passage appears is in the destructuring of lambda-list arguments to defmacro in Lisp Steele 1984, 1990. Unfortunately, the analogy, or the implementation of the analogy, leaves a lot to be desired. The principal objection is that a change to the dummy variable x on the right side of the rule f x_ :=x=3 followed by f y changes the value of the global variable y. A cleaner treatment would be to make x serve as a bound variable" within the pattern. Problems with evaluation may also be related to the design error in binding explained in the next section, or might be independent. 8.3 Binding of variables Here is a simple program:
g x_ :=Block a ,a=1+x
This is a program much simpli ed from what one might write in the course of computing an elaborate function, and it seems to be very straightforward. The result of g 3 is 4 and of g s is 1+s. But why then is g a not a+1? It is 251+Hold 1+a , with a warning message about Recursion depth exceeded". This is one result of a Mathematica feature" which can have drastic destructive e ects:2 Any time you compose a presumably more serious program with local variables, you may have a con ict with a name that occurs within the actual parameters. In the Mathematica manual, section 2.5.10, Advanced Topic: Function Arguments and Local Variables, Wolfram admits, Sometimes it can be very confusing to have the name of a local variable con ict with a value you give for a function argument. The best way to solve this problem is somehow to have the names of the local variables that appear in your functions be such that they will never appear in the values of function arguments." That is, you could try to avoid this con ict by programming, for example:
g x_ :=Block afunnyvar ,afunnyvar=1+x
But this can rapidly get tiresome; potentially recursive functions lead to constructions like Block v=Unique "s" ,... in version 1.2's Laplace transform package, and even this is not necessarily e ective.
To quote from Maeder 1988 p. 8 The more subtle ones problems become only apparent in the context of a longer session with Mathematica and are then normally very hard to nd." Actually, it appears that his proposed solution is a work-around for a bug, corrected by the introduction of Module.
2
15
Maeder suggests avoiding the accidental capture" of variable bindings by using the construction
Begin "`Private`" f x_ := Block `a ,a=x+1 End
The user's variable a, or to be more verbose, Global`a is then never the same as the identier Global`Private`a that is used inside the body of f. However, typing f `Private`a still causes an error, so the problem is not really solved. You can elaborate on the Context names to reduce the likelihood that names will con ict you then have to worry about two programmers accidentally chosing the same Context name, but you cannot really eliminate the problem as is done by, for example, Pascal. By the way, if you make the mistake of trying to look at a program in some Context, on-line, you will be deluged by the repetition of Context quali cations on every name. This is quite painful. Apparently this scoping was nally recognized as wrong, and a real solution appears in version 2.0. also described by Maeder 1991 p. 8. Using Module instead of Block one de nes
g x_ :=Module a ,a=1+x
and this works as expected: g a = 1+a. The new problem is that every time the Module is entered, new names are produced, presumably taking up system resources of time and perhaps space. This can be seen by trying the program
h x_ :=Module a ,Print a
which prints a di erent value looking like a$12 each time it is invoked. Even if they are removed eventually as Temporary objects, the overhead is certainly much higher than stack-allocation of variables in a classical Algol-like language. Consider another example, also xed in version 2.0, based on the de nition of the function h j_ := Sum h k , k,0,j-1 . This has problems with the local variable k. Mathematically speaking it is reasonable that h 0 returns 0, and it would seem that h 1 should return Sum h k ,fk,0,0g or just h 0 , which is 0. However Mathematica 1.2 wrote followed by a 230-line message related to in nite recursion. Although these speci c examples, all reported as bugs, have been xed, and Module repairs the defective Block, the misunderstanding persists. Even though the role of the variable k in Limit f k ,k- b is very much like the role of k above, in this case the global value of the symbol try k=3 interferes with its use as a bound variable in the limit. Furthermore, D Sum a i , i,1,n ,a i returns Sum 1, i,1,n while D Sum a i , i,1,n ,a j returns 0. The bound variable i is somehow capturing an instance of i from outside the summation. As a sidelight, might one really wish to write a program g x_,y_ :=x=y which when invoked as g r,3 actually sets the value of r to 3? Attitudes toward programming by side-e ects on parameters vary. Purists condemn it and point out it is not necessary. Others have grown up in a culture that permits or even encourages it. Fortran, and among computer algebra systems, Maple, make use of it. Common Lisp provides multiplevalue returns to allow a somewhat disciplined approach to returning several values from a function, reducing the need for side-e ects or returning temporary structures of values requiring destructuring merely to return extra results. In my view, Mathematica 16
General::itervar: In iterator k,0,0-1 , variable k already has a value
should strongly discourage the use of side-e ects, given that it cannot keep track of them e ectively. The next section explains the problem. 8.4 In nite evaluation The evaluation process in Mathematica has several di culties. Although it would be hard to discern from the documentation, the design a uses fallible heuristics and b injects some non-determinism into its results. As a consequence, Mathematica's evaluation techniques can be problematical. This section reviews the process and its consequences to the system semantics. When presented with an expression h a1 ; a2; the system recursively evaluates the head of the expression h to f , then the arguments3 a1 to b1 , a2 to b2 , etc., in turn. Next, attributes and transformation rules as appropriate for application to f b1; b2; are examined and additional changes may be made to the expression, perhaps through more evaluation of the right-hand sides of rules. If anything has changed from the original expression, h a1 ; a2; it is claimed that the system e ectively starts the evaluation sequence over again." In fact, it often doesn't re-evaluate, because that would be quite costly. Instead the system predicts whether an additional evaluation will change anything. Sometimes the prediction fails. I am grateful to David Jacobson, as well as public discussion by S. Wolfram for illuminating these points. The prediction is apparently not even entirely deterministic. The only user-visible hint about this problem is in the description of the Update command which can be called to re-evaluate symbols under special circumstances that rarely occur in practice." The only speci c special circumstance mentioned in the manual Wolfram 1991 is the case of change to a global value changing the outcome of a Condition de ning a symbol's value. The simplest example of this is f x_ := x ; globflag. If the global ag globflag is False and you evaluate f 3 , it evaluates to itself. A subsequent change to globflag should cause an instance of f 3 to be re-evaluated. It won't. The basic problem is that the presence of side-e ects makes it virtually impossible to tell if evaluation is really complete. Consider
count = 0 g x_ := x ; ++count 100
In an actual in nite evaluation system, g 3 would evaluate to 3. In Mathematica it evaluates to g 3 . Only after 100 forced re-evaluations will it return 3. The notion of e ective re-evaluation" is based on placing a time-stamp on each expression which evaluates to itself. Time" in this case is simply a function that increases monotonically as computing proceeds. Note that if enough objects are produced, the system might over-run a xed-width time-stamp eld. On a MIPS M 120 processor using version 1.2, executing Do k=1, 12200 takes about 1 second. If the time-stamp is stored in an unsigned 32-bit word as we speculate is done here, then executing Do k=1, 2^32 would over-run the time-stamp in about 4 days. As explained below, an inappropriate time-stamp might provoke an error, thus causing a long-running computation to fail mysteriously. Assuming we do not encounter such an unlikely problem, the result of evaluating an expression E is simply E if E previously evaluated to itself and if E 's time-stamp is su ciently up-to-date.
3
unless they are held."
17
The notion of su ciently up-to-date" appears to be implemented approximately as follows: Every symbol in the system is stored in a hashed symbol table, using buckets for chaining. Each symbol-table bucket has a eld which is updated to the current time when any symbol in the bucket is changed in any way. If the time-stamp on an expression E is at least as recent as all the symbol-table buckets of the symbols which occur in E , then that is deemed recent enough. This is clearly wrong. Even symbols not occurring explicitly in an expression can, by changing, cause an expression to no longer self-evaluate e.g. count above. Also, since any change to a symbol will update the time-stamp on all the symbol-table bucket occupants, coincidences of shared hash-buckets a ect evaluation. Such coincidences cause re-evaluation of expressions and consequent side e ects in a manner not predictable by the user, who has no way of determining which symbols share a hash-table bucket. An aside: Why use this per-hash-bucket time-stamp? Presumably to keep the checking cost down. If there are b hash-buckets, then instead of descending through the tree of an expression to identify and check all its leaves, the evaluation mechanism can be based on a vector of b bits, associated with each expression. Then if the vector v associated with E has a 1" in position p, there is some dependence on some symbol in hash-bucket p. If the time-stamp on hash-bucket p is more recent than E , a full evaluation must be done, along with a recomputation of v . Note that if the user or the system suspects that something unusual" has happened that a ects expressions involving x, then a call to Update x sets the time-stamp on the symbol-table bucket associated with x to now." This will cause any expression E using x or any other symbol in the same bucket to be fully evaluated when E is next evaluated. There are a number of tactics that can be used to make checking each of the bits' hashbuckets faster, although they may translate into much increased time when changing values. Removing the in nite evaluation heuristics especially the assumption on failed rules may force the cost of evaluation to escalate substantially. Unfortunately this may be the cost of getting an in nite-evaluation answer right. An alternative might be to forbid certain kinds of state alterations" or side-e ects in the condition part of a rule|directly or indirectly, rather than merely advising against them. Note that the side e ects might occur during the evaluation of the parameters, and hence in the evaluation of an entirely di erent function. Identifying such side-e ects automatically" would ordinarily not be computationally feasible, unfortunately. Another choice would be to use a better understood evaluation scheme that has stood the test of time in some other language. At a minimum, the documentation should clearly identify the pitfalls.
9 Numerical calculation
We start with a brief example suggested by W. Kahan:
In 1 := p=314159265358979323; q=314159265358979323.; r=314159265358979323.00000000000000000000; s=p+0.00000000000000000000; In 2 := Out 2 = In 3 := Tan s ,N Tan p ,Tan q ,Tan r
1.59981, 1.59981, ComplexInfinity, -1.1297926523089085443 p==q, q==r, r==s, r==p
18
Out 3 =
True, True, True, True
At rst sight, the values of p, q and r all look the same. Adding zero to p should not alter its value much. However, Mathematica can see di erences here, even though it claims all the values are equal. Incidentally, the last value in Out 2 is correct. To continue the example, recalling that p and q are equal,
In 4 := N Tanh p ,Tanh q General::ovfl: Overflow occurred in computation. Out 4 = 1., 0.
Since tanh x ! 1 as x ! 1, the answer 1 is correct; the over ow error message suggests that a poor method is being used for computing tanh of large arguments. Although the number 0: is the result of the over ow, the answer does not indicate it. Only by computing its Accuracy, can you discover that no digits are correct". The comparable results from version 1.2 are even worse, incidentally. Should we care about such discrepancies? Yes! Numerical calculation in the context of a symbolic system should be done in a manner that places special importance on making decisions correctly. A system which cannot deal correctly with numerical constants will ultimately have di culty with symbolic computation. You need not explicitly use its numerical subsystem to run into problems. As part of a symbolic calculation, a subroutine may determine that an expression is a constant, but it is generally through careful numerical calculation that it can tell if for example the constant is positive. Such a determination is often quite important. Mathematica version 2.0 seems to distrust its own numerical system to such an extent that it refuses to determine the sign of e , . This is p improvement over version 1.2 an which apparently gets the sign wrong since it simpli es e , 2 to e , ; in version 2.0 you must use PowerExpand to force this error. What does it mean for a system to know" all about constants, if it cannot answer simple questions about them? 9.1 Accuracy and precision Mathematica uses the terms accuracy and precision in non-standard ways. By convention, as well as by informal usage, accuracy indicates the accomplishment of a goal of closeness to the truth." If v is the correct value and v 0 is the computed value, then two measures for accuracy are absolute error: v , v 0 and relative error: v , v 0=v . Precision is the exactitude of an assertion, even though the assertion might be false. For example, the statement that = 22=7 is quite precise, although it is not very accurate. Precision is generally discussed in terms of binary or decimal digits provided in a representation of a number; it refers to the e ort used to carry out detailed operations, but not whether some goal of correctness is accomplished. Mathematica associates attributes of Precision and Accuracy with each number, unlike Fortran or C in which a variable has a precision attribute. Mathematica de nes Accuracy as the number of decimal digits to the right of the decimal point, and Precision as the total number of signi cant decimal digits. The term signi cant" is not de ned, however. In Mathematica's terminology, 3.01 meters is more Accurate but less Precise than 301.0 centimeters4 . In Mathematica, the number written as 3.0 is a representation of
4
In this discussion we ignore the fact that Mathematica generally uses machine hardware" oating-
19
any number greater than 295 100 and less than 305 100. That is, 3.0 is like an interval|a number which rounds to 3.0 when printed with two decimal digits. The numbers which you might type in as 3.0 and 3.00 are di erent, since they represent di erent intervals| although as we have seen, Mathematica does not always respect this distinction since they are equal under comparison.. Mathematica computes using arbitrary-precision arithmetic, presumably when necessary. Its decision mechanism for determining when this is necessary is not always obvious or correct. For example, what is a good approximation to sin of a large integer apparently close to a multiple of ? Sin 3141592653589793238. yields 0, to Accuracy 0; Sin 3141592653589793238.00 yields ,0:45 to Accuracy 2; but the expression N Sin 3141592653589793238 yields ,0:641653 to Accuracy 16. The most Accurate is the least accurate. In discussions with Wolfram Research employees, they defend their usage of the terms by reiterating Accuracy is de ned as the number of signi cant digits to the right of the decimal place." and that Your paradoxes result from associating the everyday meaning of accurate with the well-de ned concept of Accuracy." This is reminiscent of Lewis Carroll: `When I use a word,' Humpty Dumpty said in rather a scornful tone, `it means just what I choose it to mean|neither more nor less' " Through the Looking Glass, 1872 It is not di cult to compute these functions correctly: there are several packages for computing arbitrary-precision transcendental functions in the open literature see for example Brent 1978, Wyatt et al. 1976, Fateman 1976, Bailey 1991. As another example, this one illustrating a debatable choice for the use of Precision, consider the function f h_ :=N N Pi,2 h -N Pi,h ,3*h which one might use to look at digits of as computed by Mathematica. The function f plausibly would give the di erence between a 2h-digit approximate value for and an h-digit value, computed to 3h digits. Computing f 50 returns the value 0. Accuracy 49, Precision 0. Clearly only about h-digit computation is being done. What one might hope would work better in version 2.0 and later using SetPrecision is
g h_ :=SetPrecision N Pi,2 h ,3 h -SetPrecision N Pi,h ,3 h
but oddly enough, g 50 returns 0, to Accuracy 149. The explanation for this is presumably that the value of is apparently being cached" to 2h digits, and the subsequent lower precision" version of Pi actually has hidden digits that correspond to more places than deserved when it is re-precisioned to 3h digits. As a consequence, the value of N Pi,50 is not a constant, but depends on what has been computed before. Before discussing one alternative model, we present some remarkable puzzles|consequences of Mathematica's arithmetic|based on examples discovered by David Jacobson of HewlettPackard Laboratories. Consider the iteration si := 2si,1 , 3si,1 2. This iteration converges from a starting value of s0 = 0:3 rather rapidly toward sn = 1=3. Indeed, with any number x replacing 3 in that formula, the iteration proceeds to compute an approximation to 1=x from a suitable starting value. See what happens in Mathematica 2.0.
In 1 := s i_ :=s i =2 s i-1 - 3 s i-1 ^2; * Use "bigfloats" * In 2 := s 0 = SetAccuracy 3 10,18
point arithmetic for small" numbers such as 3.0 and 3.00; that is, all numbers 3.0...0 with fewer than 16 zeros are the same, but each is subtly di erent from 3.0...0 with 16 zeros. We assume that rules for calculation are those speci ed for the software extension to arbitrary precision in Mathematica.
20
Out 2 = 0.3 In 3 := Out 3 = s 1 ,s 2 ,s 20 ,s 30 ,s 40 0.33, 0.3333, 0.333333, 0.3, 0.
In other words, given a convergent iteration that mathematically gains accuracy at each step as it approaches 1=3, the computation begins to veer away to 0. It ends up at a stationary point at 0. Furthermore, even though they print di erently, all the iterates beyond s 4 are pairwise Equal in Mathematica. This is apparently not a bug, but a feature. WRI argues that the Accuracy of the results is decreased at each iteration. One way of thinking of this is as though the arithmetic were in some sense subject to experimental error. This is supposed to protect the user from believing inaccurate results. Such a scheme might be valuable if it worked. On the other hand its utility would seem to be rather low if one could increase the Accuracy of a number by averaging it with itself 100 times. But this is just what happens, as seen below.
In 1 := r=q=1.00000000001000000; In 2 := Do q=q+q 2, 100 In 3 := q Out 3 = 1.00000000001000000000342263224918264460733208674 In 4 := Out 4 = Accuracy r ,Accuracy q 17, 47 ;
This bug" was xed in version 2.0, so that now to exhibit the aberrant behavior, line 2 should be Do q=q+q+q+q 4, 100 ; instead. Making such errors more obscure in succeeding versions of the system will not make them go away, unfortunately. Incidentally, q+q+q+q is always re-written as q+q+q+q whether you like it or not. In either version of the system, the value of q has increased in Accuracy by 30 decimal digits. This general approach, sometimes called signi cance arithmetic, attempts to propagate fuzz" in operations on numbers like 1:23?:::? and 3:45?:::? where ? indicates an unknown digit. Unfortunately, making this system secure under operations requires rules that are quite pessimistic, and generally results in too large a loss of signi cant digits" to sustain a chain of numerical computation. Less pessimistic rules as implemented in Mathematica not only provide rather poor control of uncertainty, but can lead to nonsensical results. According to W. Kahan, it is a folk theorem" that under any implementation of signi cance arithmetic there must exist chains of operations that lead to either an unwarranted loss of signi cance, or to a gain in signi cance, or both. The real problem is then that users will be lulled into a false sense of security. Somewhat less damaging, but certainly disconcerting, users may also be given answers much worse than they deserve. The rate of gain plus the rate of loss in bits per operation must exceed one. David Jacobson's examples above illustrate Mathematica's susceptibility to the consequences of this theorem. Is there a way out of this? Certainly one can attempt to gure out by some independent calculation, the actual accuracy of every variable at the end of each iteration, and replace it by an equal, but more or possibly less Accurate" value by using SetAccuracy and or 21
SetPrecision.
But then not only has the Mathematica design failed to perform its intended automatic error analysis, it has in fact made it advisable to perform such error analysis where previously it was not needed! Indeed, in a tutorial on Mathematica's arithmetic, Keiper 1990 illustrates the necessity, in computing the value of a Chebyshev polynomial, to repeatedly compute the appropriate precision in intermediate calculations. Keiper also explicitly sets the accuracy of the nal result. Each of these resettings requires copying over the value into a newly created number, and so is not a trivial operation. Consider for all computational purposes that each number representable in a oatingpoint hardware format in the computer means an exact rational number. That's it. No special hidden precision" or accuracy". The number 3.0 has the same numerical value as the integer 3. For very long strings of digits, the hardware is inadequate, so we are inevitably forced to a software arbitrary-precision" system to extend the model of numbers. Such a system allows a number of bits in the fraction" or mantissa" of the representation beyond almost any expected useful range, and generally allows a very much larger, and perhaps arbitrary-precision, exponent range. Usually such a system controls the fraction length for all calculations by some global setting, but other techniques can be used to combine numbers with di erent fraction lengths. Some well-known systems Fortran associate precision with the name of a variable. The precision of operations can be controlled by observing the precision of the destination variable, as well as explicit requested precisions along the way. An straight-forward extension of numbers to arbitrary precision does not address accuracy explicitly in the number system, but if it is to be dealt with at all requires auxiliary information to be computed. There is no illusion that the con dence in numbers is related to their representation. The typical technique for increasing certainty in a sequence of calculations is to increase the number of bits in the fraction as represented, improve the input data if possible, and repeat the computation. Arbitrary-precision oating-point arithmetic along this model is implemented in Macsyma, Maple, and Reduce. It is used in packages by Brent 1978 and Bailey 1991. One of the advantages of this system is that it can actually be used for the treatment of inexact values better than in the default system in Mathematica. For example, if x is in the interval 2.94,3.01 , then x lies in the closed interval between the two exact not fuzzy numbers 294 100 and 301 100. Interval arithmetic has a substantial literature and is used in a number of computer packages see Moore 1979. Iterative algorithms and notions of convergence generally must be recast for interval computation. Although Mathematica version 2.0 introduces a notation for intervals, namely RealInterval, a critique of this aspect of the system will have to wait until the bugs are xed and only the features remain. On a related topic, since Mathematica allows machine oating-point numbers, it must deal with the IEEE binary oating-point standard In nity and Not-a-Number NaN representations. One can obtain such an expression by computing 0 0, which Mathematica identi es as Indeterminate. Combining two of these quantities by addition, multiplication, etc., generally yields Indeterminate. So far, so good. Unfortunately, this value has Precision and Accuracy of Infinity, and any two Indeterminates test equal ==. Consequently, regardless of what you might wish, 00 is equal to 0=0. The IEEE 754 speci cation requires that NaN symbols compare as unequal to everything, including themselves. There is also, in Mathematica, a notion of ComplexInfinity, which is returned by the Limit program, even when RealInfinity or, to use a notation included in the system DirectedInfinity 1 would be more appropriate. 22
9.2 Numerics|A summary In summary, although the immediate symptoms arise from the way Mathematica computes the Precision and Accuracy of the result of additions and multiplications, the actual problem is its version of arithmetic turns out to be a bad idea. Making the model more useful would require an admission of the incorrectness of the model being promoted. Wolfram Research is aware of the anomalies above as illustrated by the somewhat ineffective changes in version 2.0 but appears to be unmotivated to change to a model that has fewer problems, such as the one above. Mathematica fails to satisfy a reasonable set of criteria for oating-point arithmetic and dealing with uncertainty in the context of symbolic and arbitrary-precision computations. Certainly identity operations should not ratchet up or down in precision. In a well-designed system, exact numbers should not cause di culties. Systems should not make misleading representations about oating-point numeric data any more than for other data. It should be possible to independently certify numbers to be of some particular accuracy or precision, regardless of the method of their computation. Presumably machine-precision" oating-point numbers should be incorporated in a manner that will be at least as good as numerical subroutine libraries, and allow for fast computation. And special values like 1 should be handled with a view toward preservation of correct computations. Finally, lulling the user into a false sense of security is a far greater defect in a symbolicnumerical computing system than in a purely numerical system. A symbolic system has at least in principle the tools to provide correct answers.
10 Integration
Symbolic integration has always seemed of particular interest in computer algebra systems. Integration can show systems in a good light because it is now easy to compute solutions to nearly all problems encountered in a rst calculus course. Yet, success with such problems does not mean the domain is solved." On the contrary, many important problems cannot be handled either by methods taught to freshman calculus students, or by the more powerful but still only partly e ective algorithms implemented by systems. In an attempt to correct some of the aws in earlier versions, Mathematica 1.2 includes a package for de nite integrate DefiniteIntegr which, when read in, uses piecewise integration and correctly provides the answer to the problem Integrate 1 x^2, x,-1,1 , namely 1, rather than the totally incorrect answer, ,2, it provided previously. But it takes special e ort to get the right answer reading in the library, which is most unfortunate. Also unfortunate is the very large number of known instances in which Mathematica produces incorrect results for purely symbolic rather than numerical integrals. One example is all that space here allows: A computation which incorrectly returns 0 in version 1.2, is the following: Z x t I = 1 +1x, , coscos t dt 2 2x , Actually, the value of this integral depends on the value of the parameter x. For example, if x is set to 1=2, Mathematica 1.2 correctly computes the answer as 2 rather than 0. In a later version of Mathematica, the errors caused by too quickly simplifying the square-root of perfect-squares that are generated in this problem are avoided by not sim-
23
plifying roots. The answer is given as
q 1 , x2 + ,1 + x22 q : ,1 + x22 This solution is too conservative in simplifying; the remedy is apparently to advise the user to apply PowerExpand to simplify and or to commit errors. This has two bene ts for the Mathematica authors: a The analysis necessary to see if p simpli cation can the p be done is now avoided." For example, the system transforms 32 ! 3, but 2 is unchanged. b Any errors in expanding powers are now committed explicitly by the user. Neither of these bene ts is of great use to the customer. Indeed, the most likely situation is that the user is less well equipped than the system to determine the validity of the transformations in PowerExpand. The choice of branches can sometimes be made from the context of the computation, but such context is not generally available in Mathematica. Incidentally, the unsimpli ed integration answer above is still arguably wrong for x = 1. Substituting 1 for x in the answer gives Indeterminate as a result of computing a 1 0 expression. Using the Limit program gives 0 even though the directional limits from above and below give di erent answers of 0 and 2 at x = 1; and nally, if you compute the integral with x = 1 from rst principles the answer is neither 0 nor 2 but .
11 Libraries
The packaging mechanism seems to be an unhappily complex one, but perhaps no worse than that available in most other languages. Common Lisp however seems to have a more e ective technique for organization of modules in its Object System CLOS, Steele 1990. Mathematica is not unique among interactive programs intending to provide access to large libraries of scienti c programs. It does, however, fail to address adequately three issues: 1. Quality of numerical routines: It would be preferable to have only the highest quality numerical routines. Often well-known mathematical formulas su er from disasterous numerical instability, or are unreliable with respect to choice of complex branches. Because Mathematica has arbitrary-precision oating-point numbers, the design of a correct routine for some of its more esoteric functions may have to break new ground or alternatively, be unnecessarily slow or inaccurate. On the other hand, the availability of extra-precise arithmetic may ease the implementor's task. Lacking any documentation, and faced with the evidence of past bug reports, it is di cult to be con dent in the quality of the routines. Occasionally the correctness of their intentions is hard to assess. It would raise the quality of the routines if Mathematica were to adopt en masse the product of several decades of scienti c computing analysis from one of the reputable software libraries. 2. Accessibility of symbolic or other algorithms that might be in disk library les: The situation in version 2.0 has improved somewhat from the past, when programs basically had to be loaded manually before the system had any information about them. Version 2.0 introduces stubs" as a partial remedy. A stub attribute for a name causes at parse time the loading of an associated le de ning that name. Still, the system presupposes that the user knows more about names of appropriate 24
functions and packages than is likely to be the case. A better solution may be di cult, but the problem remains. 3. E ciency in retrieval of the most appropriate data: Mathematica seems to have barely scratched the surface in these matters. For example, the incorporation of large tables of integrals using Mathematica's patternmatcher is problematical in terms of e ectiveness and speed. Although it is plausible to add a handful of special integration rules by patterns, experiments have suggested that adding a large table would require very general patterns whose appropriateness can be eliminated only after the application of side-conditions. The look-up process in such a table is extremely costly and cannot easily be optimized. There are undoubtedly other approaches to incorporating information, but pattern-matching is the technique Mathematica promotes most extensively. The e ectiveness of this in the large" remains to be demonstrated. There may, of course, be alternative approaches developed for library building but Mathematica's current framework for the integration of large amounts of mathematical knowledge is rather disappointing.
12 Plotting
Mathematica's adaptive plotting of curves in a plane is somewhat e ective, choosing more points at areas of high curvative than other places, and not rendering exceptionally distant points. This is not in itself a novelty since, for example, Maple version 4.1 already had such a feature. Undoubtedly any plotting program can be fooled by some functions. De ning functions by rules, logical statements or other discontinuous expressions provides enormous opportunity for generating di cult problems. Even relatively routine-looking expressions can provide di culties. Consider the curve taken from a problem set by W. Kahan to stymie plotting functions, y x := 1+ x2 +0:0125 log j1 , 3x , 1j for 0 x 2. Mathematica does no better than other programs in nding the very narrow slot, in an otherwise parabolic function, that at x = 4=3 dives to ,1. The slot's width is on the order of 10,4 units. Although it is discovered in plotting from 0.0 to 2.0, it is entirely missed if the plot range is 0.1 to 2.0. Adaptive plotting for surfaces would presumably be even more useful, but this is not provided. The surface rendering algorithms seem to be highly e ective, and the chosen default settings often work. When they do not, some study of the options is necessary; substantial exibility is provided. There is a high level of interest in Mathematica simply for its graphics facilities, and the ease with which curves and surfaces can be speci ed. It is not unique in these capabilities since numerous other purely numerical systems such as Matlab Mathworks 1988 have high-quality plotting routines. Using PostScript as a device-independent intermediate form for plots is somewhat novel; other programs including Maple and Macsyma tend to use PostScript only for communicating with hard-copy devices. A feature that is missing from Mathematica, but is present in Milo Avitzur 1988 and Theorist Bonadio 1990 is interactive re-plotting, where one can specify a rubberband" rectangular region of interest and have that section blown up or replotted for more detail. Re-displaying a graph in Mathematica with a di erent PlotRange can rescale and clip out a section, but does not provide new intermediate points to justify the detail. Indeed providing such an interactive facility might be a challenge to Mathematica implementation, given the separation of front- and back-end processing in the design. The linkage, which might in the most general case require guring out what kinds of 25
modi cations the user has inserted in a PostScript text, or where new points must be calculated relative to mouse-positions, may be daunting, and would probably require a rather di erent representation. The interface in Theorist is simple, provides menu-driven or mouse-directed changes such as zooming in, changing the viewpoint on 3-d graphs, or coloration. Maple version V provides a less elaborate setting but also allows for the very useful real-time rotation of surface plots|a feature lacking in Mathematica.
13 Data types and operations
Mathematica is not set up to deal with data types except in a rather super cial way. It has been found quite useful in other computer algebra systems, and indeed in programming languages generally, to have a deeper notion of types. Mathematica supports surface" type checks only, and does not understand their relationships. For example, although an integer is mathematically speaking a special case of a rational number, Mathematica does not recognize this. A pattern for x_Rational will not match 3. Although Mathematica has a program for testing for the super cial appearance of a polynomial PolynomialQ, it does not know that x2 , 1=x + 1 is a polynomial in x. It does not have a polynomial data type" at all. Contrast this to other systems|Maple has a structured type polynomR,V which corresponds to the class of expressions in the variables V whose coe cients are of type R. That is, polynominteger,x speci es univariate polynomials in x whose coe cients are integers. Scratchpad handles types in a di erent way, being strongly typed. A value belongs to only one type in which a clearly de ned set of operations are available. A type is assigned to each expression by the system top-level interpreter if none is supplied by the user; types are an essential part of every manipulation. There is a real dilemma presented by Mathematica's design: Consider what would happen if you were to try to incorporate a data type for compactness of representation or speed. Let us say you wished to write a much faster PolynomialExpand command. On the one hand, the system could be changed to include a polynomial data type. It could be included the way Complex is a data type, or it could be super cially simulated by construction of a new function." That is, one could use Poly x,4,5,6 to represent 4x2 + 5 x + 6 for a discussion of such representations and speed, see Fateman 1991. But then the semantics of the Mathematica system require that a user be able to alter the simpli cation of such components as Plus and Times | how could you assure that user patterns, as well as the built-in evaluation mechanism would always match forms including Poly as appropriate? Since patterns are based on syntactic forms, and the new data type proposal has removed or changed those forms, the pattern mechanism in place will not work. Perhaps a whole new pattern matcher could be built to match the abstraction" of a polynomial? It appears that the complexity of such changes to the matcher and other parts of the system are hidden from the user, who is consequently prohibited from making such important changes. Further subsections here elaborate on the consequences of the design. 13.1 Expressions Mathematica makes no use of the well-de ned type structure that is easily imposed on a major part of the mathematical domain of the system. The manual claims that every object is an expression which can be used anywhere in the system. This is a considerable oversimpli cation of what in fact is provided, but this kind of belief has probably led the implementors to neglect checking for appropriate classes of data when needed. Such checking is, 26
in many cases, rather di cult because of the system design. Sometimes expressions either make no sense, or have to be treated rather di erently depending upon details which may be unexamined in general. For example, consider a function f x with a discontinuity at 0. Take the limit of f x as x approaches 0. Version 1.1 of Mathematica was clearly unprepared for the task since it proceeded to compute what was reasonable for analytic functions, but unreasonable here. That is, it computed a Taylor series. Limit If x==0,1,0 ,x- 0 yielded If O x ^4+False*x+O x ^3+Equal^0,0, 0,0 ,1,0 . This kind of error can be xed on a case by case basis, but the notion that the Limit evaluation program should be constructed as a collection of manipulations of data-representations, without mathematical semantics, was, and still is, fundamentally inadequate. The next version of the system xed the particular example of the limit of If, but without addressing the general problem. As a somewhat random example, but anticipating the next section, consider Limit O x ^4,x- 3 . Probably this should be unde ned. Mathematica 1.1 gave 0. Version 1.2 gives Indeterminate. Version 2.0 leaves the expression unevaluated. Resolving some of these questions once and for all" is hindered by the fact that Mathematica does not have a canonical simpli cation policy see x8.1, also Moses 1971. The Simplify command initially worked at most for algebraic not transcendental, exponential, or log expressions. Even so, --1+q 1+q withstood Simplify, and needed ExpandNumerator to reduce it to 1-q 1+q. In version 2.0, Simplify is more powerful. Yet, how many ad hoc simplify-like commands are needed? In some other systems, the appropriate use of data structures would produce simpli ed expressions as a consequence of the abstraction and representation in use. Operations on these forms would be well de ned. 13.2 Series Although Mathematica uses a single format for expressions based on pre x trees, at least one sub-area is specialized, and uses the standard form to encode data more like a list of terms. These are expressions with the head Series. Although it is my view that special mathematical notions can well deserve special forms, this one was integrated into the system in a hazardous way. Given two Series that agree to a certain order, their di erence should be equal to the next omitted term. Yet in version 1.2 the expression comes out 0, rather than Ox^3. Correcting version 1.2 by putting a rule on Subtract produced errors in other parts of the system. N Series ... fails. This bug was xed in version 2.0, but its presence is indicative of the kinds of patches that will haunt the system unless there is a more formal approach to special data types. A new data type in version 2.0, RealInterval promises to be a major problem in this regard, since all the operations of the system must be extended to a new set of objects for which numerous built-in assumptions fail. Even the latest version I've tried, which says correctly that Sin O x ^4 = O x^4 and Sin Pi 2+O x ^4 = 1+O x^4 , erroneously believes that Cos O x ^4 =1. Certainly these are bugs to be ironed out, but is there an end in sight? The semantics for big-O notation see, for example, Graham et al. 1989 can be dealt with formally, but such a formal treatment necessarily points out that the use of the big-O notation is inconsistent with the usual meaning of =". Given f x = Ox and g x = Ox, Mathematica correctly says that f x -g x is Ox but mistakenly believes that f x ==g x . 27
Series Sin x , x,0,2 - Series Tan x , x,0,2
13.3 Other Datatypes If another type of data structure such as Poisson series|a structure used in celestial mechanics that can be thought of as similar to Fourier series|were to be introduced, how would the addition of two di erent series be handled? Or even the addition of two series in di erent variables or about di erent expansion points? These are touchy areas and cannot be dismissed by allowing the user to guess which of several possible mathematical conventions might hold. The informal use of rules to de ne combinations is quite error prone and ine cient. A further di culty is that any operations on novel data-types left unde ned may fall into the pit of Mathematica's notion of generic operations. Leaving the current set of generic operations unaltered is risky; on the other hand, modifying the generic operations requires skill and may slow the system down substantially. To be practical, it may also require access to proprietary code. Is there a good solution? The correspondence between mathematical concepts and data abstractions, and then between abstractions and representations, has a rather substantial literature, especially among the Scratchpad implementors. Formal systems take this seriously, and there are successes to be observed. Multiple representations in Macsyma add to its power and complexity. While each of these approaches has problems, Mathematica seems to have taken a very super cial step toward integrating the creation of new data structures into the system. It could do better.
14 Debugging
While it is not our primary objective to complain about particular hardware software implementations of Mathematica, when a system design itself may be interfering with the ability to debug programs, it rises to the level where it should be noted in a general review. In the past, our experience in debugging user programs with the standard kernel on several early versions of the system were quite negative. If you succeeded in interrupting the computation from the keyboard, something that was not always possible, you were thrown into a break in which you could not examine values, compute values, or do anything but print the name of the usually internal program that was interrupted, or the sequence of programs without arguments, being executed. One had the choice sometimes of continuing, aborting the computation or exiting from Mathematica. These options and these pieces of information were meager. Beginning in version 1.2 and continuing in 2.0, a redesign of debugging has changed the situation. Now the On tracing command ordinarily produces excessively verbose data. By using Trace rather than On, a user may lter this ood: it is possible to provide a pattern to compare with the forms so that only matching expressions will be printed. This requires a sensitive touch and some prescience about what will be relevant to see. It is apparently possible to still nd non-interruptible loops in Mathematica, in which case no debugging is really possible, and a crash with loss of data" is about the only way to halt a computation. Finally, the spelling correction facility can painfully slow, especially when rst inadvertently invoked and especially on smaller-memory systems. Mercifully it is possible to disable it.
15 Abuse of Mathematics
While many of Mathematica's intellectual ancestors make logical hash of mathematical ambiguities or boundary values, probably no other system has documentation so bold as to assert that the system, rather than mathematical tradition, should determine the 28
meaning of for example multiple-valued inverses Wolfram 1988 page 358 or that it is the human user of the system who has primary responsibility to check the input and perhaps the output of the system Wolfram 1988 page 425: You have to be careful, however, when the integration region contains a singularity." . While one can be philosophical about this and to quote W. N. Venables, Like cars and knives and most other useful things, symbolic manipulation systems in general, and Mathematica in particular, are inherently dangerous and not for the reckless." it is certainly possible to include some safety measures. The competitors to Mathematica are not without fault; they tend, however, to be more cautious. For example, Macsyma checks to R see if n = ,1 before returning the value xn+1 =n + 1 for xn dx. Mathematica does not. DERIVE nesses this problem by returning xn+1 , 1=n + 1 which yields the correct limit as n ! ,1.
16 Which weaknesses are easily xed? Which are permanent?
Certainly one can nd errors of implementation, or ine ciencies, or situations in which programs in Mathematica were written to be e cient but insu ciently general. Improving Mathematica's algorithms may x certain problems, and if past experience is any guide, occasionally will insert new ones. In well-de ned areas such as the division of arbitrary-precision integers, the factorization of polynomials over the integers, and so on, one assumes that past identi ed bugs in Mathematica either have been or will eventually be repaired. Thus, other than reporting and classifying known problems of this kind to some maintenance person, and perhaps devising work-arounds, the customer has little choice but to wait for the x to arrive. This is, of course, the situation with almost any piece of software that is not available in source code; and even with source code, most people would not be well equipped to nd and repair bugs. The more di cult problems have to do with errors of design, implementation errors that are particularly widespread in the code and whose xes have major impact on the speed of the system, or areas where retraction of claims in the documentation are needed. Such areas are reviewed below. 16.1 Errors of Design These areas have been mentioned previously. Scope of names: blocks, packages, rules In addition to problems already indicated with local programming variables in Block which is xed by using Module, Mathematica seems to have a problem dealing with quanti ed mathematical variables. Most mathematicians would agree that the use of the literal x in limx!a f x is locally bound, and that this expression is entirely equivalent to limz!a f z . The puzzle is how to specify the result of z = f x followed by Limit z,x- a . Is the x inside f x the same as the x inside the Limit? Although each of several built-in constructions using bound variables may be patched eventually, there is still confusion inherent in Mathematica's language concerning local and global programming variables and mathematical indeterminates with the same printed name. The de nition of Hold and related commands now including Evaluate, ReleaseHold, and Unevaluated suggests more than anything else that the system is still out of control. 29
In nite Evaluation This technique is too expensive to apply correctly, and somewhat haphazard when implemented heuristically. It appears to be one of the more negative and unforeseen consequences of relying heavily on rules as though they were procedure calls. Model of numerical calculation It is possible to perform arithmetic on approximate quantities of di erent precision by using the wider precision, and padding the lower precision with zeros of whatever radix is being used in the representation. It is also possible to truncate or round the wider precision to be cruder, and perform only the crude arithmetic. As we have illustrated, Mathematica's attempt to predict numerical errors is awed. Furthermore, without interval arithmetic implemented in a sensible and subtle way, the problem of constants" becomes intractable instead of merely occasionally p unsolvable. Such questions as the resolution of x2 ! x signRealx cannot be handled until this problem is solved together with at least linear inequalities. Stark data model The data model is that of a basically uninterpreted tree. Mathematica does not support representing and computing with an expression which is to be treated as for example real-valued. It has no support for declaring" that x can assume only positive integer values, and that for example x 0 or x x + 1 is an even integer. If it is told that x 0 it does not know that x 0 is false, nor will it object to setting x to a complex number. At least primitive versions of such representations are necessary if the system is to proceed automatically in simplifying expressions such as Sqrt x^2 . Even though piecewise-de ned functions are a basic representation scheme for introducing new functions, there is no useful way of di erentiating such functions, even if they are piecewise di erentiable. Since only surface types are handled, Mathematica often does not hesitate to respond with meaningless results when given unexpected, but not necessarily incorrect, inputs. Because features which are accidents of the implementation are mostly undocumented and are not necessarily mathematically consistent they are presumably subject to undocumented revision in the future. It would appear possible in principle to add piecemeal all the rules that might be needed to provide the equivalent of the Macsyma assume" facility Mathlab Group 1983. A careful examination of such a programme quickly leads to the conclusion that much of the built-in functionality of the simpli er component of Mathematica would have to be papered over, and the e ciency would be very poor. Lack of canonical forms While we have earlier indicated problems with the decidability issues in Mathematica's simpli er, there is another consideration worth mentioning under the general topic of canonical forms." In fact, for very signi cant types of calculations, especially those with more structure than polynomials, it is sometimes imperative to provide special canonical compact encodings for e cient processing. A prime example of this is Poisson series, used for computations in celestial mechanics and areas where computations with similarly large many thousand terms sine-cosine series are important. Of the general purpose" computer algebra systems, Macsyma is apparently still unique in incorporating alternative encodings comparable to those used in special-purpose systems. 30
Other systems including Reduce and Maple, as well as Mathematica can simulate" the form of a Poisson series using a general tree-like representation similar to that used for every other object, but this simulation is orders of magnitude slower and uses many times more storage than necessary special purpose systems such as Schoonschip are described in Buchberger 1983. In Mathematica there are no canonical compressed data forms comparable to Macsyma's CRE canonical rational expression form, or Poisson series. The only case where a somewhat compressed form is used is Mathematica's Series form. This is built out of the usual list structure, but is printed as a sum with a big-O" term. This form is a mixed blessing because it is poorly integrated into the system; its problems suggest that integration of any additional special forms will also cause design di culties. Reliance on the user The user is cautioned, You have to be careful, however, when the integration region contains a singularity." Wolfram 1988, 1991. Actually, it would be helpful if the system were more careful, and not only for integration. In version 2.0, the PowerExpand function is provided so that the user can commit errors explicitly that in version 1.2 were committed automatically. Unfortunately the user may be quite unprepared to determine the necessary information about signs of subexpressions in ranges that determine the legality of the transformations. M. Monagan points out that for some functions, e.g. LinearSolve, one can pass a ZeroTest to the function. It doesn't stop Mathematica from returning wrong answers, but again merely passes the bug" on to the user. Defective model of equality Often identities, invariants, or axioms don't hold in Mathematica. If a==b or the even stronger assertion that a===b is True, then for any deterministic, side-e ectfree function f, once should expect that f a ==f b . Mathematica does not match this expectation. Documentation Although it may seem that the documentation Wolfram 1988, 1991 is quite a superior part of the Mathematica package, there are subtle omissions. For example, the precise algorithms used for most of the routines are not described or even hinted at. The rationale given by Wolfram March, 1991, in a lecture at the University of California at Berkeley was that such lack of speci cation was a positive feature. Speci city would hinder the implementors|they would not be able to replace an old routine with a new one with di erent characteristics e.g. time complexity. Wolfram asserted that it was more important to implement new features than document the old. These arguments are not especially convincing: the same arguments were o ered in the context of Macsyma's reference manual twenty years ago, and the documentation is still lacking. By contrast, Maple system documentation of algorithms is extensive and on-line. 16.2 Errors of Implementation Non-uniformity of approach. Although much of Mathematica could in principle be written in its own" language, a barrier has been placed in the implementation for reasons of e ciency and security. The user has little chance of altering the behavior of the system in fundamentally e cient ways. The only handles available are those provided by using ags already 31
anticipated by the developers. By contrast, even though Maple is also a proprietary system, most of the source code is available for examination, and is in a language the user can write in. Full-evaluation and Update. As previously noted, it appears that this is implemented with heuristics that are sometimes wrong in order to be fast. Even so, it is probably slower and certainly harder to explain than single evaluation. Bugs and Debugging. There seem be a large number of barriers to e ective debugging of user-written code and rules because of the many ways rules can interact. A programmer is welladvised to make frequent use of the Clear command to remove the e ects of partially debugged code during development. Complexity of code. It may be di cult and expensive for WRI to repair design defects that are only now being recognized. There are some 180,000 lines of C code in version 1.2 and more than 350,000 in version 2.0. If each programmer must become familiar with a substantial portion of this corpus before contributing to it, the prospect of real improvement will decline, and the prospect of introducing bugs will increase. A complex environment for developers cannot be healthy. By contrast, the comparable system Maple has a kernel of about 20,000 lines of C, with most of the higher-level command capability de ned through programs which can be easily printed in source-code form as part of the on-line documentation. In the course of evaluating Mathematica, we at Berkeley have generated voluminous correspondence nearly one megabyte|450 text pages reporting bugs in version 1.1, 1.2 and beta releases of 2.0. Many bugs survive in 2.0, and additional ones have appeared. Although we have had version 2.0 beta only a few months, about 1=3 of the bug reports are on this beta system. Some bugs are, of course, simple to repair or inconsequential. Others may be quite critical to applications. 16.3 Unjusti ed Claims From the academic point of view, this last type of error has great potential for damage: there are many problems for which Mathematica appears to claim complete solutions, but is not even as good as other programs for example, in symbolic integration, solution of equations, simpli cation, numerical evaluation. Mathematica certainly does not know" all of mathematics, nor is it apparent that it could form the basis for a program that approached such a lofty goal. A user of this system might erroneously believe that if Mathematica cannot solve a problem, no other program or human, perhaps can do so. Users of Mathematica may prematurely abandon the technology because the rst system they tried was insu cient. This phenomenon was prevalent among early FORMAC users see Tobey et al. 1965, who typically ran out of memory on the 32K word IBM 7094 rather rapidly, and assumed that if this quite powerful system for that time couldn't solve their problem, it was not solvable. In brief, there is a risk that the audience for symbolic mathematical computation will believe that whatever Mathematica has done, has been `done' as best as possible" and thereby believe that any shortfall is inherent in the technology, and not the particular program. 32
On the other hand, the general awareness of symbolic computing has been vastly improved, and this may be more than fair compensation to academics or others interested in the eld.
17 Conclusions
This review is hardly the nal word on Mathematica, a program we expect to continue to change even as it has changed substantially during the writing of this commentary. In fact, we hope that some improvements in the program were prompted by earlier drafts of this paper. In the interests of brevity, almost all mentions of earlier bugs now xed have been dropped. In a few places some discussion of features of obsolete" versions of the system remain for the following reasons: In some sense, every user has an old" version. The as-yet-unreleased version as we write, 2.0 is not yet generally available cannot be reviewed satisfactorily. Looking at the aws of the past gives some insight into the aws of the future. It is possible that having raised the consciousness of the public to symbolic mathematics, the Mathematica program will then also evolve to satisfy all the various criticisms indicated here, as well as other criticisms. Alternatively, or in addition, commercial or academic rivals" may provide new or better solutions to these problems. For those persons waiting eagerly for mathematicians to be replaced by a universal computer program that does mathematics", it is our opinion that this will require the development of technology that does not yet exist. Continuing research should learn from the Mathematica experience in combining symbolic mathematics in a general scienti c information and programming environment with applications for research and development, teaching, and even entertainment.
18 Acknowledgments
Opinions expressed in this paper are the author's and do not necessarily represent the views of government sponsors or others mentioned below. The author wishes to thank numerous persons for enlightening discussions, comments on earlier drafts of this review, and access to preprints or technical reports. These include Paul Abbott, Bruno Buchberger, Robert Campbell, Bruce Char, Steven Christensen, Gene Cooperman, James H. Davenport, Sam Dooley, David Jacobson, G. H. Gonnet, R. W. Gosper, Dan Grayson, W. Kahan, Silvio Levy, Roman Maeder, Kevin McCurley, Kevin McIsaac, Michael Monagan, Steven Omohundro, Malcolm Slaney, Neil Soi er, Ilan Vardi, William N. Venables and Stephen Wolfram. Also thanks to the four anonymous referees and the editors of JSC for extremely helpful comments, as well as patience through a number of intermediate revisions of sections. A preliminary copy of this paper was provided to Wolfram Research Incorporated WRI, and the resulting comments were used in re ning this version. The author would also like to thank WRI for providing access to beta-test versions of Mathematica 1.2 and 2.0.
Arnon, D., Beach, R., McIsaac, K. and Waldspurger, C. 1988 CaminoReal: An interactive mathematical notebook, in Document Manipulation and Typography: Proc. Intl. Conf. on Electronic Publishing, Document Manipulation, and Typography J.C. van Vliet, ed., Nice, France, April 20 22, 1988, Cambridge University Press. 1 18.
19 References
33
Avitzur, R. 1988. Milo a Macintosh computer program Paracomp Inc. San Francisco, CA. Milo has been incorporated in FrameMaker, Frame Technology Corp. San Jose CA. 1990. Bailey, D. H. 1991. MPFUN: A portable high performance multiprecision package. NAS Applied Research O ce, NASA Ames Research Ctr. Mo et Field, CA. Barton, D. and Fitch, J. P. 1972. A review of algebraic manipulation programs and their application. Comput. J. 15, 362 381. This is an extended abstract of: Applications of Algebraic Manipulative Programs in Physics, in Rep. on Prog. in Phys.35, no. 3 235 314. Bonadio, A. 1990. Allan Bonadio, Theorist a Macintosh computer program, Prescience Corp. 939 Howard St. San Francisco CA. 94103. 1990, 1991. Buchberger, B., Collins, G. E., Loos, R. eds. 1983. Computer Algebra: Symbolic and Algebraic Computation, Springer-Verlag. Buchberger, B. 1991. Grobner bases in Mathematica: Enthusiasm and Frustration. RISC-LINZ Report 3-3, J. Kepler Univ., A-4040 Linz, Austria. Brent, R. P. 1978. A Fortran multiple-precision arithmetic package, ACM Trans. on Math. Softw. 4 no. 1, 57 70. Computer Algebra Group. 1988. The Scratchpad II computer algebra system interactive environment users guide, Draft 1.1 July 19, 1988 Mathematical Sciences Dep't, IBM Research Division, T. J. Watson Res. Ctr. Yorktown Hts, NY. 382 pages see also Jenks, R. D., Trager, B. M. 1984. A primer: 11 keys to new Scratchpad. Proc. Eurosam 84, Lecture Notes in Computer Science 174, Springer-Verlag. Cooperman, G. 1986. Semantic matcher for Macsyma. Proc. 1986 ACM Symp. on Symbolic and Algeb. Comp., 132 134. Fateman, R. J. 1976. The MACSYMA Big-Floating-Point arithmetic system. Proc. of the 1976 ACM Symp. on Symbolic and Algebraic Computation, 209 213. Fateman, R. J. 1989. A review of Macsyma. IEEE Trans. on Knowledge and Data Eng. 1, no. 1. 133 145. Fateman,R. J. 1991. FRPOLY: A benchmark revisited. Lisp and Symbolic Programming, 4 153 162. Fenichel, R. 1966. An on-line system for algebraic manipulation. Ph.D. dissertation, Harvard Univ., also Report MAC-TR-35, Project MAC, M.I.T., available from the Clearinghouse, document AD657-282. Foderaro, J. K. 1983. The design of a language for algebraic computation systems. Ph.D. dissertation, EECS Dep't., Univ. Calif., Berkeley. Foster, G. DREAMS: Display REpresentation for Algebraic Manipulation Systems. Rpt. UCB CSD 84 193, Computer Science Div. Univ. of Calif, Berkeley. Foster, K. R., Bau, H. H. 1989. Symbolic Manipulation Programs for the Personal Computer. Software review Science 243, 679 684. Golden, J. P. 1977. The evaluation of atomic variables in Macsyma. Proc. 1977 Macsyma Users' Conf. Univ. of Calif, Berkeley. 109 122. Graham, R. L., Knuth, D. E., Patashnik, O. 1989. Concrete Mathematics, Addison-Wesley Publ. Co. Greif, J. M. 1985. The SMP pattern matcher. Proc. Eurocal '85, vol. 2, Lecture Notes in Computer Science 204, Springer-Verlag. 303 314. Hearn, A. C. 1984. Reduce 3 User's Manual, The RAND Corp. P.O. Box 2138, Santa Monica CA 90406. Hearn, A. C. 1976. A new REDUCE model for algebraic simpli cation. Proc. 1976 ACM Symp. on Symbolic and Algebraic Computation, 46 50. Herman, E. A. 1988. Review of Mathematica. also, discussion by Barwise, J., Uhl, J. Jr., Zorn, P. Notices of the AMS 35, no. 9, 1334 1349. Hoenig, A. 1990. Mathematica, a program for various work stations and personal computers. Review Math. Intell. 12, no. 2, 69 74. Itturiaga, R. 1967. Contributions to mechanical mathematics. Ph.D. dissertation, Comptr. Sci. Dep't., Carnegie-Mellon Univ., Pittsburgh, Pa. Jenks, R. D. 1976. A pattern compiler. Proc. 1976 ACM Symp. on Symbolic and Algebraic Computation, 60 65. Keiper, J. 1990. Numerical Computation. Tutorial Notes. Mathematica User Conference, Redwood City, CA. Knuth, D. E. 1969. The Art of Computer Programming, vol 1. Fundamental Algorithms, AddisonWesley Publ. Co. Korsvold, K. 1965. On-Line algebraic simplify program. Stanford A.I. Project Memo 37. Kudera, J. R. 1988. Physics made easy. letter to the editor, Fortune May 25, 1988.
34
Maeder, R. 1988, 1991 Programming in Mathematica. 1st, 2nd edition. Corresponding to Mathematica versions 1.2 and 2.0 resp. Addison Wesley. Mathlab Group. 1983. Macsyma Reference Manual, Lab. for Comp. Sci, MIT, Jan, 1983 2 volumes: version 10, available also from the National Energy Software Center NESC, Argonne, IL. Similar manuals are available from Symbolics, Inc., for example, version 11 Symbolics, Inc. Oct. 1985. McCurley, K. S. 1988. Book review Wolfram, Stephen 1988 Mathematica: A System for Doing Mathematics by Computer. ORSA J. on Computing 2, no. 4. 366 368. McIsaac, K. 1985. Pattern matching algebraic identities. SIGSAM Bull. 19, no. 2. 4 13. Mathworks Inc. 1988. Matlab a computer program. S. Natick, MA. Moore, R. E. 1979. Methods and Applications of Interval Analysis. SIAM, Philadephia, PA. Moses, J. 1971. Algebraic simpli cation, a guide for the perplexed. Comm. ACM. 14, no. 8. 527 538. Moses, J. 1974 Macsyma: the fth year. Proc. Eurosam 74, Stockholm, Sweden, ACM SIGSAM Bull. 8, no. 3. 105 110. Moses, J. 1977. The variety of variables in mathematical expressions. Proc. 1977 Macsyma Users' Conf. Univ. of Calif, Berkeley. Pavelle, R., Wang. P. S. 1985. Macsyma from F to G. J. Symbolic. Comp. 1, no. 1, 69 100. Pratt, V. R. 1973. Top down operator precedence. 1973 ACM Symposium on Principles of Prog. Lang., Boston, MA. See also, a detailed memo 1977 CGOL|An algebraic notation for MACLISP users, distributed with the CGOL source code. Simon, B. 1990. Four computer mathematical environments. Notices AMS 37, no. 7. 861 868. The Soft Warehouse 1991. Derive, a computer program. version 2.03 The Soft Warehouse, 3615 Harding Av., Honolul HI 96816. Soi er, N., Smith, C. J. 1986. MathScribe: A user interface for computer algebra systems. Proc. 1986 ACM Symp. on Symbolic and Algeb. Comp., 16 23. Steele, G. L. Jr. 1984, 1990. Common LISP the Language, Digital Press. 1st ed., 2nd ed. Symbolic Computation Group. 1990. Maple a computer program version V. Computer Science Dep't., Univ. of Waterloo, Waterloo, Ontario, Canada. This program is sold by Waterloo Maple Software for most computers except for the Apple Macintosh. The distributor for Macintosh is Brooks Cole Publishing Co., Paci c Grove, CA 93950 Taubes, G. A. 1988. Physics whiz goes into biz. Fortune, April 11, 1988. 90 93. Tobey, R. G., Bobrow, R. J., Zilles, S. N. 1965. R. G. Tobey, R. J. Bobrow, and S. N. Zilles. Automatic simpli cation in Formac. Proc. AFIPS 1965 Fall Joint Comput. Conf., 37 52. van Hulzen, J. A., Calmet, J. 1983. Computer algebra systems. in: Buchberger et al., 1983 221 224. Vogel, W. K. 1989. Mathematica 1.1. Biotechnology Software. Mary Ann Liebert Inc. Publ. NY July August. 2 7. Wolfram, Stephen. 1988, 1991. Mathematica|A system for doing mathematics by computer, Addison-Wesley, 1st ed., 2nd ed. Wyatt, W. T. Jr., Lozier, D. W., and Orser, D. 1976. A portable extended precision arithmetic package. ACM Trans. on Math. Softw. 2, no. 3. 209 229 |
Elementary Number Theory and Its Applications
9780321237071
ISBN:
0321237072
Edition: 5 Pub Date: 2004 Publisher: Addison-Wesley
Summary: Elementary Number Theory and Its Applicationsis noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. Computational exercises and computer projects are also provided. In addition to years of use and professor feedback, the fifth edition of this text has been thoroughly checked to ensure the quality and accuracy of the m...athematical content and the exercises. The blending of classical theory with modern applications is a hallmark feature of the text. The Fifth Edition builds on this strength with new examples and exercises, additional applications and increased cryptology coverage. The author devotes a great deal of attention to making this new edition up-to-date, incorporating new results and discoveries in number theory made in the past few years.
Rosen, Kenneth H. is the author of Elementary Number Theory and Its Applications, published 2004 under ISBN 9780321237071 and 0321237072. One hundred twenty Elementary Number Theory and Its Applications textbooks are available for sale on ValoreBooks.com, eighteen used from the cheapest price of $15.35, or buy new starting at $39 |
MAT-098 is a review of the mathematics skills that are needed for college level mathematics.The course covers operations with whole numbers, integers, fractions, decimals, percents,ratios, proportions. The course put emphasis on problem solving, reasoning, connections(relation of math concepts to other disciplines), communicating mathematical ideas, and number and operations sense.
Credit not applicable toward total credit graduation requirements
.Eligibility to enroll in this course is based on placement examination (CPT).
Learning Outcomes:
By the end of this course, students should have achieved the following learning outcomes:
1.
Students will demonstrate improvement of their pre-algebra skills by increasing their pre-test scores by at least 25% during the post-test.2.Students will demonstrate mastery level on each of the main concepts of the course (seebelow the Course Curriculum) by earning at least 70% on the assessments (i.e. quizzes)scheduled for each main concept.3.Students will demonstrate college readiness to handle the rigor of the next mathematicscourse by attaining a score of at least 70% in the post-test.
Course #Course NameCreditClass Schedule
MAT 098
Arithmetic Review 3 Credits
Wed. 8 – 9:15 AMBlended course:
Computer LabAEC and Online
New students (those who never enrolled previously in basic skills math courses)can exit this course and be placed in the next higher level math course if they score90% in the Initial Assessment that will be administered during the first week of the course.
4.
Students will demonstrate higher confidence and motivation doing mathematics byscoring on average of at least a point higher in the Likert scale administered at the startand end of the course.
ALEKS is a web-based assessment and learning math system that uses artificial intelligentprogramming to provide an individualized learning experience for every student. Theinstructional model of this course will mainly consist on students actively learning at their ownpace with the assistance of ALEKS, the online resources available in Blackboard, and the face-to-face classroom meetings.model. In a blended course, students complete the majority (60%) of the learning activitiesonline (i.e. Blackboard and ALEKS), and the other learning activities (40%) takes place in theface-to-face classroom. Here is what students should expect in this course:
Face-to-Face Meetings
: Class will meet
once a week
in the classroom (mathcenter), where students will complete learning activities led by the instructor. Studentswill complete a
class project
in every face-to-face meeting that must be
published
onthe Internet. In addition, students must visit the math center during instructor's officehours to take the course assessments (quizzes & final exam).
Asynchronous Online Meetings
: Students will login at least on two differentdays to the course shell in Blackboard and participate in the online discussions that areposted in the discussion board.
Computer assisted instruction
: A learning and assessment web-based system(i.e. ALEKS) is used to help students grasp and master the course content. ALEKS willassess students' prior knowledge of the course content and create a visualrepresentation (i.e. pie chart) of what they know and need to learn. Based on thisassessment, students work on the topics they are ready to learn. Students receiveimmediate feedback for their performance and are continuously assessed to guaranteemastery and retention of the course content.
Self-paced learning
: Students complete the course content at their own pacebased on their prior knowledge of the math concepts and skills covered in the course,and with the guidance of a suggested timeline available in this syllabus. Students havethe opportunity to complete two courses in one academic term.
Online Learning Resources
: ALEKS provides detailed explanations anddemonstrations of the concepts and skills covered in the course. It also providessupplementary resources such as videos, animations, Power Point presentations, mathdictionary, and the course textbook (i.e. e-book). In addition, students have access inBlackboard of additional instructor-made resources (i.e. handouts, Power Points,screencasts, etc.) and math links to other Internet sites that provide tutorials, virtualmanipulatives, and multimedia materials.
Available Assistance
: Students have many alternatives to seek assistance tosucceed in this course: (a) Visit the math center to get individual assistance from theinstructor (see office hours info); (b) Visit the math center during business hours to signup for a tutoring session; (c) Ask questions using the Question thread in the discussionboard of Blackboard (questions will be answered within 24 hours); (d) Contact theinstructor by phone Sunday to Thursday from 5 to 8 pm.
Course Curriculum
In this course, students will cover the following main concepts:
Whole Numbers (55)
Fractions (30)
Decimals, Proportion, Percents (40)
Measurement and Data Analysis (27)
Geometry (26)
Signed Numbers and Intro to Algebra (43)
The main topics are represented visually with a pie chart in ALEKS. Students meetthe course objectives of each main topic when they have filled the slice thatrepresents the main topic. In total, students must master 221 objectives (or topics)to pass this course. To see a detailed distribution of the 221 topics by mainconcepts, please visit the site select the course
Basic Math
.
A dark color in the pie represents how much studentshave mastered a particular main topic, and a light color represents how much of the main topicstudents still need to master.
The course textbook is an important source of reference to help students masterthe course concepts. Each main concept (slice of the pie) consists of the followingchapters in the course textbook: |
Pre-Calculus Teacher Resources
Find Pre Calculus educational ideas and activities
Title
Resource Type
Views
Grade
Rating identify positive and negative angles using the Unit Circle. In this pre-calculus lesson, students identify the different angles on a unit circle using the coordinate plane and the four different quadrants as a guide. They calculate the basic trigonometric functions.
Students explore scatter functions. In this pre-calculus activity, students model data, evaluate the function and use their model to interpolate or predict end behavior of the function. The activity employs the use of a graphing calculator.
Students explore optimization problems. In this Pre-Calculus/Calculus/Algebra II lesson, students use Geometer's sketchpad to investigate the dimensions of the largest rectangle that can be inscribed under a parabola. Students use their findings to solve optimization problems.
Students are introduced to the basic trigonometric identities. Using a diagram, they discover why the parts of the unit circle as named as they are and use equations to finalize the Pythagorean trigonometric identities. They also review basic algrebra terminology that could be useful in the present.
High schoolers practice using graphing calculators and spreadsheets as they explore numeric limits using sequences and functions. They complete a sequencing worksheet, and determine which sequence corresponds to story a story called Froggy and Wanda.
Students investigate step functions. In this Algebra II/Pre-calculus lesson, students examine various taxi fare schemes and model the fares with step functions by making use of the calculator's greatest integer function's step function properties.
Students explore an optimization problem. In this Pre-Calculus instructional activity, students investigate a problem in which the distance between three points is minimized in order to determine the location of a switchboard. Students use Geometer's Sketchpad to explore and generate data, and use that data to hypothesize a function to describe the problem.
Students will solve exponential function problems, graph exponential functions, find the value of logarithms, determine future value, and compound interest. In this Pre-calculus lesson, students will use the properties of logs to solve test problems.
In this pre-calculus learning exercise, student solve 35 multiple choice problems and then place the answers in a Sudoku puzzle form. This learning exercise would be a great warm up or fun test review.
When it comes to rating educational calculators, this calculator is always near the top of this list. Now it's available as an app. There is a lot of calculator power wrapped up in this app. Not only is this a fully functioning calculator, it is also a computer algebra system (CAS).
We landed on the moon with less computing capabilities than you can find in this app! Here is a multiple function calculator that takes all the power you get from a handheld and adds the wonderful large screen that only a tablet can provideThis is one of those apps that will impress you right from the start with its ability to recognize handwriting, generate graphs, and solve equations. Then, the more time you spend using this app, the more impressed you will become. |
EasyA Maths Edition: Higher Level Paper 1 by Robert McCarthy
Price:
$5.00 USD.
Approx. 6,520 words.
Language:
English.
Published on February 21, 2012.
Category:
Nonfiction » Education and Study Guides » Study guides - Mathematics.
EasyA Maths Edition: Higher Level is a complete summary of the maths higher level Junior Certificate exam. It contains all the information that students need to do well in their Junior Certificate exam. The information is condensed and easier to understand and it saves students time as they dont have to search a maths books for the information they want to know.
Fundamental Math by Russell Pead
Price:
$14.88 USD.
Approx. 2,100 words.
Language:
English.
Published on February 14, 2012.
Category:
Nonfiction » Education and Study Guides » Study guides - Mathematics.
Fundamental Math provides a simple and straightforward way to learn how to solve Calculus problems from those attempting Calculus for the first time to those relearning it for a practical application. This works very well for those preparing for college as well.Middle School Math Solution: Algebra and Number by Jing Guan
Price:
$39.99 USD.
Approx. 77,460 words.
Language:
English.
Published on October 27, 2011.
Category:
Nonfiction » Science and Nature » Mathematics.
Math is a special and important learning in education. Even though Math is hard to some people, it is not hard to learn if you follow a good guide. This book is a good guide that will help high/middle school students learn basic and advanced skills with important concepts and skills carefully designed into questions and solution for students to master. This book will escort you to your success. |
1.2 Talking and writing mathematics Communication is as vital in mathematics as in any language. This unit will help you to express yourself clearly when writing and speaking about mathematics. You will also learn how to answer questions in the manner that is expected by the examinerBecause of the schooled culture we have grown up in, we are likely to recognise language, mathematics and science as distinct 'ways of knowing'. The words 'language', 'mathematics' and 'science' probably prompted you to think first of the school curriculum, where they are often treated very separately from each other. One of the intentions of the unit is to explore and develop your understandings of these three subjects, which means that, inevitably, we wil Knowing mathematics Reflection on mathematics You're about to start a course in science and technology and you're wondering whether your level of maths is going to be enough to get you through. This unit will show you how to reflect on what you know, identify which skills you might need for your course, and help you to learn those skills using worked examples and activities Your Work on your study mathematics? Mathematics and you RecognizingRecognizing mathematics, continuedAdvanced Mathematics for Secondary Teachers: Course Portfolio The course the portfolio describes is a capstone course in mathematics primarily aimed at future high school mathematics teachers. Bennett decided to write a course portfolio for this course as a way to pass the course along to other faculty members that will teach it in the future. Thus, the central purpose of this portfolio is to be a course record, suitable for other faculty members in the department to use as the main resource when they teach the course. Author(s): No creator set
License information
Related content
No related items provided in this feed
The Development and Use of Representations in Teaching and Learning about Problem Solving: Exploring Tim Boerst has explored instructional approaches that foster the development of representational skill and routine use of multiple representations in problem solving. In particular he has used the 'Rule of 3' (a structure employed in calculus reform materials that highlights the use of numerical, algebraic, and/or graphic representations in mathematical learning) to see whether an emphasis on multiple representations would deepen mathematical learning opportunities for a wide variety of students Author(s): No creator set
Mathematics Under the Microscope This book inevitably asks the question "How does the mathematical brain work?" The author tries to reflect on the explosive development of mathematical cognition, an emerging branch of neurophysiology which purports to locate structures and processes in the human brain responsible for mathematical thinking. PDF file. Author(s): No creator set
License information
Related content
No related items provided in this feed
SkyMath: Mathematics for a Blue Planet SkyMath is a set of middle school mathematics modules incorporating weather data. The modules require teachers and students to acquire and use current environmental and real-time weather data in ways that embrace the dynamic and uncertain natures of these data, in order to promote the teaching and learning of significant mathematics, consistent with the standards set by the National Council of Teachers of Mathematics. The module includes 16 activities and can be incorporated as a replacement uni2 Reflection on mathematics |
Think of it as portable office hours! AUTOMATICALLY PACKAGED WITH THE TEXT, the Interactive Video Skillbuilder CD-ROM contains more than eight hours of video instruction. The problems worked during each video lesson are shown on the viewing screen so that students can try working them before watching the solution. To help students evaluate their progress, each section contains a 10-question Web quiz (the results of which can be emailed to the instructor) and each chapter contains a chapter test, with the answer to each problem on each test. This CD-ROM also includes MathCue tutorial and quizzing software, featuring a Skill Builder that presents problems to solve and evaluates answers with step-by-step explanations; a Quiz function that enables students to generate quiz problems keyed to problem types from each section of the book; a Chapter Test that provides many problems keyed to problem types from each chapter; and a Solution Finder that allows students to enter their own basic problems and receive step-by-step help as if they were working with a tutor. A new learning tool on this CD-ROM is a graphing calculator tutorial featuring examples, exercises, and video tutorials. |
NCTM – NavigationSeries G5 Geometric patterns The student uses a variety of representations to describe geometric relationships and solve problems. G5B The student is expected to use numeric and geometric patterns
NCTMNavigationSeries The NCTMNavigationSeries focus on the NCTM standards which correlate to the Florida Sunshine State Standards. These activities follow a sequential methodology regarding "how to teach" the subject matter
The NCTMnavigationseries was the textbook used and a copy of the book was provided for each participant. Each teacher was given a membership in New Hampshire Teachers of Mathematics and the National Council of Teachers of Mathematics.
including the NCTMNavigationSeries. Topics will include discrete and continuous probability models, data analysis, and statistical inference. Graphing calculators and computers will be used as tools in the study of data.
sources including the NCTMNavigationSeries. Topics will include discrete and continuous probability models, data analysis, and statistical inference. Graphing calculators and computers will be used as tools in the study of data.
NCTM (1988). Journal for Research ... NCTMNavigationSeries Navigating Through Geometry Grades 3-5; 6-8; and 9-12 Wysiwyg://4/ Equity Provide ample time for students during the exploration/discovery phase. Be sure all
NCTMNavigationSeries: Xxx - xxxx Not sure how to apply the properties in an everyday situation for my students? Would like some additional resources and sample lessons for this standard.
NCTMNavigationSeries Navigating Through Geometry Grades 3-5 Zome Geometry Book, Key Curriculum Press Equity Provide ample time for students during the exploration/discovery phase. Be sure all groups get to share their ideas.
• Reviewer for the NCTMNavigationSeries for The Oregon Mathematics Teacher (TOMT), 2004-2007 • MAA Liaison to the Mathematics Department at Western Oregon University Institutional Service • Mathematics Department Co-chair with Dr. Hamid Behmard, 2006-present• Explored the NCTMNavigationseries. • Developed Powerpoint presentations in collaboration with Dr. Hargrove • Became familiar with the new North Carolina Standard Course of Study that will begin to be used in 2004-2005 school year.
NavigationSeries The Navigations Series may be purchased through NCTM ( These books include detailed tasks for all grade levels and content strands, and a CD with blackline masters and additional readings.
School Mathematics (E-Standards) includes tools to enhance navigation of the document, as well as a more extensive set of electronic examples (e- ... A series of in-stitutes, organized by NCTM's new Academy for Professional Develop-
NCTM has produced Navigating Through Data Analysis: Grades 9 -12 to help teachers make this approach a reality. This paper briefly describes the book and discusses in detail one chapter 's activities, ... NCTM sponsored the Navigationseries,
Navigationseries, NCTM Daily Data Organize, represent, and interpret data with up to three categories, discuss how many more and how many less, use >, <, and = Number Talks Strategies for combining numbers - counting on,
NavigationSeries The Navigations Series may be purchased through NCTM ( These books include detailed tasks for all grade levels and content strands, and a CD with blackline masters and additional readings.
This two-day course explores the four main geometry standards in depth using the Navigationseries published by NCTM. This workshop is a practical road map designed to give teachers a deeper understanding of the Principles and Standards for Geometry, particularly as it pertains to
NCTMNavigation project, a series of 35 books which was developed for math classroom teachers and students, and in 2005 received the Lifetime Achievement Award from NCTM. Peggy was recognized by Zonta for her effectiveness as a female role
the Navigationseries of the NCTM. Her main interests are mathematics education and interactions between affect and cognition during mathematical problem solving. Susan K. Eddins, NBCT, MA, (Mathematics, University of Illinois, Urbana/Champaign) is a mathematics teacher, an Instructional
textbook series (for grades 9-12) dedicate several units to DM topics such as ... permutations; Hart, DeBellis, Kenney and Rosenstein have a forthcoming NCTM book titled Navigation through Discrete Mathematics in Pre-Kindergarten to Grade 12. There are also many websites available that ...
NCTM Principles and Standards - Algebra - Geometry - Measurement - Data Analysis and ... This problem is part of a series of problems that apply Math and ... Navigation, and Control - AP Calculus Keywords:
Part of the series Math in a Cultural Context: ... Star Navigation: Explorations into Angles and Measurement (Grade 6, ... (NCTM) for school mathematics as described in Principles and Standards for School Mathematics (c.f., Adams
navigation problem. ... teaching children mathematics • March 2012 401 After discussing these questions, be sure ... Understanding Series ROSE MARY ZBIEK, Series Editor NEW Reasoning and Sense-Making Problems and Activities for
NCTM Standards - Geometry - Problem Solving - Communication - Connections ... navigation of the TI-Nspire™ handheld. Background. This problem is part of a series that applies mathematical principles in |
Graphing Functions
Understanding Graphing Plus is a simple application designed to help your children understand and learn math graphing. Understanding Graphing Plus is a simple application designed to help your children understand and learn math graphing. GRAPHING Reading & Sketching Graphs Statistics Points on a Grid Transformations Relations, Equations, Functions Linear...
Math software for students studying precalculus and calculus. Can be interesting for teachers teaching calculus. Math software for students studying precalculus and calculus. Can be interesting for teachers teaching calculus. Math Center Level 2 consists of a Scientific Calculator, a Graphing Calculator 2D Numeric, a Graphing Calculator 2D Parametric, and...
DreamCalc is a virtual Scientific Graphing Calculator. DreamCalc is a virtual Scientific Graphing Calculator. You get the intuitive feel of using a real hand-held calculator on your PC or laptop! With DreamCalc, youll be able to graph functions and plot list data more simply than ever. In fact, it is...
DreamCalc is a fully featured Graphing Calculator for Windows. DreamCalc is a fully featured Graphing Calculator for Windows. With DreamCalc, you'll be able to graph functions and plot list data more simply than ever. In fact, it is a match for many dedicated graphing packages, but far easier to use. Unlike...
Magic Graph is a strong and easy to use graphing tool for plotting and analysing graphs of mathematical functions. Magic Graph is a strong and easy to use graphing tool for plotting and analysing graphs of mathematical functions. It is fully customizable, supports wide variety of functions and provides you with outstanding analitical abilities and different... |
Seven Elementary Functions and Their Graphs
In math we often encounter certain elementary functions. These elementary functions include rational functions, exponential functions, basic polynomials, absolute values and the square root function. It is important to recognize the graphs of elementary functions, and to be able to graph them ourselves. This will be especially useful when doing transformations.
Study Your Way
Easy Help. Fun Teachers.
Expert teachers who know their stuff bring personality & fun to every video. |
Course Overview: The
purpose of this course is to expand knowledge and understanding and application
of numbers, computation, estimation, measurement, geometry, statistics,
probability, patterns and algebra. The concepts and procedures of mathematics
are investigated and developed through the defining and solving of problems,
the use of reasoning, the communication of knowledge and the understanding and
the connecting of mathematical ideas within mathematics to other disciplines
and to real life.
Materials: Everyday
you are expected to have each of the following in class:
·2-3 sharpened pencils
·2 red pens
·notebook
Grading Policy: 10 % Daily warm ups
30% Daily Assignments and Homework
60 % Assessments (20 % quizzes, 40%
chapter tests and finals)
Grading Scale:
100 – 90 A
89 – 80 B
79 – 70 C
69 – 65 D
64 – 0 F
Tardy Policy: Students
are expected to arrive on time to each class during the day. Students are
expected to be in their seats, with all required materials, when the bell
rings. Students who receive 3 tardies will receive 2
days of break detention.
Late Work Policy: Late
work will be accepted up until the chapter test is taken and will be given a 10
point deduction for each day that it is late. If you are absent (for an
approved reason) YOU are responsible for collecting the work missed the same
day you return to school. All late work for you math class can be found on a
calendar on the bulletin board. You then have the same number of days to
complete the work as the number of days you were absent from school.
Classroom Rules:
·Respect yourself, others and their things at all
times
·Respect school property
·Raise your hand for permission to speak
·Arrive on time and be prepared for class
·Always do your best work
Also, please note that no cell phones are allowed out at
school and will be confiscated and turned into the front office.
Consequences:
If your child chooses to break a rule
·First
time = verbal warning
·Second
time = change of seat
·Third
time = Detention
·Fourth
time = Referral to front office
·Severe
infractions (fighting, blatant insubordination, etc.) may result in an
automatic referral.
Homework Procedures:
Homework will be corrected in class the day after it was assigned. If the
assignment is not completed on time you can choose to either get partial credit
for the few problems completed, or receive 75 % credit for that assignment by
turning it in the next day. There is NO extra credit available in math, so
please be sure to complete your homework every day. If you received a low score
on your homework and would like extra help, you may come to after school for
tutoring and will be allowed to redo the assignment for full credit.
Unit 1 – Statistics
and Matrices – Students will learn several tools for the display and
analysis of statistical data and learn to choose an appropriate display. They
will also use measures of central tendency and measures of variation to
determine whether graphs and statistics are misleading.
Unit 2 –Probability –
Students will investigate probability and combinatorics
and calculate outcomes, permutations, and combinations. Students will also
learn how experimental probability differs from theoretical probability.
Unit 3 –Real and
Rational Numbers – Students will learn how to solve equations involving
rational numbers, develop proficiency using powers, exponents, and scientific
notation. Also examine square roots, real number system, and the Pythagorean
Theorem.
Unit 4 – Proportions,
Ratios and Percents - Students will explore how to find the rate of change
between two points on a graph by calculating slope, solve proportions, and
review the connections among fractions, decimals, and percents.
Unit 5 – Geometry – Students
will investigate geometry and measurement. Learn how to find the circumference
and area of circles as well as the surface are and volume of 3 dimensional
figures. They will also study relationships among lines and angles and learn to
classify them.
Unit 6 –Equations and
Inequalities – Students will explore the use of algebraic equations to represent
functions. Learn how to simplify equations and to write and solve 2 step
equations and solve inequalities.
Unit 7 – Linear and
Non-Linear Functions and Polynomials – Students will learn how to identify
and graph functions; calculate the slope of a line, and use the slope intercept
form of a linear equation to find the slope and the y intercept. Students will
also learn to distinguish between linear and non-linear functions. |
Linear systems theory is the cornerstone of control theory and a well-established discipline that focuses on linear differential equations from the perspective of control and estimation. In this textbook, João...
$ 10.99
Statistics Essentials For Dummies not only provides students enrolled in Statistics I with an excellent high-level overview of key concepts, but it also serves as a reference or refresher for students in upper-level...
$ 6.99
Many colleges and universities require students to take at least one math course, and Calculus I is often the chosen option. Calculus Essentials For Dummies provides explanations of key concepts for students...
$ 10.29
This comprehensive text introduces readers to the most commonly used multivariate techniques at an introductory, non-technical level. By focusing on the fundamentals, readers are better prepared for more advanced...
$ 100.99
Most people don't think about numbers, or take them for granted. For the average person numbers are looked upon as cold, clinical, inanimate objects. Math ideas are viewed as something to get a job done or...
$ 8.29
MATHEMATICAL FOOTPRINTS takes a creative look at the role mathematics has played since prehistoric times, and will play in the future, and uncovers mathematics where you least expect to find it from its many...
$ 8.29
Whether it's stuff in your kitchen or garden, stuff that powers your car or your body, stuff that helps you work, communicate or play, or stuff that you've never heard of you can bet that mathematics is there....
$ 9.49
From nutritional labels and box office statistics to terabytes and megapixels, the 21st century world is awash in numbers. How can the average Joe or Jane make sense of all that data? The key, Theoni Pappas...
$ 12.49
Part of the joy of mathematics is that it is everywhere-in soap bubbles, electricity, da Vinci's masterpieces, even in an ocean wave. Written by the well-known mathematics teacher consultant, this volume's collection...
$ 8.29
A treasure trove of stories that make mathematical ideas come to life. Explores math concepts and topics such as real numbers, exponents, dimensions, the golden rectangle in both serious and humorous ways. Stories...
$ 8.29
The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay--a phenomenon...
$ 61.99Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called...
$ 71.79
This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that...
$ 58.99
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in...
$ 66.49
This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth...
$ 71.79
This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository... |
The most helpful favourable review
The most helpful critical review
2 of 2 people found the following review helpful
5.0 out of 5 starsGreat
Extremely informative book on what has become an important topic in Special Educational Needs. Very difficult to find information about Dyscalculia I have now started to recommend this to parents of students with Dyscalculia.
Published on 10 Dec 2011 by Luthilu
15 of 15 people found the following review helpful
3.0 out of 5 starsIt has a lot of uses, and could help you understand maths....
printing errors, which would make this less than an ideal book for someone wanting to resolve this problem in their own... a few of the most interesting pages have things like - for more information, please go to (**)... which of course should have been replaced before publishing.
Early pictures were confusing - oddly the later pictures were much better organised and easier to read.
Overall an interesting, informative book which requires a bit of concentration to fully appreciate it.
As someone who grew up literally terrified of maths and who has avoided maths throughout adulthood I was forced to look for a cure when suddenly faced with having to pass a maths test for work aged 26! Sadly a lot of the books I looked at were written for adults helping children, with adult maths difficulties largely ignored.
This book, however, manages to take maths right back to the very basic principles without patronising the reader and explains most of the fundamental concepts in two or more ways, so if one method doesn't click with you then the other will. Some of the exercises sound a bit strange but believe me they are worth doing. Interestingly I found that what the book was ultimately doing was buiding my confidence in using maths.
The overwhelming sense of relief I felt when I found out I'd passed that maths test is probably entirely due to this book.
My only complaint with this book (and the sole reason it has been demoted to 3 stars!) is that it is appallingly produced. Quite often the author refers to a method on page () and the page number is missing! There are a myriad of other equally annoying publishing errors. Whilst these are frustrating they don't interfere with the author's message too much and are easily overcome, however the question has to be asked why the publisher apparently didn't proof-read the book before allowing it to go to print!
To summarise; if you're desperate for a book that understands your issues with maths, and which offers clear, practical advice you won't go far wrong with this book.
Extremely informative book on what has become an important topic in Special Educational Needs. Very difficult to find information about Dyscalculia I have now started to recommend this to parents of students with Dyscalculia. |
Single Variable Calculus
9780495012337
ISBN:
0495012335
Edition: 6 Pub Date: 2007 Publisher: Thomson Learning
Summary: Study smarter and work toward the grade you want with this helpful guide. You'll find a short list of key concepts; a short list of skills to master; a brief introduction to the ideas of each section; an elaboration of the concepts and skills, including extra worked-out examples; and links in the margin to earlier and later material in the text and Study Guide.
Stewart, James is the author of Single Variable... Calculus, published 2007 under ISBN 9780495012337 and 0495012335. Seven hundred sixty five Single Variable Calculus textbooks are available for sale on ValoreBooks.com, two hundred forty four used from the cheapest price of $0.75, or buy new starting at $5 |
1. Setting the Goal: Modern Vegetation of North America--Composition and Arrangement of Principal Plant Formations. 2. Cause and Effect: Factors Influencing Composition and Distribution of North American Plant Formations through Late Cretaceous and Cenozoic Time. 3. Context. 4. Methods, Principles, Strengths, and Limitations. 5. Late Cretaceous throughDo you shy away from using numbers? Basic Mathematics can help. An easy-to-follow guide, it will ensure you gain the confidence you need to tackle maths and overcome your fears. It offers simple explanations of all the key areas, including decimals, percentages, measurements and graphs, and applies them to everyday situations, games and puzzles to... |
1 WHY GEOMETRIC
ALGEBRA?
This book is about geometric algebra, a powerful computational system to describe and
solve geometrical problems. You will see that it covers familiar ground—lines, planes,
spheres, rotations, linear transformations, and more—but in an unfamiliar way. Our
intention is to show you how basic operations on basic geometrical objects can be done
differently, and better, using this new framework.
The intention of this first chapter is to give you a fair impression of what geometric
algebra can do, how it does it, what old habits you will need to extend or replace, and
what you will gain in the process.
1.1 AN EXAMPLE IN GEOMETRIC ALGEBRA
To convey the compactness of expression of geometric algebra, we give a brief example of
a geometric situation, its description in geometric algebra, and the accompanying code
that executes this description. It helps us discuss some of the important properties of the
computational framework. You should of course read between the lines: you will be able
to understand this example fully only at the end of Part II, but the principles should be
clear enough now.
1
2 WHY GEOMETRIC ALGEBRA? CHAPTER 1
Suppose that we have three points c1 , c2 , c3 in a 3-D space with a Euclidean metric, a line
L, and a plane Π. We would like to construct a circle C through the three points, rotate
it around the line L, and then reflect the whole scene in the plane Π. This is depicted
in Figure 1.1. Here is how geometric algebra encodes this in its conformal model of
Euclidean geometry:
1. Circle. The three points are denoted by three elements c1 , c2 , and c3 . The oriented
circle through them is
C = c1 ∧ c2 ∧ c3 .
The ∧ symbol denotes the outer product, which constructs new elements of com-
putation by an algebraic operation that geometrically connects basic elements (in
this case, it connects points to form a circle). The outer product is antisymmetric:
if you wanted a circle with opposite orientation through these points, it would
be −C, which could be made as −C = c1 ∧ c3 ∧ c2 .
2. Rotation. The rotation of the circle C is made by a sandwiching product with an
element R called a rotor, as
C → R C/R.
C
c1
RC/R
c2 c3
L M n
π = p (n`) p
−π C / π
F i g u r e 1.1: The rotation of a circle C (determined by three points c1 , c2 , c3 ) around a line L,
and the reflections of those elements in a plane Π.
SECTION 1.1 AN EXAMPLE IN GEOMETRIC ALGEBRA 3
The product involved here is the geometric product, which is the fundamental
product of geometric algebra, and its corresponding division. The geometric prod-
uct multiplies transformations. It is structure-preserving, because the rotated circle
through three points is the circle through the three rotated points:
R (c1 ∧ c2 ∧ c3 )/R = (Rc1 /R) ∧ (R c2 /R) ∧ (R c3 /R).
Moreover, any element, not just a circle, is rotated by the same rotor-based formula.
We define the value of the rotor that turns around the line L below.
3. Line. An oriented line L is also an element of geometric algebra. It can be con-
structed as a "circle" passing through two given points a1 and a2 and the point at
infinity ∞, using the same outer product as in item 1:
L = a1 ∧ a2 ∧ ∞.
Alternatively, if you have a point on L and a direction vector u for L, you can make
the same element as
L = a1 ∧ u ∧ ∞.
This specifies exactly the same element L by the same outer product, even though it
takes different arguments. This algebraic equivalence saves the construction of many
specific data types and their corresponding methods for what are geometrically the
same elements.
The point at infinity ∞ is an essential element of this operational model of
Euclidean geometry. It is a finite element of the algebra, with well-defined
algebraic properties.
4. Line Rotation. The rotor that represents a rotation around the line L, with rotation
angle , is
R = exp( L∗ /2).
This shows that geometric algebra contains an exponentiation that can make ele-
ments into rotation operators. The element L∗ is the dual of the line L. Dualization
is an operation that takes the geometric complement. For the line L, its dual can be
visualized as the nest of cylinders surrounding it.
If you would like to perform the rotation in N small steps, you can interpolate the
rotor, using its logarithm to compute R1/N , and applying that n times (we have done
so in Figure 1.1, to give a better impression of the transformation). Other transfor-
mations, such as general rigid body motions, have logarithms as well in geometric
algebra and can therefore be interpolated.
5. Plane. To reflect the whole situation with the line and the circles in a plane Π, we first
need to represent that plane. Again, there are alternatives. The most straightforward
is to construct the plane with the outer product of three points p1 , p2 , p3 on the plane
4 WHY GEOMETRIC ALGEBRA? CHAPTER 1
and the point at infinity ∞, as Π = p1 ∧ p2 ∧ p3 ∧ ∞. Alternatively, we can instead
employ a specification by a normal vector n and a point p on the plane. This is a
specification of the dual plane π ≡ Π∗ , its geometric complement:
π = p (n∞) = n − (p · n) ∞.
Here is a contraction product, used for metric computations in geometric alge-
bra; it is a generalization of the inner product (or dot product) from vectors to the
general elements of the algebra. The duality operation above is a special case of the
contraction.
The change from p to p in the equation is not a typo: p denotes a point, p is its
location vector relative to the (arbitrary) origin. The two entities are clearly distinct
elements of geometric algebra, though computationally related.
6. Reflection. Either the plane Π or its geometric complement π determine a reflection
operator. Points, circles, or lines (in fact, any element X) reflect in the plane in the
same way:
X → − π X/π.
Here the reflection plane π, which is an oriented object of geometric algebra, acts
as a reflector, again by means of a sandwiching using the geometric product. Note
that the reflected circle has the proper orientation in Figure 1.1.
As with the rotation in item 2, there is obvious structure preservation: the
reflection of the rotated circle is the rotation of the reflected circle (in the
reflected line). We can even reflect the rotor to become R ≡ π exp( L∗ /2)/π =
exp(− (−πL∗ /π)/2), which is the rotor around the reflected line, automatically
turning in the opposite orientation.
7. Programming. In total, the scene of Figure 1.1 can be generated by a simple C++
program computing directly with the geometric objects in the problem statement,
shown in Figure 1.2. The outcome is plotted immediately through the calls to the
multivector drawing function draw(). And since it has been fully specified in terms
of geometric entities, one can easily change any of them and update the picture.
The computations are fast enough to do this and much more involved calculations
in real time; the rendering is typically the slowest component.
Although the language is still unfamiliar, we hope you can see that this is geometric pro-
gramming at a very desirable level, in terms of quantities that have a direct geometrical
meaning. Each item occurring in any of the computations can be visualized. None of the
operations on the elements needed to be specified in terms of their coordinates. Coor-
dinates are only needed when entering the data, to specify precisely which points and
lines are to be operated upon. The absence of this quantitative information may suggest
that geometric algebra is merely an abstract specification language with obscure opera-
tors that merely convey the mathematical logic of geometry. It is much more than that: all
expressions are quantitative prescriptions of computations, and can be executed directly.
Geometric algebra is a programming language, especially tailored to handle geometry.
SECTION 1.1 AN EXAMPLE IN GEOMETRIC ALGEBRA 5
// l1, l2, c1, c2, c3, p1 are points
// OpenGL commands to set color are not shown
line L; circle C; dualPlane p;
L = unit_r(l1 ^ l2 ^ ni);
C = c1 ^ c2 ^ c3;
p = p1 << (e2 ^ ni);
draw(L); // draw line (red)
draw(C); // draw cicle (green)
draw(p); // draw plane (yellow)
draw( — p * L * inverse(p)); // draw reflected line (magenta)
draw( — p * C * inverse(p)); // draw reflected circle (blue)
// compute rotation versor:
const float phi = (float)(M_PI / 2.0);
TRversor R;
R = exp(0.5f * phi * dual(L));
draw(R * C * inverse(R)); // draw rotated cicle (green)
// draw reflected, rotated circle (blue)
draw( — p * R * C * inverse(R) * inverse(p));
// draw interpolated circles
pointPair LR = log(R); // get log of R
for (float alpha = 0; alpha < 1.0; alpha += 0.1f)
{
// compute interpolated rotor
TRversor iR;
iR = exp(alpha * LR);
// draw rotated circle (light green)
draw(iR * C * inverse(iR));
// draw reflected, rotated circle (light blue)
draw( — p * iR * C * inverse(iR) * inverse(p));
}
F i g u r e 1.2: Code to generate Figure 1.1.
You may be concerned about the many different products that occurred in this application.
If geometric algebra needs a new product for every new operation, its power would
be understandable, but the system would rapidly grow unwieldy. This is perhaps the
biggest surprise of all: there is only one product that does it all. It is the geometric product
6 WHY GEOMETRIC ALGEBRA? CHAPTER 1
(discovered by William Kingdon Clifford in the 1870s), which we used implicitly in the
example in the sandwiching operations of rotation and reflection. The other
products (∧, , ∗ , sandwiching) are all specially derived products for the purposes of span-
ning, metric projection, complementation, and operating on other elements. They can all
be defined in terms of the geometric product, and they correspond closely to how we think
about geometry classically. That is the main reason that they have been given special sym-
bols. Once you get used to them, you will appreciate the extra readability they offer. But
it is important to realize that you really only need to implement one product to get the
whole consistent functionality of geometric algebra.
Because of the structural properties of geometric algebra, this example can be extended
in many ways. To name a few:
• Spherical Reflection. If we had instead wanted to reflect this situation in a sphere,
this is done by
X → − σ X/σ.
Here σ is the dual representation of a sphere (it encodes a sphere with center c pass-
ing through p as the representational vector p (c ∧ ∞)). We depict this in Figure 1.3.
C
c1
RC/R
c3
c2
−σ C / σ p
M
σ
L
F i g u r e 1.3: The rotation of a circle C (determined by three points c1 , c2 , c3 ) around a line L,
and the reflections of those elements in a sphere σ.
SECTION 1.2 HOW IT WORKS AND HOW IT'S DIFFERENT 7
The only thing that is different from the program generating Figure 1.1 is that the
plane π was replaced by the sphere σ, not only geometrically, but also algebraically.
This generates the new reflection, which reflects the line L to become the circle
M = −σ L/σ. It also converts the reflected rotor around M into the operation σ R/σ,
which generates a scaled rotation around a circle, depicted in the figure. The whole
structure of geometric relationships is nicely preserved.
• Intersections. The π-based reflection operator of item 6 takes the line L and pro-
duces the reflected line π L/π, without even computing the intersection point of the
line and the plane. If we had wanted to compute the intersection of line and plane,
that would have been the point π L = Π∗ L. This is another universal product,
the meet, which computes the intersection of two elements Π and L.
• Differentiation. It is even possible to symbolically differentiate the final expression
of the reflected rotated circle to any of the geometrical elements occurring in it. This
permits a sensitivity analysis or a local linearization; for instance, discovering how
the resulting reflected rotated circle would change if the plane π were to be moved
and tilted slightly.
1.2 HOW IT WORKS AND HOW IT'S DIFFERENT
The example has given you an impression of what geometric algebra can do. To under-
stand the structure of the book, you need a better feeling for what geometric algebra is,
and how it relates to more classical techniques such as linear algebra.
The main features of geometric algebra are:
• Vector Spaces as Modeling Tools. Vectors can be used to represent aspects of geom-
etry, but the precise correspondence is a modeling choice. Geometric algebra offers
three increasingly powerful models for Euclidean geometry.
• Subspaces as Elements of Computation. Geometric algebra has products to com-
bine vectors to new elements of computation. They represent oriented subspaces of
any dimension, and they have rich geometric interpretations within the models.
• Linear Transformations Extended. A linear transformation on the vector space dic-
tates how subspaces transform; this augments the power of linear algebra in a struc-
tural manner to the extended elements.
• Universal Orthogonal Transformations. Geometric algebra has a special represen-
tation of orthogonal transformations that is efficient and universally applicable in
the same form to all geometric elements.
• Objects are Operators. Geometric objects and operators are represented on a par,
and exchangeable: objects can act as operators, and operators can be transformed
like geometrical objects.
8 WHY GEOMETRIC ALGEBRA? CHAPTER 1
• Closed Form Interpolation and Perturbation. There is a geometric calculus that can
be applied directly to geometrical objects and operators. It allows straightforward
interpolation of Euclidean motions.
In the following subsections, we elaborate on each of these topics.
1.2.1 VECTOR SPACES AS MODELING TOOLS
When you use linear algebra to describe the geometry of elements in space, you use a real
vector space Rm . Geometric algebra starts with the same domain. In both frameworks,
the vectors in an m-dimensional vector space Rm represent 1-D directions in that space.
You can think of them as denoting lines through the origin. To do geometry flexibly, we
want more than directions; we also want points in space. The vector space Rm does not
have those by itself, though its vectors can be used to represent them.
Here it is necessary to be more precise. There are two structures involved in doing geo-
metrical computations, both confusingly called "space."
• There is the physical 3-D space of everyday experience (what roboticists call the task
space). It contains the objects that we want to describe computationally, to move
around, to analyze data about, or to simply draw.
• Mathematics has developed the concept of a vector space, which is a space of abstract
entities with properties originally inspired by the geometry of physical space.
Although an m-dimensional vector space is a mathematical generalization of 3-D physical
space, it does not follow that 3-D physical space is best described by a 3-D vector space.
In fact, in applications we are less interested in the space than in the geometry, which
concerns the objects residing in the space. That geometry is defined by the motions that
can freely move objects. In Euclidean geometry, those motions are translations, rotations,
and reflections. Whenever two objects differ only by such transformations we refer to
them as the same object, but at a different location, with a different orientation, or viewed
in a mirror. (Sometimes scaling is also included in the permitted equivalences.)
So we should wonder what computational model, based in a vector space framework, can
conveniently represent these natural motions of Euclidean geometry. Since the motions
involve certain measures to be preserved (such as size), we typically use a metric vector
space to model it. We present three possibilities that will recur in this book:
1. The Vector Space Model. A 3-D vector space with a Euclidean metric is well suited to
describe the algebra of directions in 3-D physical space, and the operation of rotation
that transforms directions. Rotations (and reflections) are orthogonal linear trans-
formations: they preserve distances and angles. They can be represented by 3 × 3
orthogonal matrices or as quaternions (although the latter are not in the linear alge-
bra of R3 , we will see that they are in the geometric algebra of R3 ).
2. The Homogeneous Model. If you also want to describe translations in 3-D space,
it is advantageous to use homogeneous coordinates. This employs the vectors of a
SECTION 1.2 HOW IT WORKS AND HOW IT'S DIFFERENT 9
4-D vector space to represent points in physical 3-D space. Translations now also
become linear transformations, and therefore combine well with the 3-D matrix
representation of rotations.
The extra fourth dimension of the vector space can be interpreted as the point at the
origin in the physical space. There is some freedom in choosing the metric of this
4-D vector space, which makes this model suitable for projective geometry.
3. The Conformal Model. If we want the translations in 3-D physical space represented
as orthogonal transformations (just as rotations were in the 3-D vector space model),
we can do so by employing a 5-D vector space. This 5-D space needs to be given a
special metric to embed the metric properties of Euclidean space. It is expressed as
R4,1 , a 5-D vector space with a Minkowski metric.
The vectors of the vector space R4,1 can be interpreted as dual spheres in 3-D phys-
ical space, including the zero-radius spheres that are points. The two extra dimen-
sions are the point at the origin and the point at infinity.
This model was used in the example of Figure 1.1. It is called the conformal model
because we get more geometry than merely the Euclidean motions: all conformal
(i.e., angle-preserving) transformations can be represented as orthogonal transfor-
mations. One of those is inversion in a sphere, which explains why we could use a
spherical reflector in Figure 1.3.
Although these models can all be treated and programmed using standard linear algebra,
there is great advantage to using geometric algebra instead:
• Geometric algebra uses the subspace structure of the vector spaces to construct
extended objects.
• Geometric algebra contains a particularly powerful method to represent orthogonal
transformations.
The former is useful to all three models of Euclidean geometry; the latter specifically works
for the first and third. In fact, the conformal model was invented before geometric alge-
bra, but it lay dormant. Only with the tools that geometric algebra offers can we realize its
computational potential. We will treat all these models in Part II of this book, with special
attention to the conformal model. In Part I, we develop the techniques of geometric alge-
bra, and prefer to illustrate those with the more familiar vector space model, to develop
your intuition for its computational capabilities.
1.2.2 SUBSPACES AS ELEMENTS OF COMPUTATION
Whatever model you use to describe the geometry of physical space, understanding vector
spaces and their transformations is a fundamental prerequisite. Linear algebra gives you
techniques to compute with the basic elements (the vectors) by using matrices. Geometric
algebra focuses on the subspaces of a vector space as elements of computation. It constructs
these systematically from the underlying vector space, and extends the matrix techniques
10 WHY GEOMETRIC ALGEBRA? CHAPTER 1
to transform them, even supplanting those completely when the transformations are
orthogonal.
The outer product ∧ has the constructive role of making subspaces out of vectors. It uses
k independent vectors vi to construct the computational element v1 ∧ v2 ∧ · · · ∧ vk , which
represents the k-dimensional subspace spanned by the vi . Such a subspace is proper (also
known as homogeneous): it contains the origin of the vector space, the zero vector 0. An
m-dimensional vector space has many independent proper subspaces: there are m sub- k
spaces of k dimensions, for a total of 2m subspaces of any dimension. This is a considerable
amount of structure that comes for free with the vector space Rm , which can be exploited
to encode geometric entities.
Depending on how the vector space Rm is used to model geometry, we obtain different
geometric interpretations of its outer product.
• In the vector space model, a vector represents a 1-D direction in space, which can
be used to encode the direction of a line through the origin. This is a 1-D proper
subspace. The outer product of two vectors then denotes a 2-D direction, which
signifies the attitude of an oriented plane through the origin, a 2-D proper subspace
of the vector space. The outer product of three vectors is a volume. Each of those
has a magnitude and an orientation. This is illustrated in Figure 1.4(a,b).
• In the homogeneous model, a vector of the vector space represents a point in the
physical space it models. Now the outer product of two vectors represents an ori-
ented line in the physical space, and the outer product of three vectors is interpreted
as an oriented plane. This is illustrated in Figure 1.4(c,d). By the way, this represen-
u
tation of lines is the geometric algebra form of Pl¨ cker coordinates, now naturally
embedded in the rest of the framework.
• In the conformal model, the points of physical space are viewed as spheres of radius
zero and represented as vectors of the vector space. The outer product of three points
then represents an oriented circle, and the outer product of four points an oriented
sphere. This is illustrated in Figure 1.4(e,f). If we include the point at infinity in
the outer product, we get the "flat" elements that we could already represent in the
homogeneous model, as the example in Section 1.1 showed.
It is very satisfying that there is one abstract product underlying such diverse construc-
tions. However, these varied geometrical interpretations can confuse the study of its alge-
braic properties, so when we treat the outer product in Chapter 2 and the rest of Part I,
we prefer to focus on the vector space model to guide your intuitive understanding of
geometric algebra. In that form, the outer product dates back to Hermann Grassmann
(1840) and is the foundation of the Grassmann algebra of the extended quantities we call
proper subspaces. Grassmann algebra is the foundation of geometric algebra.
In standard linear algebra, subspaces are not this explicitly represented or constructed.
One can assemble vectors vi as columns in a matrix [[V]] = [[v1 v2 · · · vk ]], and then treat
the image of this matrix, im([[V]]), as a representation of the subspace, but this is not an
SECTION 1.2 HOW IT WORKS AND HOW IT'S DIFFERENT 11
b
b
a
a
c
(a) (b)
b b
a
a c
(c) (d)
b
b
d
a
a
c
c
(e) (f)
F i g u r e 1.4: The outer product and its interpretations in the various models of Euclidean
geometry. (a,b): the vector space model; (c,d): the homogeneous model; and (e,f): the con-
formal model.
12 WHY GEOMETRIC ALGEBRA? CHAPTER 1
integral part of the algebra; it is not a product in the same sense that the dot product is. If
the matrix is square, we can take the determinant det([[V]]) to represent the amount of area
or volume of the subspace and its orientation, but if it is not square, such measures are less
easily represented. Subspaces are simply not well represented in standard linear algebra.
1.2.3 LINEAR TRANSFORMATIONS EXTENDED
Linear transformations are defined by how they transform vectors in the vector space Rm .
As these vectors transform, so do the subspaces spanned by them. That fully defines how
to extend a linear transformation to the subspace structure.
If one uses a matrix for the representation of the linear transformation on the vector space
level, it is straightforward and automatic to extend this to a matrix that works on the sub-
space levels. You just take the outer product of its action on the basis vectors as its defini-
tion on the basis for subspaces. Now you can perform the same linear transformation on
any subspace.
This way of thinking about linear transformations, with its use of the outer product,
already provides structural advantages over the usual coordinate-based methods. Pro-
grams embedding this automatic transference of a vector space mapping to its subspaces
are simpler. Moreover, they permit one to choose a representation for geometric elements
that transforms most simply within this framework. An example is the representation of
the attitude of a plane through the origin; its representation by a normal vector has more
complicated transformations than its equally valid representation by an outer product of
two vectors.
Within the subspace representation, a general product can be given for the intersection
of subspaces (the meet product), which also transform in a structure-preserving manner
under the extended linear transformations (the transform of an intersection is the inter-
section of the transforms). This uses more than the outer product alone; it also requires
the contraction, or dualization.
The resulting consistent subspace algebra is good to understand first. Its subspace
products are the algebraic extensions of familiar techniques in standard linear algebra.
Seeing them in this more general framework will improve the way you program in lin-
ear algebra, even if you do not make explicit use of the extended data structures that the
subspaces provide. Therefore we begin our journey with the treatment of this subspace
algebra, in Chapters 2 to 5.
1.2.4 UNIVERSAL ORTHOGONAL TRANSFORMATIONS
In the vector space model and the conformal model, orthogonal transformations are used
to represent basic motions of Euclidean geometry. This makes that type of linear transfor-
mation fundamental to doing geometry in those models. (General linear transformations
are still useful to represent deformations of elements, on top of the basic motions of the
geometry, but they are not as crucial).
SECTION 1.2 HOW IT WORKS AND HOW IT'S DIFFERENT 13
Geometric algebra has a special way to represent orthogonal transformations, more
powerful than using orthogonal matrices. These are versors, and the example in
Section 1.1 showed two instances of them: a rotor and a reflector. A versor V transforms
any element X of the geometric algebra according to the versor product:
X → (−1)xv V X/V,
where the sign factor depends on the dimensionality of X and V, and need not concern us
in this introduction. This operator product transcends matrices in that it can act directly
on arbitrary elements: vectors, subspaces, and operators.
The product involved in the sandwiching of the versor product is the geometric product;
as a consequence, subsequent operators multiply by the geometric product. For instance,
R2 (R1 X/R1 )/R2 = (R2 R1 ) X/(R2 R1 ). This product is linear, associative, and invertible,
but not commutative. That matches its geometric interpretation: orthogonal transforma-
tions are linear, associative, and invertible, but their order matters.
The two-sidedness of the versor product of an operator may come as a bit of a surprise,
but you probably have seen such two-sided products before in a geometrical context.
• When the vectors of a space transform by a motion represented by [[M]] (so that [[x]]
becomes [[M]] [[x]]), a matrix [[A]] transforms to become [[M]] [[A]] [[M]] −1 . Note that
in linear algebra, vectors and operators transform differently, whereas in geometric
algebra they transform in the same manner.
• Another classical occurrence of the two-sided product is the quaternion repre-
sentation of 3-D rotations. Those are in fact rotors and, therefore, versors. In
the classical representation, you need to employ three imaginary numbers to repre-
sent them properly. We will see in Chapter 7 how geometric algebra simply uses
the real subspaces of a 3-D vector space to construct quaternions. Quaternions
are not intrinsically imaginary! Moreover, when given this context, they become
universal operators, capable of rotating geometric subspaces (rather than only
being applicable to other quaternions).
The versor form of an orthogonal transformation automatically guarantees the preserva-
tion of algebraic structure (more technically known as covariance). Geometrically, this
implies that the construction of an object from moved components equals the movement of
the object constructed from the original components. Here, "construction" can be the con-
nection of the outer product, the intersection of the meet, the complementation of the
duality operation, or any other geometrically significant operation.
You have seen in the example how this simplifies constructions. In traditional linear
algebra, one can only transform vectors properly, using the matrices. So to move any
construction one has built, one has to move the vectors on which it was based and rebuild
it from scratch. With geometric algebra, it is possible to move the construction itself: the
lines, circles, and other components, and moreover all of these are moved by the same
versor construction with the same versor representing the motion.
14 WHY GEOMETRIC ALGEBRA? CHAPTER 1
You need no longer to be concerned about the type of element you are moving; they all
automatically transform correctly. That means you also do not need to invent and build
special functions to move lines, planes, or normal vectors and can avoid defining a motion
method for each data structure, because all are generic. In fact, those differing methods
may have been one of the reasons that forced you to distinguish the types in the first place.
Now even that is not necessary, because they all find their place algebraically rather than
by explicit construction, so fewer data types are required. This in turn reduces the number
of cases in your program flow, and therefore may ultimately simplify the program itself.
1.2.5 OBJECTS ARE OPERATORS
In geometric algebra, operators can be specified directly in terms of geometric elements
intrinsic to the problem.
We saw in Section 1.1, item 6, how the dual plane π (i.e., an object) could be used imme-
diately as the reflector (i.e., an operator) to produce the reflected line and circles. We also
constructed the rotor representing the rotation around the line L by exponentiating the
line in item 4.
Geometric algebra offers a range of constructions to make versors. It is particularly simple
to make the versors representing basic motions as ratios (i.e., using the division of the
geometric product): the ratio of two planes is a rotation versor, the ratio of two points is
a translation versor, and the ratio of two lines in 3-D is the screw motion that turns and
slides one into the other. These constructions are very intuitive and satisfyingly general.
As you know, it is much harder to define operators in such direct geometrical terms using
linear algebra. We summarize the usual techniques:
• There are several methods to construct rotation operators. Particularly intricate are
various kinds of standardized systems of orientating frames by subsequent rota-
tions around Euler angles, a source of errors due to the arbitrariness of the coordi-
nate frames. One can construct a rotation matrix from the rotation axis directly (by
Rodrigues' formula), and this is especially simple for a quaternion (which is already
an element of geometric algebra). Unfortunately, even those are merely rotations at
the origin. There is no simple formula like the exp( L∗ /2) of geometric algebra to
convert a general axis L into a rotation operator.
• Translations are defined by the difference of vectors, which is simple enough, but
note that it is a different procedure from defining rotations.
• A general rigid body motion converting one frame into another can be artificially
split in its rotational aspects and translational aspects to produce the matrix. Unfor-
tunately, the resulting motion matrix is hard to interpolate. More rewarding is
a screw representation, but this requires specialized data structures and Chasles'
theorem to compute.
SECTION 1.3 PROGRAMMING GEOMETRY 15
The point is that these linear algebra constructions are specific for each case, and
apparently tricky enough that the inventors are often remembered by name. By contrast,
the geometric algebra definition of a motion operator as a ratio is easily reinvented by any
application programmer.
1.2.6 CLOSED-FORM INTERPOLATION AND PERTURBATION
In many applications, one would like to apply a motion gradually or deform it
continuously (for instance, to provide smooth camera motion between specified views).
In geometric algebra, interpolation of motions is simple: one just applies the correspond-
ing versor V piecemeal, in N steps of V1/N . That N th root of a motion versor V can be
determined by a logarithm, in closed form, as exp(log(V )/N ). For a rotor representing a
rotation at the origin, this retrieves the famous "slerp" interpolation formula of quater-
nions, but it extends beyond that to general Euclidean motions. Blending of motions can
be done by blending their logarithms.
By contrast, it is notoriously difficult to interpolate matrices. The logarithm of a matrix
can be defined but it is not elementary, and not in closed form. A straightforward way to
compute it is to take the eigenvalue decomposition of the rigid body motion matrix in
the homogeneous coordinate framework, and take the Nth root of the diagonal matrix.
Such numerical techniques makes the matrix logarithm expensive to compute and hard
to analyze.
Perturbations of motions are particularly easy to perform in geometric algebra: the small
change in the versor-based motion VX/V to any element X can be simply computed as
X x B, the commutator product of X with the bivector logarithm of the perturbing versor.
This is part of geometric calculus, an integrated method of taking derivatives of geometric
elements relative to other geometric elements. It naturally gets into differential geom-
etry, a natural constituent of any complete framework that deals with the geometry of
physical space.
1.3 PROGRAMMING GEOMETRY
The structural possibilities of the algebra may theoretically be rich and inviting, but that
does not necessarily mean that you would want to use it in practical applications. Yet we
think you might.
1.3.1 YOU CAN ONLY GAIN
Geometric algebra is backwards-compatible with the methods you already use in your
geometrical applications.
Using geometric algebra does not obviate any of the old techniques. Matrices, cross prod-
u
ucts, Pl¨ cker coordinates, complex numbers, and quaternions in their classical form are
16 WHY GEOMETRIC ALGEBRA? CHAPTER 1
all included in geometric algebra, and it is simple to revert to them. We will indicate these
connections at the appropriate places in the book, and in some applications we actually
revert to classical linear algebra when we find that it is more efficient or that it provides
numerical tools that have not yet been developed for geometric algebra. Yet seeing all these
classical techniques in the context of the full algebra enriches them, and emphasizes their
specific geometric nature.
The geometric algebra framework also exposes their cross-connections and provides uni-
versal operators, which can save time and code. For example, if you need to rotate a line,
and you have a quaternion, you now have a choice: you can convert the quaternion to a
rotation matrix and apply that to the positional and directional aspects of the line sepa-
rately (the classical method), or you view the quaternion as a rotor and apply it immedi-
ately to the line representation (the geometric algebra method).
1.3.2 SOFTWARE IMPLEMENTATION
We have made several remarks on the simpler software structure that geometric
algebra enables: universal operators, therefore fewer data types, no conversions between
formalisms, and consequently a simpler data flow.
Having said that, there are some genuine concerns related to the size of geometric algebra.
If you use the conformal model to calculate with the Euclidean geometry of 3-D space,
you use a 5-D vector space and its 25 = 32 subspaces. In effect, that requires a basis of 32
elements to represent an arbitrary element. Combining two elements could mean 32 × 32
real multiplies per geometric product, which seems prohibitive.
This is where the actual structure of geometric algebra comes to the rescue. We will explain
these issues in detail in Part III, but we can reassure you now: geometric algebra can
compete with classical approaches if one uses its algebraic structure to guide the imple-
mentation.
• Elements of geometric algebra are formed as products of other elements. This
implies that one cannot make an arbitrary element of the 32-dimensional frame-
work. Objects typically have a single dimensionality (which is three for circles and
lines) or a special structure (all flats contain the point at infinity ∞). This makes
the structure of geometrically significant elements rather sparse. A good software
implementation can use this to reduce both storage and computation.
• On the other hand, the 32 slots of the algebra are all used somehow, because they are
geometrically relevant. In a classical program, you might make a circle in 3-D and
would then have to think of a way to store its seven parameters in a data structure. In
geometric algebra, it automatically occupies some of the 5 = 10 slots of 3-vector
3
elements in the 5-D model. As long as you only allocate the elements you need, you
are not using more space than usual; you are just using the pre-existing structure to
keep track of them.
SECTION 1.4 THE STRUCTURE OF THIS BOOK 17
• Using linear algebra, as you operate on the composite elements, you would have
had to invent and write methods (for instance, to intersect a circle and a plane).
This would require special operations that you yourself would need to optimize for
good performance. By contrast, in geometric algebra everything reduces to a few
basic products, and their implementation can be optimized in advance. Moreover,
these are so well-structured that this optimization can be automated.
• Since all is present in a single computational framework, there is no need for conver-
sions between mathematically different elements (such as quaternions and rotation
matrices). Though at the lower level such conversions may be done for reasons of
efficiency, the applications programmer works within a unified system of geomet-
rically significant elements of computation.
Using insights and techniques like this, we have implemented the conformal model and
have used it in a typical ray-tracing application with a speed 25 percent slower than the
optimized classical implementation (which makes it about as costly as the commonly used
homogeneous coordinates and quaternion methods), and we believe that this overhead
may be reduced to about 5 to 10 percent. Whether this is an acceptable price to pay for a
much simpler high-level code is for you to decide.
We believe that geometric algebra will be competitive with classical methods when we also
adapt algorithms to its new capabilities. For instance, to do a high-resolution rendering,
you now have an alternative to using a much more dense triangulation (requiring many
more computations). You could use the new and simple description of perturbations to
differentially adapt rays of a coarse resolution to render an ostensibly smoother surface.
Such computation-saving techniques would easily compensate for the slight loss of speed
per calculation.
Such algorithms need to be developed if geometric algebra is to make it in the real world.
We have written this book to raise a new generation of practitioners with sufficient fun-
damental, intuitive, and practical understanding of geometric algebra to help us develop
these new techniques in spatial programming.
1.4 THE STRUCTURE OF THIS BOOK
We have chosen to write this book as a gradual development of the algebraic terms in tan-
dem with geometric intuition. We describe the geometric concepts with increasing preci-
sion, and simultaneously develop the computational tools, culminating in the conformal
model for Euclidean geometry. We do so in a style that is not more mathematical than
we deem necessary, hopefully without sacrificing exactness of meaning. We believe this
approach is more accessible than axiomatizing geometric algebra first, and then having
to discover its significance afterwards.
The book consists of three parts that should be read in order (though sometimes a
specialized chapter could be skipped without missing too much).
18 WHY GEOMETRIC ALGEBRA? CHAPTER 1
1.4.1 PART I: GEOMETRIC ALGEBRA
First, we get you accustomed to the outer product that spans subspaces (and to the
desirability of the "algebraification of geometry"), then to a metric product that extends
the usual dot product to these subspaces. These relatively straightforward extensions from
linear algebra to a multilinear algebra (or subspace algebra) already allow you to extend
linear mappings and to construct general intersection products for subspaces. Those capa-
bilities will extend your linear algebra tool kit considerably.
Then we make the transition to true geometric algebra with the introduction of the
geometric product, which incorporates all that went before and contains more beyond
that. Here the disadvantage of the approach in this book is momentarily annoying, since
we have to show that the new definitions of the terms from the earlier chapters are "back-
wards compatible." But once that has been established, we can rapidly move on from
considering objects (the subspaces) to operators acting on them. We then easily absorb
tools you may not have expected to encounter in real vector spaces, such as complex num-
bers and quaternions. Both are available in an integrated manner, real in all normal senses
of the word, and geometrically easily understood.
Part I concludes with a chapter on geometric differentiation, to show that differential
geometry is a natural aspect of geometric algebra (even though we will use it only inci-
dentally in this book).
1.4.2 PART II: MODELS OF GEOMETRY
In Part II the new algebra will be used as a tool to model aspects of mostly Euclidean
geometry. First, we treat directions in space, using the vector space model, already familiar
from the visualizations used in Part I to motivate the algebra.
Next, we extend the vector embedding trick of homogeneous coordinates from practical
computer science to the complete homogeneous model of geometric algebra, which
u
includes Pl¨ cker coordinates and other useful methods.
Finally, in Chapter 13 we can begin to treat the conformal model of the motivating exam-
ple in Section 1.1. The conformal model is the model that has Euclidean geometry as an
intrinsic part of its structure; all Euclidean motions are represented as orthogonal trans-
formations. We devote four chapters to its definition, constructions, operators, and its
ability to describe general conformal transformations.
1.4.3 PART III: IMPLEMENTATION OF GEOMETRIC ALGEBRA
To use geometric algebra, you will need an implementation. Some are available, or you
ı
may decide to write your own. Na¨ve implementations run slow, because of the size of the
algebra (32-D for the basis of the conformal model of a 3-D Euclidean space).
SECTION 1.5 THE STRUCTURE OF THE CHAPTERS 19
In the third part of this book, we give a computer scientist's view of the algebraic
structure and describe aspects that are relevant to any efficient implementation, using its
multiplicative and sparse nature. We end with a simple ray tracer to enable comparison
of computational speeds of the various methods in a computer graphics application.
1.5 THE STRUCTURE OF THE CHAPTERS
Each regular chapter consists of an explanation of the structure of its subject. We explain
this by developing the algebra with the geometry, and provide examples and illustrations.
Most of the figures in this book have been rendered using our own software package,
GAViewer. This package and the code for the figures are available on our web site,
We recommend that you download the software, install it, and follow the instructions to
upload the figures. You can then interact with them. At the very least you should be able to
use your mouse for 3-D rotation and translation to get the proper spatial impression of the
scene. But most figures also allow you to interactively modify the independent elements
of the scene and study how the dependent elements then change.
You can even study the way we have constructed them1 and change them on a command
line; though if you plan to do that we suggest that you first complete the GAViewer tutorial
on the web site. This will also allow you to type in formulas of numerical examples.
If you really plan to use geometric algebra for programming, we recommend doing the
drills and programming exercises with each chapter. The programming exercises use a
special library, the GA sandbox, also available from the web site. It provides the basic data
structures and embedding of the products so that you can program directly in geometric
algebra. This should most closely correspond to how you are likely to use it as an extension
of your present programming environment.
We also provide structural exercises that help you think about the coherence of the geo-
metric algebra subject in each chapter and ask you to extend some of the material or
provide simple proofs. For answers to these exercises, consult the web site.
Historically, geometric algebra has many precursors, and we will naturally touch upon
those as we develop our concepts and terms. We do not meticulously attribute all results
and thoughts, but occasionally provide a bit of historic perspective and contrast with tra-
ditional approaches. At the end of each chapter, we will give some recommended literature
for further study.
1 But be warned that some illustrative figures of the simpler models may use elements of the conformal model in
their code, since that is the most natural language to specify general elements of Euclidean geometry |
Developmental Math students enter college needing more than just the math, and this has directly impacted the instructor's role in the classroom. Instructors have to teach to different learning styles, within multiple teaching environments, and to a student population that is mostly unfamiliar with how to be a successful college student. Authors Andrea Hendricks and Pauline Chow have noticed this growing trend in their combined 30+ years of teaching at their respective community colleges, both in their face-to-face and online courses. A... MOREs a result, they set out to create course materials that help today's students not only learn the mathematical concepts but also build life skills for future success. Understanding the time constraints for instructors, these authors have worked to integrate success strategies into both the print and digital materials, so that there is no sacrifice of time spent on the math. Furthermore, Andrea and Pauline have taken the time to write purposeful examples and exercises that are student-centered, relevant to today's students, and guide students to practice critical thinking skills. Intermediate Algebraand its supplemental materials, coupled with ALEKS or Connect Math Hosted by ALEKS, allow for both full-time and part-time instructors to teach more than just the math in any teaching environment without an overwhelming amount of preparation time or even classroom time. |
The sculptures by Brent Collins was an eye catch with the different shapes, this was the first that I have see of this type...
see more
The sculptures by Brent Collins was an eye catch with the different shapes, this was the first that I have see of this type of math. I also like the utility curve for mathematics - Bit-Player by Brian Hayes – the curve for life. It's like the title of our booklet "MATH IS EVERYWHERE".
This site offers assistance for all types of math from basic math through calculus. There are text and video lessons as well...
see more
This site offers assistance for all types of math from basic math through calculus. There are text and video lessons as well as practice tests. These tutorials are based on a large variety of textbooks available from middle school through post-secondary
A parabola is the set of points that are equally distant from the focus point and the directrix. This Tab Tutor program will...
see more
A parabola is the set of points that are equally distant from the focus point and the directrix. This Tab Tutor program will help you learn about the equation of a parabola and how to use it to derive the focus, vertex, and orientation. A useful glossary also introduces you to other features like the latus rectum and the axis of symmetry.
With this Tab Tutor program, you'll learn about the equation of a horizontal ellipse and how to use it to derive the foci,...
see more
With this Tab Tutor program, you'll learn about the equation of a horizontal ellipse and how to use it to derive the foci, vertices, and minor axis. A useful glossary also introduces you to other features like the latus rectum and major axis.
With this Tab Tutor program, you'll learn about the equation of a vertical ellipse and how to use it to derive the foci,...
see more
Learn about the equation of a horizontal hyperbola with this Tab Tutor program. With step-by-step instruction and an...
see more
Learn about the equation of a horizontal
Learn about the equation of a vertical hyperbola with this Tab Tutor program. With step-by-step instruction and an...
see more
Learn about the equation of a vertical |
Book Search
One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to...
Published October 29th 2007 |
Conic Sections - Circles - Yay Math This video is an introduction to circles. The instructor explores the effect of changing parameters in the equation and graphing circles on a co-ordinate plane. He also instructs how to find the radius, center, and points on circumference. White board in a class setting, some interaction, engaging, several examples of increasing complexity. The discussion is clear and understandable. (06:49)Introduction to Matrices - Yay Math A quick introductory look at how matrices work. The video reviews naming, adding and multiplying matrices. White board in a class setting, some interaction, engaging, several examples of increasing complexity. The discussion is clear and understandable. Preview - full version at by Robert Ahdoot, yaymath.org approach |
Mathematics for Business - 8th edition
Summary: The Eighth Edition of Mathematics for Business continues to provide solid, practical, and current coverage of the mathematical topics students must master to succeed in business today. The text begins with a review of basic mathematics and goes on to introduce key business topics in an algebra-based context.
Chapter 1, Problem Solving and Operations with Fractions, starts off with a section devoted to helping students become better problem solvers and critical...show more thinker while reviewing basic math skills. Optional scientific calculator boxes are integrated throughout and financial calculator boxes are presented in later chapters to help students become more comfortable with technology as they enter the business world. The text incorporates applications pertaining to a wide variety of careers so students from all disciplines can relate to the material. Each chapter opener features a real-world application.
Features
Current financial data used throughout the text.
Real-world applications within exercise sets are now called out by topical headings for each problem so that students immediately see the relevance of the problems to their lives.
Introduction to problem solving in Section 1.1 helps students learn how to think through solving common problems. The emphasis on problem-solving skills is carried through the text so that students can enter the business world with critical thinking skills and apply what they have learned.
Chapter openers now incorporate an application with a real-world graph or figure so students can understand how the chapter content pertains to actual business situations.
Financial calculator boxes that explain how to solve examples using a financial calculator are now integrated into later chapters to familiarize students with the technology they will be using in the business world.
A Metric System Appendix, complete with examples and exercises, explains the metric system and teaches students to convert between US and metric units of measurement.
'Net Assets emphasize the World Wide Web and keep students current on how businesses adapt to technology.
Cumulative Reviews help students review groups of related chapter topics and reinforce their understanding of the material.
Dust Cover MissingVery89321357434
$9.45 +$3.99 s/h
Good
Quality School Texts OH Coshocton, OH
2006-06-22 |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.
Math 117 Lecture 3 notes page 1Math 117 Lecture 3 Notes: Geometry comes from two Greek words, ge meaning earth and metria meaning measuring. The approach to Geometry developed by the Ancient Greeks has been used for over 2000 years as the basis of
Non-injectivity of the map from the Witt group of a variety to the Witt group of its function eldBurt TotaroFor a regular noetherian scheme X with 2 invertible in X, let W (X) denote the Witt group of X [7]. By denition, the Witt group is the quot
Research Statement Michael D. BarrusMy main research interests lie in graph theory and combinatorics. A principal theme in my research is the study of structures associated with graphs. Such a structure can be viewed as a collection of parameters or
CUTE: A Concolic Unit Testing Engine for CKoushik Sen, Darko Marinov, Gul AghaDepartment of Computer Science University of Illinois at Urbana-Champaign{ksen,marinov,agha}@cs.uiuc.eduABSTRACTIn unit testing, a program is decomposed into units wh
WORKSHEET FOR 1/23/2009 - SOLUTIONS (1) Suppose that 0 f (t)dt = 3x2 + ex cos x. Find f (2). Solution: We must use a version of the fundamental theorem of calculus, that states that: x d f (t)dt = f (x). dx a With this in mind, we can dierentiate bo
(1) Suppose that events A and B are independent. Show that A and B c (Ac and B c ) are independent. (2) Assume that events A and B are independent. Let P (A B) = 1. Show that P (A) = 1 or P (B) = 1. (3) A tennis player must win two games in a row to
1. Slot-size bound for chaining (Problem 11-2 in [CLRS]) Suppose that we have a hash table with n slots, with collisions resolved by chaining, and suppose that n keys are inserted into the table. Each key is equally likely to be hashed to each slot.
Chapter X Special Data Structures (cont)X.1 A Data Structure for Disjoint SetsMany appplications need a data structure that maintains a collection of disjoint sets under the following set of operations: MakeSet(x): creates a set that contains x Un
Chapter VIII Minimum Spanning Tree: Randomized Linear Time AlgorithmA general approach to design an algorithm with linear running time is to use recursion (one or more times) on problems of total size (over all recursions) at most a fraction of the
Chapter IX Special Data Structures (cont)IX.1 Binomial and Fibonacci HeapsRecall that a heap is a data structure that stores a set of items/elements that have an associated priority so that the element with highest priority can be accessed quickly
Chapter XI FingerprintingXI.1 Comparing Two StringsConsider the problem of comparing two strings x, y, for simplicity in {0, 1}. They are very long and we would rather not compare them directly. A case where this is meaningful is if they are store
Chapter XV Linear ProgrammingA linear program consists of an objective function and a number of linear constraints. The goal is to optimize (minimize or maximize) the objective function under the constraints. Though highly restrictive in that all th
Chapter I Quick-Sort, Treaps, Skip Lists, etcWe revisit quicksort and study two related search data structures: treaps and skip lists. We also consider the selection problem nding the k-th smallest element. The main common ingredient is the use of r
Chapter X Hashing and ApplicationsA hash table allows to store a set of keys so that search and updates can be performed in constant time. This would be trivial with a table that is as large as the universe, but we also require the size of the table
III.0Chapter IV NP-CompletenessWe have found already several problems, like the traveling salesman tour (TST) problem, for which it seems hard to come up with an ecient algorithm. But what means ecent ? To formalize this, it has been suggested, an
Chapter VII Network FlowGiven a directed graph network, with two special vertices the source and the sink, and with a capacity associated with each edge, the question is to compute a maximum ow from s to t. This is the simplest of a class of graph
Chapter VI All Pair Shortest Paths and Matrix MultiplicationVI.1 APSPs and Matrix MultiplicationThere is a close similarity between the inner loop in the APSP algorithm and matrix multiplication. Recall that for nn matrices A = (aij ) and B = (bij
Chapter III Greedy AlgorithmsWe consider algorithms for optimization problems that make greedy choices according to some local criteria. The resulting algorithms are often simple and fast. However, they rarely give an optimal solution. We consider s
Chapter III Closest Pair of PointsWe consider the problem nding a closest pair of points in a given set of points in the plane. More precisely, we are given a set of n points in R2 each is a pair x = (x1 , x2 ) and we are interested in a pair of p
Chapter I IntroductionWe consider several variants of the scheduling problem, and see how dierent techniques can be used. During the course we will further explore all these techniques.I.1Interval Scheduling: Greedy AlgorithmINTERVAL SCHEDULIN
Chapter IV Divide and Conquer: Integer Multiplication and FFTIV.1 Integer Multiplication: Karatsubas AlgorithmConsider the problem of multiplying two positive integer numbers a and b in binary representation. Through most of this course, we simply
Chapter VIII Special Data StructuresVIII.1 Amortized AnalysisOften a data structure is used in an algorithm and accordingly one is not interested in the worst-case time needed to perform a certain operation; it is sucient if the performance is goo
CPSC 226EXAM IFall, 2004The number of points (out of a total of 150) that each question is worth is indicated in parentheses. For multiple choice questions, provide the BEST answer. Answer essay and short answer questions thoroughly but concise
CPSC 226EXAM IIFall, 2002The number of points that each question is worth is indicated in parentheses. For multiple choice questions, provide the BEST answer. Answer essay and short answer questions thoroughly but concisely; extraneous informat
CPSC 226FINAL EXAMFall, 2002The number of points that each question is worth is indicated in parentheses. The exam is worth 304 points (that includes 4 points of extra credit Merry Christmas!) For each question, provide the BEST answer. Good l
CPSC 226EXAM IIFall, 2005The number of points (out of a total of 150) that each question is worth is indicated in parentheses. For multiple choice questions, provide the BEST answer. Answer essay and short answer questions thoroughly but concis
CPSC 226EXAM IIFall, 2007Provide the BEST answer for each question. Each multiple-choice question is worth 4 points; each matching blank is worth 2 points. Good luck! 1. Your name: _ 2. Which of the following is NOT one of the four methods of w
CPSC 226FINAL EXAMFall, 2006For each question, provide the BEST answer. Each question is worth four points, except the final matching question, which is worth a total of 30 points. The exam is worth a total of 306 points (six holiday bonus poin
CPSC 226EXAM IIFall, 2004The number of points (out of a total of 150) that each question is worth is indicated in parentheses. For multiple choice questions, provide the BEST answer. Good luck! 1. (3) Your name: _ 2. (3) A primary advantage of
CPSC 226EXAM IFall, 2002The number of points that each question is worth is indicated in parentheses. For multiple choice questions, provide the BEST answer. Answer essay and short answer questions thoroughly but concisely; extraneous informati
CPSC 226FINAL EXAMFall, 2007For each question, provide the BEST answer. Each question is worth five points, except the final matching question, which is worth a total of 30 points. The exam is worth a total of 305 points (five holiday bonus poi
CPSC 226EXAM IFall, 2003The number of points (out of a total of 150) that each question is worth is indicated in parentheses. For multiple choice questions, provide the BEST answer. Answer essay and short answer questions thoroughly but concise
CPSC 226 - Plant Families Quiz September 9, 2003 (1 point) Your name _ For the following multiple-choice questions, circle the letter of the BEST answer. Each question is worth two points. Good luck! 1. Members of Polygonaceae a. have ocrea b. have a
CPSC 226Competition QuizFall, 2003Your name _ You have just completed an experiment to determine the critical weed free period for pumpkins. Your treatments and the resulting yields are as follows: Treatment 1. maintained weed free until harves
CPSC 226Competition QuizFall, 2005Your name _ You have just completed an experiment to determine the critical weed free period for Illini Xtra Sweet sweet corn. You started out with a tilled field, planted your sweet corn, and implemented 9 dif
CPSC 226 - Plant Families Quiz September 7, 2004 5, 2006 4, 2007 (1 point) Your name _ In the blank following each number, write in the letter corresponding to the family that best matches the description. Not all letters will be used, but no letter should be used m
CPSC 226FINAL EXAMFall, 2004For each question, provide the BEST answer. Multiple-choice questions are each worth three points; the matching question is worth a total of 30 points. The exam is worth a total of 306 points (thats 6 points of extra
CPSC 226FINAL EXAMFall, 2005For each question, provide the BEST answer. Each question is worth three points, except the final matching question, which is worth a total of 30 points. The exam is worth a total of 306 points (six holiday bonus poi
CPSC 226 - Plant Families Quiz September 6, 2005 (1 point) Your name _ For the following multiple-choice questions, circle the letter of the BEST answer. Each question is worth two points. Good luck! 1. A plant with ocrea belongs to the family a. Cyp |
Problems in Geometry at Putnam County High School
by
Debra Newsome
In response to calls for reform, in both the teaching and the learning
of school mathematics, the National Council of Teachers of Mathematics has
taken a major stand on the content and emphasis of the mathematics curriculum.
In producing its Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989) and other documents, professional educators have assumed leadership
roles in two critical areas: (1) the creation of a vision of mathematics
in an increasingly technological society with a diverse variety of needs
and requirements and (2) the design of a set of standards to guide curriculum
revision within this vision. These goals as well as the broader goals of
getting students to value math, gain confidence in their own ability, and
become comfortable as mathematical thinkers place the teacher in the role
as a coach and catalyst for knowledge acquisition.
The state of Georgia has taken a close look at the NCTM Standards in producing
its Quality Core Curriculum (QCC). Indeed, the Geometry or Informal Geometry
guide states:
Geometry provides students with a way to link their
perceptions of the real world with the mathematics that
allows them to solve a variety of problems they will
encounter not only in other disciplines but also in their
lives....Geometry should provide students with visual and
concrete representations that help them gain insight into
important areas of mathematics and their applications to
the real world...High school geometry must extend beyond
the traditional treatment of geometry as a deductive
system and provide students with a broad view of
geometry and its applications...(p. 1).
This small sample of statements from the introduction seeks to link a Georgia
high school course in geometry directly to Standards 1, 2, 3, 7, 8, and
14, and indirectly to the remaining Standards. Despite these affirmations,
the Putnam County High School geometry courses for college-bound students
and the geometry topics for general or vocational students have continued
to be taught predominantly through textbooks and strongly teacher-centered
instruction, especially for lower-achieving students. Manipulatives have
been limited to 3-dimensional models of geometric solids to be held up in
the front of the classroom. Tools have been limited to compass and straight
edge and then used exclusively to complete paper-and-pencil constructions
for college-bound students. Instructors outside the field have rarely been
sought and interdisciplinary lessons have been severely limited by the knowledge
base of individual teachers.
The role of the teacher as a facilitator rather than as a dictator is perhaps
the greatest distinction between the "traditional" mathematics
classrooms at PCHS and those envisioned by the writers of the standards.
Too many secondary teachers are uncomfortable granting students permission
to move around the room, not to mention allowing interaction with their
classmates! Students are complacently resigned to assume the passive role
of being "talked at" by the teacher. Many adults still view mathematics
classrooms as being the same as they were during personal educational experiences.
Students must now accept a far greater and active role in their own education
to receive far greater rewards. Through group and individual projects, assignments,
and the like, the student is able to explore and see the integration of
mathematical topics that emphasize the body of mathematics as a whole "greater
than the sum of its parts."
The current role of the teacher in many classrooms at PCHS is that of an
information-giver, despite personal desires to create active rather than
passive learners. Teachers need to accept new roles as leaders and inventors
and be facilitated in these roles, but time is lacking to share with and
support other teachers. The mathematics and science classrooms at PCHS are
located across the hall from each other and opportunities to collaborate
have been initiated during the past school year and should continue to be
actively sought. Interdisciplinary teams are nonexistent in most high schools
although the practice of common planning is now part of the middle school
concept. Many primary teachers enhance learning of specific content with
instructional units, integrating a variety of subjects, as opposed to the
isolated content areas in the high school curriculum. Students should be
accustomed to the integration of topics based on experiences at the primary
and middle school levels.
Although publishers have sought to meet the varied mathematical needs of
teachers and students, textbooks can not continue as the main source of
geometry instruction. It is not in the spirit of the Standards nor the interest
of the students to approach applications as side bars or "extra"
exercises. Emphasis on pencil and paper exercises reinforced by correct
answers will not facilitate full appreciation of a subject so visual in
content and scope, yet this has been the nature of the resources used by
mathematics teachers at Putnam County High School. Despite the adoption
of new textbooks which include a variety of supplemental materials, the
focus remains on the textbook as a primary source of instruction. The complaint
raised continues to be lack of time to explore a variety of presentation
methods and materials prior to classroom implementation. Students must be
shown that there is more to geometry than those topics which appear in a
textbook but this engagement can occur only when teachers accept the responsibility
to facilitate rather than dictate learning.
One reason for the existence of a narrow approach to the teaching of high
school geometry is tradition. Teachers tend to adopt teaching styles and
methods most like those to which they themselves were exposed. Many people
hold the childhood views of school as authoritarian institutions in which
somebody smart stands in front of a room and tries to pass information on
to large groups of students. The content of textbooks, too, is often a reproduction
of those used by previous generations despite the current penchant for visual
displays. Emphases in content topics may change, but the typical classroom
activities carry on as they always have. Society must realize that not only
can students (and teachers) learn from each other, but also from manipulatives,
other tools and technologies, a spectrum of qualified personnel, and a wide
variety of situations.
Tradition breeds the idea that schools should operate on the assumptions
of the past and therefore schools are having a difficult time adapting to
the ever-changing needs of the present. Even with the large volume of, the
retrieval speed of, and the variety of sources of information available
today, our school finds it difficult to stay abreast of current trends.
Rural schools, like Putnam County High, are especially at risk for lagging
behind, since they are often further removed by geography from major business
and industry influences. Information as to the current and future needs
of employers and post-secondary institutions may be disseminated by informal
or other means which may be unavailable or inaccessible to rural districts.
Smaller population centers lack the financial and human resources that are
taken for granted in urban and surrounding suburban centers. The low population
density in rural areas limits the attention given to education by the media
and hence even fewer resources may be identified or accessed (Bracey, 1992).
Availability of financial resources continues to be a major player in the
determination of curriculum and its implementation. The diversity in size
and resources impinges negatively on rural communities in the applications
of geometry to which students are exposed. Smaller systems or those less
financially able, have had to be content with less technology, specifically
computers, software, and multimedia applications. Rural schools face strong
competition for resources from suburban schools whose populations include
a larger tax base to provide money for materials, experimental and/or innovative
programs, and instructors of the highest caliber. Furthermore, a wider variety
of business and industry personnel in non rural areas provides resourceful
contacts that can increase motivation for students to learn geometry, especially
if knowledge acquired can be applied in a local job or is directly related
to a field chosen for further study.
The world apart from school depends on the successful conception, implementation,
and completion of projects that involve the cooperation of many individuals.
Students are too often left to forage through the same old curriculum, in
the same old manner, with the same old results, namely poor student achievement,
motivation, and inspiration. An education system suffused with individual
and group projects, particularly apprenticeships and hands-on experiences
can fill the void in genuine student understanding. Assessment, although
often looked upon with disfavor by students, must become part of the learning
process rather than the objective. Alternative assessment advocates maintain
that the proof of a person's capacity is found in their ability to perform
or produce, not in their ability to answer on cue. Students value the opportunity
to discover for themselves what they have mastered, without the need for
teacher approval. Mathematics teachers at PCHS are seeking new ways to assess
students, but these processes require further study since no department
member is experienced in alternate assessment methods.
Geometry offers a wonderful opportunity for students to expand their knowledge,
gain confidence, and explore new interests. Projects completed independently
of the classroom or cooperatively within the classroom have allowed students
at PCHS to explore new areas of geometric applications while pursuing their
own interests. Art, architecture, construction, design, drafting, forestry,
history, map-making, photography, research, teaching, and many other areas
all provide opportunities for students to use geometry in a field of interest.
My students have explored a variety of topics including tessellations, scale
models, art, blueprints, fractals, geometry in the workplace, and bridges.
These projects, when presented to classmates, offered other opportunities
for personal growth and inspiration for peers. As their teacher, it has
been a joy to experience the interest of this aspect of my students' geometric
experience.
Teachers, too, can benefit from the experiences of their peers and expand
their knowledge base to provide interesting activities for their students.
Multi-disciplinary committees can be formed to plan for units that would
reinforce concepts in all areas. Math conference participants have shared
a variety of projects and plans for implementing programs designed to provide
innovative opportunities for students to connect and integrate concepts
and activities within the disciplines of mathematics, science, and technology
education. If business personnel can be persuaded to provide input, perhaps
offering real problems occurring on the job, students may experience the
power of geometry through sources outside of the classroom. Study for an
advanced degree has offered opportunities to extend and explore standard
topics within the framework of technological innovation. The door that has
been opened by this knowledge will benefit teachers and students at PCHS
for years to come.
With graphing calculators and computer software, students can now experiment
with parameter changes that previously had to be time-consumingly drawn
by hand and visualize how the graphs are effected without the tedium of
re-drawing. Many maintain that these visualizations allow students to make
abstract connections, yet students are only now beginning to experience
technology at Putnam County High School. Graphing and computer technologies
which have been available at other schools throughout the state have only
now been made available to students at PCHS, and presently only on a severely
limited basis. As recently as the past school year, students have been graphing
exclusively using paper-and-pencil, which has had obvious limitations to
the depth of mathematical experiences. Technology is quite expensive and
financial considerations have often been given as justification for lack
of spending in Putnam and other rural counties. Extra effort must be made,
however, to supplement direct expenditures on technology with alternate
sources of materials acquisition such as aid-in-kind. Putnam County leaders
must overcome their aversion to seeking financial sources outside of the
local budget. Grant money may be sought, business leaders may donate or
lend materials and/or speakers, or appeals to the community may result in
volunteers who can share their real-world uses of geometry with students.
Despite statements in curriculum guides or mandates from administration
as to the content to be taught in geometry, teachers make the ultimate decisions
as to the depth and scope of these objectives and how they will be carried
out in the classroom. It is evident that student-centered activities must
come from teacher leadership positions. Moves to broaden the approach to
teaching geometry will not occur without a direct effort by teachers in
the classroom to refocus the direction and methods of instruction. Most
teachers in Putnam County have not been trained in the use nor the application
of technology toward teaching methodology in the classroom. This training
process, though slow to be implemented, will surely benefit all parties.
Many sources offer suggestions on how to make mathematics more interesting.
The current high school student seems to be under the influences of "entertainment".
Leisure and fun are pervasive in their lives in the form of fast-paced computer
and video technology and students seem to expect education to come in the
form of entertainment. Many books activities have been written to make mathematics
fun. The "hook" of fun is designed to increase success and encourage
the further study of mathematics. Activities can be designed that are fun
but also require application of concepts learned in the classroom. For example,
rope and chalk can be used outside to build a hopscotch or a basketball
court after a unit in geometry on compass and straight edge constructions.
Origami activities produce objects of beauty and geometric significance.
Tessellation exploration and creation can be as practical as quilting or
as creative as design opportunities.
The use of manipulatives is recommended by professional mathematics teachers
and their associations (NCTM, 1989). Teachers are inundated with catalogs
offering manipulative merchandise for sale, but these accouterments cost
money. Creative teachers and students, however, can generate many items
with available scrap materials, donations, and redesign of projects. For
example, a hypsometer can be used to measure the angle of elevation or depression
but a crude representation can be constructed of tag board or adapted from
a protractor and a paper clip. Low budget activities such as paper folding,
geoboard lessons, tangrams, etc. are often readily shared among teachers
at professional conferences or work sites.
Practice lab experiences, although normally confined to science classrooms,
can provide opportunities for students to brainstorm problem solving techniques
and explore results. For example, Hunt (1978) describes three methods that
are commonly used to determine the height of an object. Working cooperatively,
students are likely to determine these and other methods through discovery
or research. The same problem can be solved in geometry after a unit on
basic trigonometric functions, employing a hypsometer or its equivalent.
Labs can be involved and require additional resources or as basic as paper
folding explorations. Explorations using The Geometer's Sketchpad await
the geometry students at PCHS for this school year. NCTM yearbooks and addenda
offer other resources for exploration. Materials included with adopted textbooks
also offer suggestions for lab activities and these may be fully accessible
to student groups.
Opportunities for changing the traditional narrow approach to high school
geometry exist. Students, teachers, business and community leaders, and
parents can exert a positive influence over the materials and methods used
in the classroom to provide innovative and interesting explorations in geometry.
Every successful innovation has the power to spark the imagination of a
mind.
Bibliography
Bracey, G. W. The second Bracey report on the condition of public education.
Phi Delta Kappan, 74, 104-117,1992. |
Online Algebra
0.00 (0 votes)
Document Description
Tutor
Help with Online Algebra Ii Tutors
Av
Algebra 2 Lessons Covered
Tutorvista provides intensive tutoring for Algebra from expert online tutors and following are the concepts mastered by the tutors and those covered in our help program:
* Radical Equations
* Exponential And Logarithmic Functions
* Graphing functions
* Quadratic functions
* Third Degree Polynomials
Our online tutoring is one on one and personalized and as it is interactive and thus learning with tutorvista becomes fun. So get your algebra 2 answers from an expert tutor now.
TutorVista offers Algebra 2 help online. Solve problems and get answers, work on basic
concepts as well as take help with algebra 1 and get Algebra 2 homework too with TutorVista.
Study with highly ...
Get answers to all Algebra word problems online with TutorVista. Our online Algebra tutoring
program is designed to help you get all the answers to your Algebra word problems giving you
the desired ...
Having difficulty working out Math problems? Stuck with your homework and having nightmares
before your next Math test? TutorVista's tutors can help you. TutorVista's Online Help features ...
Content Preview
Online Algebra Tutor Help with Online Algebra Ii Tutors Av Algebra 2 Lessons Covered Tutorvista provides intensive tutoring for Algebra from expert online tutors and following are the concepts mastered by the tutors and those covered in our help program: * Radical Equations * Exponential And Logarithmic Functions * Graphing functions * Quadratic functions * Third Degree Polynomials Our online tutoring is one on one and personalized and as it is interactive and thus learning with tutorvista becomes fun. So get your algebra 2 answers from an expert tutor now.
Learn More about Equation solver Algebra 2 Answers TutorVista makes getting Online Algebra answers really easy. Connect with a tutor anytime, anywhere and get all the answers online. In fact you can also get free help to your homework problems as well. Tutorvista online algebra tutors make your learning on algebra two effective as well as interesting. We help you to learn how to do algebra and thus you ensure a complete learning from an online helper. We provide tutoring for all the grades, chose you grade and avail this help now! By Grades Grade 12 Grade 11 Grade 10 Grade 9 Grade 8 Grade 7 Grade 6 Grade 5 Grade 4 Grade 3 Free Help from Experts Undergo a free demo session from tutorvista and experience our live algebra 2 tutoring. Interact with the best tutor online by using the whiteboard and talking over the phone. Work out your problems, get instant answers with steps of explanation, avail to homework help and also gain exam prep help. So, Enrol for TutorVista's online tutoring today and give your grades a boost! Read More on Math Problem Solver |
Introduction to Algebra Author(s): No creator setIntroduction to Fractions Lesson on fractions. Fractions are introduced and the basics are taught in this video. Examples problems are shown, solved, and explained with picture representations. More lesson can be found at |
A Problem Solving Approach to Mathematics for Elementary School Teachers
edition: 11th
Author(s): Billstein, Rick; Libeskind, Shlomo; Lott, Johnny W.
ISBN: 9780321756664 / HardcoverThis is a loose-leaf edition book (Same content, just cheaper!!) 2nd day shipping available, ships same or next day. This book contains mild water damage that does not hinder the readability of the book. Books may not contain access codes or supplementary material. |
Professional Commentary: This three-part activity illustrates the use of iteration, recursion, and algebra to model and analyze the changing amount of medicine in an athlete's body. The activity is adapted from High School Mathematics at Work, a publication from the National Research Council....
Professional Commentary: In this 4-lesson unit students develop and analyze exponential models for the behavior of light passing through water. Students begin by considering how light intensity changes from near the surface to the bottom of the ocean....
Professional Commentary: In this investigation students explore the possible reasons behind the observation that northwestern crows consistently drop a type of mollusk called a whelk from a height of 5 meters to break its shell. Students are given activity sheets and a graphics calculator....
Professional Commentary: How does the amount of weight that can be supported by a spaghetti bridge relate to the width (number of spaghetti strands) and the length of a bridge? Students gather data comparing the amount of weight that can be supported, the number of strands of spaghetti used, and the length of the bridge....
Professional Commentary: The major goal of this lesson is to collect data from a variety of experiments, determine what type of model best fits the data, and explain why. Students explore a variety of relationships using pennies, pressure, temperature, light, and pendulums to determine the algebraic equation that best represents the pattern modeled by the variables involved in...
Professional Commentary: This multi-day activity has students look for functions within a given set of data. After analyzing patterns in the data, students should be able to determine the type of function that best represents the data....
Professional Commentary: In this week-long unit, students examine the problem of space pollution caused by human-made debris in orbit to develop an understanding of functions and modeling. The unit provides students an opportunity to use spreadsheets, graphing calculators, and computer graphing utilities....
Professional Commentary: Graphing calculators are sophisticated devices that can run small computer programs and draw the graph represented by complex equations in an instant. In the last few years, they have become mandatory in many high school mathematics classes and can be used on the SAT and advanced placement exams and other standardized tests....
Professional Commentary: How does one go about finding the volume of an irregular shape? Students get a preview of integral calculus as they compute volume of a solid as the sum of volumes of slices of the solid....
Professional Commentary: The purpose of this activity is to discover relationships among the volumes of simple solids such as cones, cylinders, pyramids, cubes, and spheres. A discussion of the underlying mathematical ideas is included.... |
rapidly expanding area...New. This item is printed on demand. The rapidly expanding area of algebraic graph theory uses two different branches of algebra to explore various aspects of graph theory: linear algebra (for spectral theory) and group theory (for studying graph symmetry). |
On CueMath
JEE Math Book
Take your preparation for the IIT JEE Mathematics to the next level. Get your copy of the Mathematics for the JEE Advanced, published by Pearson and written by one of CueMath's creators. This book contains hundreds of new and interesting problems, along with detailed step-by-step solutions.
IIT JEE Math Lessons & Problems
CueMath has 700+ very high quality lessons and thousands of problems on all IIT JEE Mathematics topics from Algebra, Geometry and Calculus. This material has been prepared by IIT JEE top rankers who also happen to have years of teaching experience. In the interest of the IIT aspirant community, this material has been freely made available online. |
Pre-Calculus Help
In this section you'll find study materials for pre-calculus help. Use the links below to find the area of pre-calculus you're looking for help with. Each study guide comes complete with an explanation, example problems, and practice problems with solutions to help you learn pre-calculus. Systems of Equations
A system of equations is a collection of two or more equations whose graphs might or might not intersect (share a common point or points). If the graphs do intersect, then we say that the solution to the system is the point or ... |
book is designed for researchers, students and practitioners interested in using fast and efficient iterative methods to approximate solutions of nonlinear equations. The following four major problems are addressed. Problem 1: Show that the iterates are well defined. Problem 2: concerns the convergence of the sequences generated by a process and the question of whether the limit points are, in fact solutions of the equation. Problem 3: concerns the economy of the entire operations. Problem 4: concerns with how to best choose a method, algorithm or software program to solve a specific type of problem and its description of when a given algorithm succeeds or fails. The book contains applications in several areas of applied sciences including mathematical programming and mathematical economics. There is also a huge number of exercises complementing the theory. |
Roadmap to 8th Grade Math
(Paperback)
Roadmap to 8th Grade Math
(Paperback)
Roadmap to 8th Grade Math Book Description
If Students Need to Know It, It's in This Book This book develops the Math skills of eighth graders. It builds skills that will help them succeed in school and on the New York State test. Why The Princeton Review? We have more than 20 years of experience helping students master the skills needed to excel on standardized tests. Each year, we help more than 2 million students score higher and earn better grades. We Know the New York State Testing Program Our experts at The Princeton Review have analyzed the New York State test, and this book provides the most up-to-date, thoroughly researched practice possible for the Grade 8 Mathematics test. We break down the test into individual skills to familiarize students with the test's structure, while increasing their overall skill level. We Get Results We know what it takes to succeed in the classroom and on tests. This book includes strategies that are proven to improve student performance. We provide - content review, detailed lessons, and practice exercises modeled on the skills tested by the New York State Grade 8 Mathematics test - engaging puzzles and word problems - 2 complete practice New York State Math tests
About the Author :
Karen Lurie has contributed to Roadmap to 8th Grade Math as an author.
Lurie is a professional researcher and pop culture writer who has written extensively about television.
Popular Searches
The book Roadmap to 8th Grade Math by Karen Lurie
(author) is published or distributed by Princeton Review [0375763554, 9780375763557].
This particular edition was published on or around 2003-11-11 date.
Roadmap to 8th Grade Math has Paperback binding and this format has 228 number of pages of content for use.
This book by Karen Lurie |
Outlines and Highlights for Excursions in Modern Mathematics by Peter Tannenbaum, Isbn : 9780321568038
Student Resource Guide To Accompany Excursions In Modern Math
Videos on DVD with Optional Subtitles for Excursions in Modern Mathematics
Summary
Student Resource Guidecontains full worked out solutions to odd-numbered exercises from the text, "selected hints" that point the reader in#xA0;one of many#xA0;directions leading to a solution and keys to student success including lists of skills#xA0;that will#xA0;help prepare for chapter exams. |
Bienvenidos! Here we offer information and
materials
on using original historical sources in teaching mathematics. This
includes our own experiences and materials, and those of others who are
teaching with original
sources. We welcome suggestions for other links to include here,
and comments and suggestions for improvements.
FIRST BOOK AVAILABLE MATHEMATICAL EXPEDITIONS: CHRONICLES BY THE EXPLORERS (see below for details on our courses
and this first book, or order from Springer) NEW BOOK AVAILABLE MATHEMATICAL MASTERPIECES:
FURTHER CHRONICLES BY THE EXPLORERS (see below for details on our courses,
the new second
book, or order from Springer)
Our odyssey
of teaching with original sources
Our journey towards utilizing original texts as the primary object
of
study in undergraduate and graduate courses began at the senior
undergraduate
level. In 1987 we read William Dunham's article A "Great
Theorems"
Course in Mathematics (American Mathematical Monthly93
(1986), 808-811), in which he describes a course based on mathematical
masterpieces from the past, viewed as works of art. His ideas and
materials went on to become the well known best-seller Journey
Through
Genius: Great Theorems of Mathematics. We were inspired to
develop
a similar course, at the senior level, but with one crucial difference:
Whereas Dunham presents his students with his own modern rendition of
these
masterpieces, our idea was to use the original texts themselves.
With assistance from New Mexico State University's honors program,
dean,
and mathematics department, we developed and team taught the course Great
Theorems: The Art of Mathematics, and it has now found a
successful
and permanent niche in the university's curriculum, serving as a lively
capstone course for students majoring in a number of diverse
disciplines.
It is the only mathematics course certified to meet the university's
"Viewing
a Wider World" upper division general education requirement. Our
experiences
with this senior level course convinced us that teaching with
original
sources could be both successful and inspiring for us and our
students.
The course is described in detail in Mathematical
Masterpieces: Teaching With Original Sources (html) (or dvi
or ps)
(in
Vita
Mathematica: Historical Research and Integration with Teaching, R.
Calinger (ed.), MAA, Washington, 1996, pp. 257-260). We also involved
other
faculty in teaching and contributing material for this course. Our four
author second bookMathematical
Masterpieces: Chronicles by the Explorers has emerged from
this
course, written with two of these colleagues at New Mexico
State
University, Arthur Knoebel and Jerry Lodder.
We came to believe that this approach to teaching and learning could
also help provide the motivation, perspective, and overview so lacking
in typical lower division courses, since it is being increasingly
recognized
that an historical point of view can address these deficiencies.
As Niels Henrik Abel observed: "It appears to
me that if one wants to make progress in mathematics, one should study
the masters and not the pupils." We have written an
article
Recovering
Motivation in Mathematics: Teaching with Original Sources (html)
(or dvi or ps)
(UME Trends
6, September 1994) espousing our reasons and
philosophy for this teaching approach. We were inspired to try to
use the study of original texts as a teaching pedagogy introducing
lower
division students to important currents of mathematical thought.
Thus we developed the course Spirit and
Evolution
of Mathematics, again with support from the New Mexico State
University mathematics department and honors program, allowing us to
team
teach the course while under development. It provides an
"introduction
to great problems of mathematics" for students with a good high school
background in mathematics, and is intended both to attract and retain
mathematics
majors, and to give non majors a rich experience in the nature and
content
of mathematical thought, satisfying a lower division university
mathematics
general education requirement (the course is one of only a handful
certified
for this). In fact, the true prerequisite is a certain level of
mathematical
maturity and ability, rather than courses with specific content.
Thus, a much broader audience has access to an interesting course with
serious mathematical content. Our experiences, after teaching
this
course numerous times, have shown that with careful selection of
original
texts, supplemental prose readings, and appropriate format for
classroom
activities and assignments, this approach can be a tremendous
success.
Students find the study of original sources fascinating, especially
when
combined with prose readings supplying cultural and historical context,
giving the course something of an interdisciplinary flavor. The
benefits
for instructors and students alike are a deepened appreciation for the
origins and nature of modern mathematics, as well as the lively and
stimulating
class discussions engendered by the interpretation of original
sources.
The course is described in detail in our article
Great
Problems of Mathematics: A Course Based on Original Sources (html)
(or dvi
or ps)
(American Mathematical Monthly99 (1992), 313-317). Our first
bookMathematical Expeditions: Chronicles by
the Explorersgrew out of this
course.
Since then we have expanded the use of original sources into high
school
courses as well as graduate courses. Work with high school
students
during two summer workshops at Colorado College with Mike Siddoway is
described
in Great
Problems of Mathematics: A Workshop for High School Students (html)
(or dvi
or ps)
(College Mathematics Journal25 (1994), 112-114).
We
also conducted a graduate course at New Mexico State University for
high
school teachers on using original sources in the high school
curriculum.
Our graduate students showed great interest in this, and it has evolved
into a regular graduate course The
Role of History in Teaching Mathematics, providing part of a
growing
mathematics education component in the mathematics
graduate program at New Mexico State University. The paper A
graduate course on the role of history in teaching mathematics
describes
the course and its origins. The
course syllabus considers the use of history, in particular
original
sources, throughout the mathematics curriculum. Our graduate students
in
this course develop and critique
major teaching units based on history, often on original sources, and
we
now have quite a collection of the historical
teaching modules they have written. A number of these have been
tested
in the classroom. Their level ranges from middle school through
the
advanced undergraduate curriculum. Write to us if you want copies of
any
of these. Our long-term dream is that the entire mathematics curriculum
should be historically based, with original sources playing a role
throughout,
and we ourselves are endeavoring to incorporate both history and
original
sources into all the courses we teach.
More recently David has teamed up with other colleagues from
mathematics and computer science in
applying our approach to the teaching of discrete mathematics, broadly
conceived. We are
combining
the pedagogy of student projects (introduced into our
calculus classes years ago) with the pedagogy of using original
historical
sources, in a NSF-funded program to develop and test student projects
written
using primary sources for teaching discrete
mathematics.
Teaching with historical sources has also led us to several research
projects in the history of mathematics, as shown in our articles
listed below.
emerged from the lower division Spirit and
Evolution
of Mathematics course. This book was written with
support
from the National Science Foundation's Division of Undergraduate
Education,
and is available fromSpringer in
paperback or hardcover in their
Undergraduate
Texts in Mathematics/Readings in Mathematics series.
The cover features portraits of five mathematicians whose original
writings
are at the heart of our five chapters, overlain with Sophie Germain's
handwriting
from a letter she wrote to Gauss in May of 1819 on her work on
Fermat's
Last Theorem, also featured in the book. See if you can read what
Germain
wrote to Gauss, or identify the people in the portraits. The book
includes
translations of Germain's letter and manuscripts, and ninety-four
portraits, mosaics, artwork, facsimiles of handwritten
manuscripts
and letters, and figures.
From the back cover
This book contains the stories of five mathematical journeys into
new realms, told through the writings of the explorers
themselves.
Some were guidedby mere curiosity and the thrill of
adventure,
while others had more practical motives. In each case the outcome
was a vast expansion of the known mathematical world and the
realization
that still greater vistas remained to be explored. The authors
tell
these stories by guiding the reader through the very words of the
mathematicians
at the heart of these events, and thereby provide insight into the art
of approaching mathematical problems.
The book can be used in a variety of ways. The five
chapters
are completely independent, each with varying levels of mathematical
sophistication.
The book will be enticing to students, to instructors, and to the
intellectually
curious reader. By working through some of the original sources
and
supplemental exercises, which discuss and solve -- or attempt to solve
-- a great problem, this book helps the reader discover the roots of
modern
problems, ideas, and concepts, even whole subjects. Students will also
see the obstacles that earlier thinkers had to clear in order to make
their
respective contributions to five central themes in the evolution of
mathematics.
Mathematical
Expeditions is suitable for several types of college courses:
Here you can also view the book's
preface
(which discusses teaching uses for the book), the table of contents,
some
chapter synopses, and some excerpts from various sections. (The figures
and photos don't show up here, the page numbers don't match those
in the table of contents, and page breaks and spacing are different
from
the actual published book.) We welcome your questions, or requests for
further excerpts you would like to see. We will add other synopses or
excerpts
from time to time.
Together with our colleagues Arthur
Knoebel and Jerry Lodder,
and
with further support from the National Science Foundation, we have
written an elder sibling for Mathematical
Expeditions. The new book Mathematical
Masterpieces contains annotated original
sources
from our upper division course Great
Theorems: The Art of Mathematics, presented as a capstone
for the undergraduate mathematics curriculum.
The book is available now fromSpringer, in
hardcover or paperback, in their
Undergraduate
Texts in Mathematics/Readings in Mathematics series.
The cover features portraits of
mathematicians whose original
writings
are at the heart of our four chapters. See if you can identify the
people in the portraits. The cover also shows a figure by Huygens
from the construction of an evolute in his Horologium Oscillatorium (The
Pendulum Clock), in our chapter on the development of the concept of
curvature. And we display Chinese text by Qin Jiu-Shao on approximating
roots of polynomials, from our chapter on numerical solutions of
equations. The book
has many portraits, artwork, facsimiles of original works, and
figures.
From the back cover
Experience the
discovery of mathematics by reading the original work of
some of the greatest minds throughout history. Here are
the stories of
four mathematical adventures, including the Bernoulli numbers as
the
passage between discrete and continuous phenomena, the search for
numerical solutions to equations throughout time, the discovery of
curvature and geometric space, and the quest for patterns in prime
numbers. Each story is told through the words of the pioneers of
mathematical thought. Particular advantages of the historical
approach
include providing context to mathematical inquiry, perspective to
proposed conceptual solutions, and a glimpse into the direction
research has taken. The text is ideal for an undergraduate
seminar,
independent reading, or a capstone course, and offers a wealth of
student exercises with a prerequisite of at most multivariable
calculus.
Mathematical Masterpieces
is suitable for several types of college courses:
a history of mathematics course
at the upper division,
a capstone course for mathematics majors,
upper division enrichment for majors in secondary mathematics
education, engineering, or the sciences.
Mathematical Masterpieces has
been reviewed
by the Mathematical Association of America.
David is part of a team of mathematicians and computer scientists at
this and other universities,
who are applying this approach to the teaching of discrete mathematics,
broadly conceived. We are
melding
the pedagogy of teaching with student projects (introduced into our
calculus classes years ago) with the pedagogy of using original
historical
sources, in a NSF-funded program to develop, test, evaluate, and
disseminate student projects
written
using primary historical sources for courses in discrete mathematics,
combinatorics, abstract algebra, logic, and algorithmic thought in
computer science. See our web
pages
Teaching
Discrete Mathematics via Primary Historical Sources for the
pedagogy and results of our Phase I NSF pilot grant, including the
classroom projects developed and published under that grant through
year 2006. See our web pages Learning Discrete
Mathematics and Computer Science via Primary Historical Sources for
the work commencing in year 2008 under our Phase II NSF expansion
grant, including the many new projects being created under that grant.
We welcome those who would like to use or test our student projects. |
Beecher, Penna, and Bittinger's Algebra and TrigonometryThe Bittinger Graphs and Models Series helps readers learn algebra by making connections between mathematical concepts and their real-world applications. Abundant applications, many of which use real data, offer students a context for learning the math. The authors use a variety of tools and techniques—including graphing calculators, multiple approaches to problem solving, and interactive features—to engage and motivate all types of learners |
Product Description
Now in Algebra, we use a lot of variables, which we can write as letters.Our algebra mission in this module is to know these unknowns! In algebra, we don't dilly-dally or mess about - we find things out. Topics include: Cost Functions Input and Output Trend Lines Graphs Domain and RangeIncludes a DVD plus a CD-ROM with teacher's guide, quizzes, graphic organizers and classroom activities. Teaching Systems programs are optimized for classroom use and include "Full Public Performance Rights".Grade Level: 8 - 12. 26 minutes |
Equations, Inequalities, & VICs eBook Description
The Equations, Inequalities, & VICs Guide covers algebra in all its various forms (and disguises) on the GMAT. It will help you master fundamental techniques and nuanced strategies to help you solve for unknown variables of every type.Each chapter builds comprehensive content understanding by providing rules, strategies, and in-depth examples of how the GMAT tests a given topic and how you can respond accurately and quickly. The Guide contains a total of 162 "In-Action" problems of increasing difficulty with detailed answer explanations. The content of the book is aligned to the latest Official Guides from GMAC (12th edition).Purchase of this book includes 6 months of access to Manhattan GMAT's online practice exams and Equations, Inequalities, & VICs question bank.
Popular Searches
The book Equations, Inequalities, & VICs by Manhattan Gmat
(author) is published or distributed by MG Prep Inc. [, 9781935707271-BEEPB].
This particular edition was published on or around 2009-5-1 date.
Equations, Inequalities, & VICs is available for use in eBook binding.
This book by Manhattan Gmat |
Product Description
In this program we are going to raise the hood and show you how algebra works! With some things in algebra, it's just a step-by-step process to get to where you want to go and that's exactly what we are going to work on today, the mechanics of algebra! Topics Covered: Rule of Equality Solving for Solving Equations with Variables on Both Sides Absolute Value Simplifying Expressions Solving InequalitiesIncludes a DVD plus a CD-ROM with teacher's guide, quizzes, graphic organizers and classroom activities. Teaching Systems programs are optimized for classroom use and include "Full Public Performance Rights". Grade Level: 8 - 12. 26 |
This collection is included inLens:Community College Open Textbook Collaborative By: CC Open Textbook Collaborative
Comments:
"Reviewer's Comments: 'I recommend this book as a "required primary textbook." This text attempts to lower the barriers for students that take courses such as Principles of Electrical Engineering, […]"
Click the
"College Open Textbooks"
link to see all content they endorseLinear Algebra: Matrices
Note:
This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.
The word matrix dates at least to the thirteenth century, when it was used to describe the rectangular copper tray, or matrix, that held individual leaden letters that were packed into the matrix to form a page of composed text. Each letter in the matrix, call it aijaij, occupied a unique position (ij)(ij) in the matrix. In modern day mathematical parlance, a matrix is a collection of numbers arranged in a two-dimensional array (a rectangle). We indicate a matrix with a boldfaced capital letter and the constituent elements with double subscripts for the row and column:
Figure 1: Matrix
In this equation AA is an mxnmxn matrix, meaning that AA has mm horizontal rows
and nn vertical columns. As an extension of the previously used notation, we
write A∈RmxnA∈Rmxn to show that AA is a matrix of size
mxnmxn with
aij∈Raij∈R.
The scalar element
aijaij is located in the matrix at the
ithith row and the
jthjth
column. For example, a23a23 is located in the second row and the third column
as illustrated in Figure 1.
The main diagonal of any matrix consists of the elements aiiaii. (The
two subscripts are equal.) The main diagonal runs southeast from the top left
corner of the matrix, but it does not end in the lower right corner unless the
matrix is square (∈Rm×m)(∈Rm×m).
The transpose of a matrix A∈Rm×nA∈Rm×n is another matrix BB whose
element in row jj and column ii is bji=aijbji=aij for 1≤i≤m1≤i≤m and 1≤j≤n1≤j≤n.
We write B=ATB=AT to indicate that BB is the transpose of AA. In MATLAB,
transpose is denoted by A'. A more intuitive way of describing the transpose
operation is to say that it flips the matrix about its main diagonal so that
rows become columns and columns become rows.
Exercise 1
If A∈Rm×nA∈Rm×n, then AT∈?_AT∈?_. Find the transpose of the matrix
A=215479.A=215479.
Matrix Addition and Scalar Multiplication. Two matrices of the same size (in both dimensions) may be added or subtracted in the same way as
vectors, by adding or subtracting the corresponding elements. The equation C=A±BC=A±B means that for each i and j,cij=aij±bijj,cij=aij±bij. Scalar multiplication of a matrix multiplies each element of the matrix by the scalar:
Matrix Multiplication. A vector can be considered a matrix with only one column. Thus we intentionally blur the distinction between Rn×1Rn×1 and Rn. Also a matrix can be viewed as a collection of vectors, each column of the matrix being a vector:
In the transpose operation, columns become rows and vice versa. The transpose of an n×1n×1 matrix, a column vector, is a 1 ×n×n matrix, a row vector:
x=x1x2|xn;xT=[x1x2xn].x=x1x2|xn;xT=[x1x2xn].
(3)
Now we can define matrix-matrix multiplication in terms of inner products of vectors. Let's begin with matrices A∈Rm×nA∈Rm×n and B∈Rn×pB∈Rn×p. To find the product AB, first divide each matrix into column vectors and row vectors as follows:
Figure 2
Thus aiai is the ithith column of A and
αjTαjT is the jthjth row of A. For matrix
multiplication to be defined, the width of the first matrix must match the
length of the second one so that all rows αiTαiT and columns bibi have the same
number of elements nn. The matrix product, C=C= AB, is an m×pm×p matrix
defined by its elements as cij=(αi,bj)cij=(αi,bj). In words, each element of the product
matrix, cijcij, is the inner product of the ithith row of the first matrix and the jthjth
column of the second matrix.
For n-vectors xx and yy, the matrix product xTyxTy takes on a special
significance. The product is, of course, a 1×11×1 matrix (a scalar). The special
significance is that xTyxTy is the inner product of xx and yy:
Another special case of matrix multiplication is the outer product. Like
the inner product, it involves two vectors, but this time the result is a matrix:
Figure 4
In the outer product, the inner products that define its elements are between one-dimensional row vectors of xx and one-dimensional column vectors of yTyT, meaning the (i,j)(i,j) element of AA is xiyjxiyj.
Exercise 2
Find C=ABC=AB where AA and BB are given by
A=1-12305,B=0-21-342202-231;A=1-12305,B=0-21-342202-231;
A=1001,B=12345678A=1001,B=12345678 ;
A=1-1-11-11111,B=036147258A=1-1-11-11111,B=036147258.
There are several other equivalent ways to define matrix multiplication, and a careful study of the following discussion should improve your
understanding of matrix multiplication. Consider A ∈Rm×n,B∈Rn×p∈Rm×n,B∈Rn×p, and C=ABC=AB so that C∈Rm×pC∈Rm×p. In pictures, we have
m[Cp]=m[An]pBn.m[Cp]=m[An]pBn.
(4)
In our definition, we represent CC on an entry-by-entry basis as
cij=(αi,bj)=Σk=1naikbkj.cij=(αi,bj)=Σk=1naikbkj.
(5)
In pictures,
Figure 5
You will prove in Exercise 3 that we can also represent CC on a column
basis:
cj=Σk=1nakbkj.cj=Σk=1nakbkj.
(6)
Figure 6
Finally, CC can be represented as a sum of matrices, each matrix being an
outer product:
C=Σi=1naiβiTC=Σi=1naiβiT
(7)
Figure 7
A numerical example should help clarify these three methods.
Example 1
Let
A=121321243321,B=121221132211.A=121321243321,B=121221132211.
(8)
Using the first method of matrix multiplication, on an entry-by-entry basis, we have
as we had in each of the other cases. Thus we see that the methods are
equivalent-simply different ways of organizing the same computation!
Exercise 3
Prove that Equations 9, 11, and 13 are equivalent definitions of matrix multiplication. That is, if C=ABC=AB where A∈Rm×nA∈Rm×n and
B∈Rn×pB∈Rn×p, show that the matrix-matrix product can also be defined by
cij=∑k=1naikbkj,cij=∑k=1naikbkj,
(18)
and, if ckck is the kthkth column of CC and akak is the kthkth column of AA, then
cj=∑k=1nakbkj.cj=∑k=1nakbkj.
(19)
Show that the matrix CC may also be written as the ' "sum of outer products"
C=∑k=1nakβkT.C=∑k=1nakβkT.
(20)
Write out the elements in a typical outer product akβkTakβkT.
Exercise 4
Given A∈Rm×n,B∈Rp×qA∈Rm×n,B∈Rp×q, and C∈Rr×sC∈Rr×s, for each of the following postulates, either prove that it is true or give a counterexample showing that it is false:
(AT)T=A(AT)T=A.
AB=BAAB=BA when n=pn=p and m=qm=q. Is matrix multiplication commutative?
A(B+C)=AB+ACA(B+C)=AB+AC when n=p=rn=p=r and q=sq=s. Is matrix multiplication distributive over addition?
((AB )T=BTAT)T=BTAT when n=pn=p.
(AB)C=A(BC)(AB)C=A(BC) when n=pn=p and q=rq=r. Is matrix multiplication associative?
Example 2: Rotation
We know from the chapter on complex numbers that a complex number z1=x1+jy1z1=x1+jy1 may be rotated by angle θθ in the complex plane by forming the product |
Lecturer: Keith Stroyan, The University of Iowa
Lecture 1: Some Intuitive Proofs with "Small" quantities This
lecture gives intuitive arguments to "prove" some basic theorems
like the Extreme Value Theorem, the Fundamental Theorem of Integral
Calculus, and the Inverse Function Theorem. We discuss why more
precise foundations are needed - at least in the teacher's mind.
Modern infinitesimals can help make fundamental theoretical reasoning
more accessible to students.
Lecture 2: Keisler's Foundations for Infinitesimal Analysis
Keisler's Axioms as a solution to doing calculus with infinitesimal
numbers. Completion of the proofs of lecture 1 (that teachers
should know to confidently encourage beginning students.)
Lecture 3:
Local Linearity and Infinitesimal Microscopes Everyone
knows the main idea of differential calculus is that "smooth
functions are locally linear." This lecture uses infinitesimals to
make this more precise in various concrete cases up to an
infinitesimal view of Stokes' theorem.
Lecture 4:
Higher Level Analysis
Modern analysis can use infinitesimals in infinite dimensional
spaces
by using foundations for "more" of mathematics than Keisler's Axioms.
This talk briefly describes suitable settings, Nelson's Idealization
Principle, and saturation with the example of differential calculus in
locally convex topological vector spaces where there is no topology
for the derivative maps |
additive inverse
arithmetic sequence
A sequence of numbers or terms in which each term after the first is obtained from the term that precedes it by adding the same constant, called the common difference. The nth term of an arithmetic sequence is a`n = a`1 + (n-1)d where a`1 is the first term and d is the common difference.
factorial n
field
A set of numbers and two operations that are defined on the set such that there exist identity and inverse elements for each operation and the closure, commutative, associative, and distributive properties hold. The set of real numbers forms a field with respect to the operations of addition and multiplication.
FOIL
function
fundamental theorem of algebra
A polynomial equation has at least one root, real or nonreal. An nth-degree polynomial equation has exactly n roots, provided each root is counted as many times as it occurs. The roots may be real or imaginary. If a polynomial equation with real coefficients has imaginary roots, then they occur in conjugate pairs.
geometric sequence
A sequence of numbers or terms in which each term after the first is obtained by multiplying the term that precedes it by the same constant, called the common ratio. The nth term of a geometric sequence is a`n = a`1r^(n-1), where a`1 is the first term and r is the common ratio.
geometric series
greatest common factor (GCF)
This of two or more monomials is the monomial with the greatest coefficient and the variable factors of the greatest degree that are common to all the given monomials. The GCF of 8a^2b and 20ab^2 is 4ab.
index
inequality
A sentence that expresses an inequality relation by using a symbol such as < (is less than), <(w/ line under; is less than or equal to), > (is greater than), > (with line under; is greater than or equal to), or =(with line through; is unequal to).
matrix addition
matrix multiplication
If A is an m x n matrix and B is an n x p matrix, then AB is an m x p matrix whose entry in the ith row and jth column is the sum of the products obtained by multiplying the elements in the ith row of matrix A by the corresponding elements in the jth column of matrix B. |
Product Details:
From the Publisher: This book demonstrates scientific computing by presenting twelve computational projects in several disciplines including Fluid Mechanics, Thermal Science, Computer Aided Design, Signal Processing and more. Each follows typical steps of scientific computing, from physical and mathematical description, to numerical formulation and programming and critical discussion of results. The text teaches practical methods not usually available in basic textbooks: numerical checking of accuracy, choice of boundary conditions, effective solving of linear systems, comparison to exact solutions and more. The final section of each project contains the solutions to proposed exercises and guides the reader in using the MATLAB scripts available online.
Description:
The mathematical implications of personal beliefs and values in science
and commerce Amid a worldwide resurgence of interest in subjectivist statistical method, this book offers a fresh look at the role of personal judgments in statistical analysis. Frank Lad ...
Description:
Offering a clear, precise, and accessible presentation, complete with MATLAB
programs, this new Third Edition of Elementary Numerical Analysis gives students the support they need to master basic numerical analysis and scientific computing. Now updated and revised, this significant ...
Description:
Each chapter provides an exploration section with problems that deal
with software evaluation, selection, modificationm and the solution of not entirely open ended problems.Book Format: Hardcover. Number of Pages: 0473. Publisher: Jones & Bartlett Publishers |
Main menu
Learn more about our programs
Big Ideas Math
Our middle school math program, Big Ideas Math, is the only comprehensive math program developed for the Common Core State Standards that delivers instruction for all students spanning 6th grade topics through Algebra 1.
Learn more about our programs
Algebra Within Reach
The AlgebraWithinReach.com companion site provides a robust selection of resources and tools to assist you in your study of Algebra. This 100% free website accompanies Ron Larson's Algebra Within Reach series of textbooks.
Learn more about our programs
Larson Precalculus
The LarsonPrecalculus.com companion site provides a robust selection of resources and tools to assist you in your study of Precalculus. This 100% free website accompanies Ron Larson's Precalculus series of textbooks.
Learn more about our programs
Larson Calculus
The LarsonCalculus.com companion site provides a robust selection of resources and tools to assist you in your study of Calculus. This 100% free website accompanies Ron Larson's 10th edition of Calculus.
About Larson Texts, Inc.
Larson Texts, Inc. produces student-friendly, market-leading math textbooks for sixth grade through college calculus that are used by over five million students each year. The textbooks are published and distributed by Big Ideas Learning, Cengage Learning, Houghton Mifflin Harcourt, Pearson, and W. H. Freeman. |
Bring two spiral bound notebooks to class each day. One of the notebooks is for class notes and the other notebook is for the homework assignments. Bring to class your homework in your math homework notebook. You should bring a red pen to class to do your math corrections. Textbook and your TI 30 scientific calculator should be brought to class daily.
ASSESSMENTS:
• Combination of quizzes, tests and mastery outcome assessments
• Teacher and/or student can initiate one further attempt on tests, quizzes or assessments within one week. There may be circumstances under which further attempts may not be offered.
• A student is eligible for a further attempt only if he/she demonstrates that additional effort was put forth to improve learning. (e.g. study/learning)
TECHOLOGY:
Texas Instrument Scientific Calculator TI-30
HOMEWORK:
Students will have homework as needed to reinforce math concepts learned. This is usually 4 to 5 nights a week. Each homework assignment should take approximately 20-30 minutes per night.
RESOURCES:
University of Chicago School Mathematics Project: Algebra , Third Edition
Published by Wright Group/McGraw Hill
I have been teaching math at Pleasantdale Middle School for twenty-one years. I received a Bachelor of Science degree from the University of Illinois at Chicago and then continued my education and received a Masters of Science degree from Florida State University.
Focus Statement:Students will solve multi-step linear equations and inequalities using the algebraic properties of the real number system.They will also graph linear equations using slope and intercept on the coordinate plane.
M.8.1Students will create a diagram of the real number system and apply the basic properties of the real number system to simplify algebraic expressions.
M.8.12.4Multiply a polynomial by a monomial using the Distributive Property.
M.8.12.5Multiply a binomial by a binomial using FOIL.
M.8.12.6Factorpolynomials with a greatest common monomial factor only.
M.8.12.7Factor trinomials using reverse FOIL.
Above-level 8th grade Math
Focus Statement:Students will solve multi-step linear, quadratic and absolute value equations and inequalities using the algebraic properties of the real number system.They will also graph linear, quadratic and absolute value equations and inequalities
MA.8.1 Students will create a diagram of the real number system and apply the basic properties of the real number system to simplify algebraic expressions. |
Constant Rule Teacher Resources
Find Constant Rule educational ideas and activities
Title
Resource Type
Views
Grade
Rating
Twelfth graders investigate derivatives. In this calculus lesson, 12th graders use technology to explore the basic derivatives and how to choose the proper formula to use them. The lesson requires the use of the TI-89 or Voyage and the appropriate application.
In this time constant worksheet, students answer 52 questions about the rate of current changes and voltage changes in capacitors. They analyze circuits and determine the time it takes for capacitors to change voltage and they specify voltages at specified times.
In this time constant circuits worksheet, students answer 23 questions about the design of circuits that need time delays, about capacitors and inductors, and about resistors and the design of circuits.
Pupils practice the concept of graphing associated to a function with its derivative. They define the concepts of increasing and decreasing function behavior and explore graphical and symbolic designs to show why the derivative can be used as an indicator for the behavior.
Students analyze graphs and determine their general shape. In this calculus lesson, students solve functions by taking the derivative, sketch tangent lines and estimate the slope of the line using the derivative. They graph and analyze their answers.
In this Calculus worksheet, 12th graders are provided with practice problems for their exam. Topics covered include limits, derivatives, area bounded by a curve, minimization of cost, and the volume of a solid of revolution. The four page document contains seventeen multiple choice questions. Answers are not included.
In this AP Calculus Practice Exam instructional activity, students solve seventeen multiple-choice questions using a graphing calculator. This BC practice test is designed to be finished in fifty minutes.
In this AP Calculus practice test activity, students prepare for the BC version on the test by solving seventeen multiple-choice questions using a calculator. The test should be timed and 50 minutes in length.
In this calculus activity, students use integration to solve word problems they differentiate between integration and anti derivatives, and between definite and indefinite integrals. There are 3 questions with an answer key.
In this calculus worksheet, 12th graders differentiate and integrate basic trigonometric functions, calculate rates of change, and integrate by substitution and by parts. The twenty-two page worksheet contains explanation of the topic, numerous worked examples, and sixteen multi-part practice problems. Answers are not provided.
In this calculus worksheet, students evaluate functions and solve problems using the derivative. They apply the rules of limits to solve functions where the limit of x approaches zero. There are 12 problems to solve.
Twelfth graders investigate the capabilities of the TI-89. In this calculus lesson, 12th graders explore the parametric equation for a circle, for arc length of curves, and for trajectories. Students investigate the symbolic and graphical representation of vectors. Students use polar functions of investigate the area bounded by a curve. Students investigate a 3D graphing application.
Students analyze implicit differentiation using technology. In this calculus lesson, students solve functions dealing with implicit differentiation on the TI using specific keys. They explore the correct form to solve these equations.
In this circuits worksheet, students answer 25 questions about passive integrator circuits and passive differentiator circuits given schematics showing voltage. Students use calculus to solve the problems. |
Numerical Computing with MATLAB
By Cleve Moler, MathWorks
How much did the reduced air resistance in Mexico City contribute to Bob Beamon's extraordinary performance in the long jump at the 1968 Olympic games? What is the effect of burning fossil fuels on the carbon dioxide concentration in the Earth's atmosphere? How does the expected return for the game of blackjack change if you remove all the kings from the deck of cards? How does Google rank Web pages?
These are examples of the programming projects that tie together several of the topics covered in Numerical Computing with MATLAB, my new textbook for an introductory course in numerical methods, MATLAB, and technical computing. I am pleased to describe the book for you here.
The emphasis in the book is on informed use of mathematical software. I want students to learn enough about the mathematical functions in MATLAB that they will be able to use them correctly, appreciate their limitations, and modify them when necessary to suit their own needs.
The Chapters
Introduction to MATLAB
Linear Equations
Interpolation
Zero and Roots
Least Squares
Quadrature
Ordinary Differential Equations
Random Numbers
Fourier Analysis
Eigenvalues and Singular Values
Partial Differential Equations
George Forsythe initiated such a software-based numerical methods course at Stanford University in the late 1960s. The 1977 textbook by Forsythe, Malcolm, and Moler and the 1989 textbook by Kahaner,Moler, and Nash that evolved from the Stanford course were based upon libraries of Fortran subroutines. This new book can be thought of as a modern, MATLAB oriented replacement for those texts.
Two editions of the book are available, an electronic edition published by The MathWorks and a conventional print edition published by the Society for Industrial and Applied Mathematics (SIAM).
The book is intended for students in engineering and science who want to have a better understanding of the numerical methods implemented in MATLAB and similar mathematical software systems. The prerequisites for the course, and the book, include:
Calculus
Some familiarity with ordinary differential equations
Some familiarity with matrices
Some computer programming experience
If students have never used MATLAB, the first chapter will help them get started. If they are already familiar with MATLAB, they can glance over most of the first chapter quickly. Everyone should read the section in the first chapter about floating-point arithmetic.
Regular readers of Cleve's Corner will find the book familiar. Several sections of the book, including the ones on floating-point arithmetic, stiffness, random number generation, and the L-shaped membrane, are expanded versions of previous columns in MATLAB News & Notes.
A collection of more than 70 M-files, which I refer to as NCM, forms an essential part of the book. These are available from the MathWorks Web site devoted to the book. There are three types of NCM files:
gui files: interactive graphical demonstrations
tx files: textbook implementations of built-in MATLAB functions
others: miscellaneous files, primarily associated with exercises
For example, one of the of the tx files, lutx, shows the algorithm used by the built-in function involved in the most important computation in MATLAB: the solution of simultaneous linear equations. Another tx file, ffttx, provides a compact implementation of the fast algorithm for the finite Fourier transform. The original Forsythe et al. text was successful in part because its Fortran programs were small enough to be read and understood. In retrospect, I think the codes distributed with Kahaner et al. were too large and unwieldy. MATLAB enables us to return to programs that can be printed in the book and discussed in class.
The book makes extensive use of computer graphics.When you have NCM available, the MATLAB statement
ncmgui
produces a window where each of 20 thumbnail plots launches a graphical demonstration of some problem, algorithm, or mathematical application (Figure 1).Most of these GUI programs are interactive. You may already be familiar with eigshow, because it is distributed with MATLAB. The other GUIs have the same spirit as eigshow. For example, with lugui, you choose the pivots in Gaussian elimination.With fzerogui, the plot zooms in on the point where a function crosses the x-axis.
Figure 1: Each of 20 thumbnail plots launches a graphical demonstration of some problem, algorithm, or mathematical application. Click on image to see enlarged view.
There are more than 200 exercises. Some of them involve modifying and extending the programs in NCM. I don't want students to write their own programs from scratch. I would rather they start with the programs that I've written. For example, one exercise asks them to modify lutx to keep track of the sign of the permutation and then compute the determinant. Another exercise has students remove the step size calculation from the ODE solver, ode23tx, and implement the classical Runge-Kutta algorithm with fixed-step size.
Some of the exercises involve ill-posed problems.What does bslashtx do with a singular system of linear equations? How does quadtx behave if the integral doesn't exist?
An exercise from Forsythe et al. that involved U. S. Census data from 1900 through 1970 now has three more data points and a GUI. Exercises in the chapter on least squares examine the Statistical Reference Datasets from the National Institute of Standards and Technology.
A Cleve's Corner about Google, "The World's Largest Matrix Computation," has been expanded to become a section on "PageRank and Markov Chains" and several exercises about sparse matrices and Web graphs. The example about touch-tone dialing that has been in MATLAB for many years, phone.m, has been expanded into the section introducing the FFT.
I've been working on this book for several years. Along with a few colleagues, I've had a chance to use it in both undergraduate and graduate university courses, and in two-day MathWorks training courses. I hope that students and teachers will find it useful for their own courses, and in their own individual study. |
ALEX Lesson Plans
Title: Zero Product Property
Description:
The zero product property allows you to solve a quadratic equation by converting it into two linear equations. This is a common strategy of algebra- to break down a problem into simpler parts, each solved by previously learned methods.
Standard(s): [MA2010] AL1 (9-12) 18: Solve quadratic equations in one variable. [A-REI4]
Subject: Mathematics (9 - 12) Title: Zero Product Property Description: The zero product property allows you to solve a quadratic equation by converting it into two linear equations. This is a common strategy of algebra- to break down a problem into simpler parts, each solved by previously learned methods. Now, where did THAT come from? Deriving the Quadratic Formula
DescriptionStandard(s): [MA2010] AL1 (9-12) 18: Solve quadratic equations in one variable. [A-REI2 (9-12) 4: Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] [MA2010] ALC (9-12) 6: Use the extreme value of a given quadratic function to solve applied problems. (Alabama)
Subject: Mathematics (9 - 12) Title: Now, where did THAT come from? Deriving the Quadratic Formula Description represented Classifying Complex Numbers
Description:
This lesson helps students distinguish between strictly complex numbers, strictly real numbers and strictly imaginary numbers while learning that real numbers and imaginary numbers are subsets of the set of complex numbers.
Standard(s): [MA2010] AL1 (9-12) 18: Solve quadratic equations in one variable. [A-REI4] [MA2010] AL2 (9-12) 1: Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. [N-CN1] [MA2010] ALT (9-12) 1: Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. [N-CN1] [MA2010] MI1 (9-12) 4: Explain the development and uses of sets of numbers, including complex, real, rational, irrational, integer, whole, and natural numbers. (Alabama) [MA2010] PRE (9-12) 1: (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. [N-CN4] |
This paper addresses the question: how do mathematics teachers make meaning from curriculum statements in relation to their...
see more
RSA Laboratories' Frequently Asked Questions About Today's Cryptography is an extensive collection of questions about modern...
see more
RSA Laboratories' Frequently Asked Questions About Today's Cryptography is an extensive collection of questions about modern cryptography, cryptanalysis, and related issues. The information is presented in question and answer form and covers eight chapters of topics including cryptography techniques, applications, readings, and an appendix containing relevant mathematical concepts from abstract algebra and number theory.
This activity would be done at the end of the school year in a pre-algebra class. It is a way to introduce algebra and its...
see more
This activity would be done at the end of the school year in a pre-algebra class. It is a way to introduce algebra and its history, putting some personality into the abstractness of the subject by researching the individuals behind algebraic concepts. It was initially found on the following site five years ago when I first did it with my classes: It has since disappeared, however, so the specific modifications I made at the time are fuzzy at best, but I have made recent adjustments to every portion.Introduction:Algebra, what does it mean? Where did it come from? Who thought up this stuff? Have you ever wondered what the word algebra means or when and where algebra was developed or who developed algebraic concepts? In this project your group will go on a journey through time and the history of mathematics to discover the answers to these questions.Task:Each group will go on a quest to find the mathematicians' histories that have named as being the fathers or founders of algebra. On this journey your group will collect information about the mathematician responsible for developing the algebraic concept assigned to your group, create a timeline to show when the concept was developed in relation to other significant events in history, and find examples of the algebraic concept. Each group will prepare a Powerpoint to present the information to the class.Group I The Father of Algebra (Algebraic thought and equations)Group II Founder of Cartesian Plane and Graphing EquationsGroup III Developer of PolynomialsGroup IV Set Notation and Venn Diagrams DesignerEach group will need a Researcher, Recorder, Mathematician, and a Reporter.Researcher - Using the resources below, work with the Recorder to find and record needed information for your topic.Recorder - Record information on your topic and citation for where the information was found. Work with the Researcher and the Reporter to prepare a report of the findings of your group.Mathematician - Work with the Researcher and the Recorder to find examples of mathematical problems from your assigned topic. Choose two examples that you can share, with which you can demonstrate the topic for the class.Reporter - Work with the other members of your group to create a presentation, using PowerPoint, which you will present to the class.
The final project is a joint effort between Gary Bengier and Mitchell Oster.We shall attempt to explain a bit about celestial...
see more
The final project is a joint effort between Gary Bengier and Mitchell Oster.We shall attempt to explain a bit about celestial mechanics. Specifically, we will begin with the very basics——orbits as ellipses—the main parameters that define an ellipse—Kepler's laws, and how they relate planetary motion to elliptical orbits—how the mathematician Gauss predicted the location of the asteroid Ceres in 1801 (thereby becoming world famous), and the techniques that he used—how astrophysicists can predict the orbits of bodies in space, based upon only a few measurements (2-3 observations of position at specific times)We will use a few "hands-on" mathematical models that you can manipulate to see at the least the basics of how this works in practice.
The web site "Learning Disabilities OnLine, the educators page", is a wonderful tool to have when dealing with...
see more
The web site "Learning Disabilities OnLine, the educators page", is a wonderful tool to have when dealing with disabled students. This web site offers a variety of topics such as Differentiating instruction, Assessment, Language and reading skills, Learning strategies, Teaching mathematics, Teaching writing , Using technology in the classroom, as well as Accommodations , Adolescents , Inclusion , Behavior and social skills, Working with parents , Special Ed requirements . This web site proposes a wide spectrum of articles about those issues as well as a glossary and Q&A. It is a very good resource that shows that disabled students can have an opportunity to learn and succeed.
Assists teachers in understanding and interpreting the properties of numbers and provides a background to the numerous proofs...
see more
Assists teachers in understanding and interpreting the properties of numbers and provides a background to the numerous proofs and solutions to various mathematical equations. Material is crucial for the teaching of secondary school mathamatics.Compulsory Readings for Mathematics II: Number Theory (PDF)
In this paper, I illuminate the notion of mathematics for teaching (its matter) and argue that it matters (it is important),...
see more
In this paper, I illuminate the notion of mathematics for teaching (its matter) and argue that it matters (it is important), particularly for mathematics teacher education. Two examples from studies of mathematics classrooms in South Africa are described, and used to illustrate what mathematics teachers use, or need to use, and how they use it in their practice: in other words, the substance of their mathematical work. Similarities and differences across these examples, in turn, illuminate the notion of mathematics for teaching, enabling a return to, and critical reflection on, mathematics teacher education.
This course includes lecture notes, assignments, problems for group work in recitation, and a full set of lecture videos....
see more
This course includes lecture notes, assignments, problems for group work in recitation, and a full set of lecture videos. These video lectures of Professor Arthur Mattuck teaching 18.03 were recorded live in the Spring 2003. Professor Mattuck has inspired and informed generations of MIT students with his engaging lectur
This archive is designed as a resource for enriching your courses with mathematical Fun Facts! It is designed to pique the...
see more
This archive is designed as a resource for enriching your courses with mathematical Fun Facts! It is designed to pique the interest of students in different areas of mathematics. The fun facts were originally conceived as five minute warm ups at the beginning of lectures so that non mathematics majors would not think math was just calculus. Presentation suggestions are also given. |
Part of the Classroom Assessment for School Mathematics, K–12 Series, this
book considers new ways to assess students, focuses on assessment tasks, offers
ways to plan and conduct a coherent classroom assessment program, and discusses
assessment data.
This book focuses on algebra as a language of process, expands the notion
of variable, develops ideas about the representation of functions, and extends
students' understanding of algebraic equivalence and change.
This book examines the study of geometry in the middle grades as a pivotal point in the mathematical learning of students and emphasizes the geometric thinking that can develop in grades 6–8 as a result of hands-on exploration.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. |
Math in Our World - 2nd edition
ISBN13:978-0077356651 ISBN10: 0077356659 This edition has also been released as: ISBN13: 978-0072982534 ISBN10: 0072982535
Summary: The author team of Dave Sobecki, Angela Matthews, and Allan Bluman have worked together to create the second edition of Mathematics in Our World, an engaging text catered to the needs of todays liberal arts mathematics students. This revision focuses strict attention to a clear and friendly writing style, integration of numerous relevant real-world examples and applications, and implementation of the step-by-step approach used for years in Blumans Elementary Statistics: A Step by Ste...show morep Approach. The result is an exceptionally engaging text that is able to both effectively and creatively convey the basic concepts fundamental to a liberal arts math curriculum for even the most hesitant student67103206 |
Use Wolfram|Alpha to Solve Calculus Problems and…...
Use Wolfram|Alpha to Solve Calculus Problems and… Everything Else.
Wolfram|Alpha is like Google on crack. However, it is not technically a search engine; it is a "computational knowledge" engine. They use a huge collection of trustworthy, built-in data to get the user the information or knowledge they are looking for. When you search for an item, Wolfram|Alpha gives you all of the relevant knowledge they have on that specific search query. For example, here is the results for the search "when did the Beatles break up?" Not only do you get the date the Beatles broke up, you also get how long away that date is from today and other noteworthy events that occurred on the same day. Here is another example, for the search "carbon footprint driving 536 miles at 32mpg" that tells you the amount of fuel consumed and the amount of c02 and carbon emitted.
Because Wolfram|Alpha is just retrieving answers from its huge database of information and formulas, you have to be specific and ask non-opinionated questions. For example, the website does not know which Lil Wayne song is the best. However, it does know things that are not opinions, like the nutritional facts of 10,000 big macs and how many planes are currently flying directly over you.
I find Wolfram|Alpha to be better than Google when I am quickly looking for specific answers. I just typed in "Countries that border France" on both Wolfram|Alpha and Google. Wolfram|Alpha quickly showed me a list of the 8 countries and a map with of France with its bordering countries highlighted. Google on the other hand sent me over to Yahoo Answers…
Other than a fun search engine, Wolfram|Alpha can also be used as a highly effective tool for college. Like the title mentions, the knowledge engine can in fact solve any calculus problem. It can easily solve any math problem thrown its way, from a basic algebra problem to whatever this is.
Wolfram|Alpha can also be used for many other college courses such as biology, astronomy, history, etc.
As Wolfram|Alpha can be kind of confusing and hard to get the hang of at first, I suggest going through this short tour and looking at some examples to help give you a better sense of how to use it. Even if you find it a little bit confusing at first, keep trying because Wolfram|Alpha really is a great way to "hack college." |
More About
This Textbook
Overview
This book fills an educational void by adapting unique classroom-tested techniques that students find most congenial...that strip the shroud of mystery from an esoteric subject...that prepare students for applications of calculus in later courses.
"...fills an educational void by adapting unique, classroom- tested techniques that will prepare students for the applications of calculus in later courses."
Related Subjects
Read an Excerpt
The Calculus Tutoring Book
By Carol Ash Robert B. Ash
John Wiley & Sons
ISBN: 0-7803-1044-6
Chapter One
1/FUNCTIONS
We begin calculus with a chapter on functions because virtually all problems in calculus involve functions. We discuss functions in general, and then concentrate on the special functions which will be used repeatedly throughout the course.
1.1 Introduction
A function may be thought of as an input-output machine. Given a particular input, there is a corresponding output. This process may be represented by various schemes, such as a table or a mapping diagram listing inputs and outputs (Fig. 1). Functions will usually be denoted by single letters, the most common being f and g. If the function g produces the output 3 when the input is 2, we write g(2) = 3.
The input of a function f is called the independent variable, while the output is the dependent variable. We say that the function fmaps x toƒ(x), and call ƒ(x) the value of the function at x. The set of inputs is called the domain of f, and the set of outputs is the range.
A function ƒ(x) is not allowed to send one input to more than one output. Figure 2 illustrates a correspondence that is not a function. For example, it is illegal to write g(x) = ± [square root of (2[x.sup.2] + 3), since each value of x produces two outputs. It certainly is legal to write and use the expression ± [square root of 2[x.sup.2] + 3], but it cannot be named g(x) and called a function.
Functions often arise when a problem is translated into mathematical terms. The solution to the problem may then involve operating on the functions with calculus. Before continuing with functions in more detail we'll give an example of a function emerging in practice. Suppose a pigeon is flying from point A over water to point B on the beach (Fig. 3), and the energy required to fly is 60 calories per mile over water but only 40 calories per mile over land. (The effect of cold air dropping makes flying over water more taxing.) The problem is to find the path that requires minimum energy. The direct path from A to B is shortest, but it has the disadvantage of being entirely over water. The path ACB is longer, but it has the advantage of being mostly over land. In general, suppose the bird first flies from A to a point P on the beach x miles from ITLITL, and then travels the remaining 10 - x miles to B. The value x = 0 corresponds to the path ACB, and x = 10 corresponds to the path AB. The total energy E used in flight can be calculated as follows:
E = energy expended over water + energy expended over land (1)
= calories per water mile × water miles + calories per land mile × land miles
Thus the energy is a function of x. Calculus will be used in Section 4.2 to finish the problem and find the value of x that minimizes E.
In deriving (1), we restricted x so that 0 [less than or equal to] x [less than or equal to] 10 since we assumed that to minimize energy the bird should fly to a point P between C and B as indicated in Fig. 3. Since problems often restrict the independent variable in a similar fashion, certain notation and terminology has become standard.
The set of all x such that a [less than or equal to] x [less than or equal to] b is denoted by [a, b] and called a closed interval (Fig. 4). With this notation, the variable x in (1) lies in the interval. The set of all x such that a < x < b is denoted by (a, b) and called an open interval. Similarly we use [a, b) for the set of x where a [less than or equal to] x < b,(a, b] for a < x [less than or equal to] b, [a, [infinity]) for x [greater than or equal to] a, (a, [infinity]) for x > a, (-[infinity], a] for x [less than or equal to] a, and (-[infinity], a) for x < a. In general, the square bracket, and the solid dot in Fig. 4, means that the endpoint belongs to the set; a parenthesis, and the small circle in Fig. 4, means that the endpoint does not belong to the set. The notation (-[infinity], [infinity]) refers to the set of all real numbers.
As another example of a function, consider the greatest integer function: Int x is defined as the largest integer that is less than or equal to x. Equivalently, Int x is the first integer at or to the left of x on the number line. For example, Int 5.3 = 5, Int 5.4 = 5, Int 7 = 7, Int(-6.3) = - 7 . Note that for positive inputs, Int simply chops away the decimal part. The domain of Int is the set of all (real) numbers. (Elementary calculus uses only the real number system and excludes nonreal complex numbers such as 3i and 4 + 2i.) The range of Int is the set of integers. Frequently, Int x is denoted by [x]. Many computers have an internal Int operation available. To illustrate one of its uses, suppose that a computer obtains a numerical result, such as x = 2.1679843, and is instructed to keep only the first 4 digits. The computer multiplies by 1000 to obtain 2167.9843, applies Int to get 2167, and then divides by 1000 to obtain the desired result 2.167 or, in our functional notation, 1/1000 Int(1000 x).
Most work in calculus involves a few basic functions, which (amazingly) have proved sufficient to describe a large number of physical phenomena. As a preview, and for reference, we list these functions now, but it will take most of the chapter to discuss them carefully. The material is important preparation for the rest of the course, since the basic functions dominate calculus.
3. The number [x.sub.0] is called a fixed point of the function f if f([x.sub.0]) = [x.sub.0]; i.e., a fixed point is a number that maps to itself. Find the fixed points of the following functions: (a) |x|/x (b) Int x (c) [x.sup.2] (d) [x.sup.2] + 4.
6. A charter aircraft has 350 seats and will not fly unless at least 200 of those seats are filled. When there are 200 passengers, a ticket costs $300, but each ticket is reduced by $1 for every passenger over 200. Express the total amount A collected by the charter company as a function of the number p of passengers.
1.2 The Graph of a Function
Information can usually be perceived more easily from a diagram than from a set of statistics or a formula. Similarly, the behavior of a function can often be better understood from its graph, which is drawn in a rectangular coordinate system by using the inputs as x-coordinates and the outputs as y-coordinates; i.e., the graph of ƒ is the graph of the equation y = f(x). In sketching a graph it may be useful to make a table of values of the input x and the corresponding output y.
The graph of the function f(x) = -2x + 3 is the line with equation y = -2x + 3 (Fig. 1). It has slope -2 and passes through the point (0,3).
The graph of Int x is shown in Fig. 2 along with a partial table of values used to help plot the graph. The graph shows for instance that as x increases from 2 toward 3, Int x, the y-coordinate in the picture, remains 2; when x reaches 3, Int x suddenly jumps to 3.
Example 1 The graph of a function g is given in Fig. 3. Various values of g can be read from the picture: since the point (0,6) is on the graph, we have g(0) = 6; similarly, g(4) = 11, g(10) = 4. Since P is lower than Q, we can tell that g(2) < g(3). If g(x) represents the final height of a tree when it is planted with x units of fertilizer, then using no fertilizer results in a 6-foot tree, using 10 units of fertilizer overdoses the tree and it grows to only 4 feet, while 4 units of fertilizer produces an 11-foot tree, the maximum possible height according to the data.
The vertical line test Not every curve can be the graph of a function. The curve in Fig. 4 is disqualified because one x is paired with several y's, and a function cannot map one input to more than one output. In general, a curve is the graph of a function if and only if no vertical line ever intersects the curve more than once. In other words, if a vertical line intersects the curve at all, it does so only once.
Equations versus functions The hyperbola in Fig. 5 is the graph of the equation xy = 1. It is also (solve for y) the graph of the function f(x) = 1/x. The hyperbola in Fig. 6 is the graph of the equation [y.sup.2] - 2[x.sup.2] = 6. It is not the graph of a function because it fails the vertical line test. However, the upper branch of the hyperbola is the graph of the function [square root of (2[x.sup.2] + 6)] (solve for y and choose the positive square root since y > 0 on the upper branch), and the lower branch is the graph of the function - [square root of (2[x.sup.2] + 6)].
Continuity If the graph of f breaks at x = [x.sub.0], so that you must lift the pencil off the paper before continuing, then f is said to be discontinuous at x = [x.sub.0]. If the graph doesn't break at x = [x.sub.0], then f is continuous at [x.sub.0].
The function -2x + 3 (Fig. 1) is continuous (everywhere). On the other hand, Int x (Fig. 2) is discontinuous when x is an integer, and 1/x (Fig. 5) is discontinuous at x = 0.
Many physical quantities are continuous functions. If h(t) is your height at time t, then h is continuous since your height cannot jump.
One-to-one functions, non-one-to-one functions and nonfunctions A function is not allowed to map one input to more than one output (Fig. 7). But a function can map more than one input to the same output (Fig. 8), in which case the function is said to be non-one-to-one. A one-to-one function maps different inputs to different outputs (Fig. 9).
The function [x.sup.2] is not one-to-one because, for instance, inputs 2 and -2 both produce the output 4. The function [x.sup.3] is one-to-one since two different numbers always produce two different cubes.
A curve that passes the vertical line test, and thus is the graph of a function, will further be the graph of a one-to-one function if and only if no horizontal line intersects the curve more than once (horizontal line test). The function in Fig. 10 fails the horizontal line test and is not one-to-one because [x.sub.1] and [x.sub.2] produce the same value of y.
Constant functions If, for example, f(x) = 3 for all x, then f is called a constant function. The graph of a constant function is a horizontal line (Fig. 11). The constant functions are among the basic functions of calculus, listed in the table in Section 1.1.
Power functions Another group of basic functions consists of the power functions x' such as
To sketch the graph of [x.sup.3], we make a table of values and plot a few points. When the pattern seems clear, we connect the points to obtain the final graph (Fig. 12). The connecting process assumes that [x.sup.3] is continuous, something that seems reasonable and can be proved formally. In general, x' is continuous wherever it is defined. If r is negative then x' is not defined at x = 0 and is discontinuous there; the graph of 1/x, that is, the graph of [x.sup.-1], is shown in Fig. 5 with a discontinuity at the origin. Figure 13 gives the graph of [x.sup.2] (a parabola) and of [x.sup.4]. For -1 < x < 1, the graph of [x.sup.4] lies below the graph of [x.sup.2] since the fourth power of a number between -1 and 1 is smaller than its square; otherwise [x.sup.4] lies above [x.sup.2]. Figure 14 gives the graph of y = [square root of (x)], the upper half of the parabola x = [y.sup.2].
Increasing and decreasing functions Suppose that whenever a > b, we have f(a) > f(b); that is, as x increases, f(x) increases also. In this case, f is said to be increasing. The graph of an increasing function rises to the right (Figs. 12 and 14).
Suppose that whenever a > b, we have f(a) < f(b); that is, as x increases, f(x) decreases. In this case, f is decreasing. The graph of a decreasing function falls to the right (Fig. 1).
The functions [x.sup.2] and [x.sup.4] (Fig. 13) decrease on the interval (-[infinity], 0] and increase on [0, [infinity]); overall, on (-[infinity], [infinity]), they are neither increasing nor decreasing. The function 1/x (Fig. 5) decreases on the intervals (-[infinity], 0) and (0, [infinity]) but is neither decreasing nor increasing on the interval (-[infinity, [infinity]).
(Continues...)
Excerpted from The Calculus Tutoring Book by Carol Ash Robert B. Ash |
Through
the study of algebra, a student develops an understanding of the symbolic language
of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations. Topics covered include: single and two step equations, word problems, graphing, solving systems and introduction to geometry.
...I have a background as an actuary and MBA training. Therefore, I have experience with many of the areas of discrete math typically encountered in introductory college coursework: set theory, combinatorics, probability theory, matrices and operations research. Actuarial Science is the field of ... |
e2020 Algebra I ... page | 2. Introduction. The purpose of this study is to present
the procedure ... To correctly answer this item, students must first identify like
terms, then combine them by performing.
of its students were failing Algebra and Geometry courses, yet there were
insufficient supports in place ... The Key: Early Intervention ... it would be critical to
use e2020 as an early intervention. ... Page 2 ...
Page 2 ... Enable e2020's reader application ... this information he or she will
press the enter key on the keyboard or click the enter ... For example, in math the
student will click on either algebra 1A or.
E2020 has responded by creating a course designed with the online student ...
skills, and strategies for remembering key information. The Diagnostic ... Fire-
Algebra mm! .... open while they answer questions in the Online Content
Assignment. |
Book Description: Tips for simplifying tricky operationsGet the skills you need to solve problems and equations and be ready for algebra classWhether you're a student preparing to take algebra or a parent who wants to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations.* Understand fractions, decimals, and percents* Unravel algebra word problems* Grasp prime numbers, factors, and multiples* Work with graphs and measures* Solve single and multiple variable equations |
In this section we mainly set up some useful notation. While the ideas here are not very central to the study of algebra, they do come up from time to time, so pay attention! It is exactly because these ideas don't reoccur on every page that they can be confusing when they suddenly come up later on. So be prepared to revisit this section as necessary to refresh your memory.
Contents
A set is a collection of things. Examples might be the set of letters used in the English alphabet, or the set of books written by John Steinbeck. For us, the sets we will discuss will usually be collections of numbers because these are the sets that are important in algebra. Each of the things in a set is called an element of the set. The number of elements in a set could be finite or could be infinite. The only requirement is that the elements of a set should be described explicitly in some way either now or in the future (after we solve some problem). There may be no elements in a set; such a set is called an empty set or a null set. In this book we will mostly try to use capital letters as the symbols for sets whereas lower case letters are often (but not always!) used for variables.
A set can be written by putting braces, that is { and }, around a list of the elements of the set, with each element being separated by a comma. For example, a set S containing natural or whole numbers from 1 to 10 could be shown as follows:
It is not always possible to list out all the numbers in a set. In these cases we rely on English to describe the set. That's right! Words are an important part of math too. The last and probably most common notation involves using variables and algebraic expressions, together with a description of what values the variables may take. For example, to describe the set of numbers that are a perfect square, we might write:
or we can even use other sets in the description like:
Sometimes we want to explicitly make clear that a particular number (or thing) is in a set. Keeping S from the example above, we know that 2 is an element of S, rather than writing this out in English, sometimes people use the shorthand 2 ∈ S. The symbol ∈ is chosen because it looks like an E, and E is the first letter of the word "element". If we want to express that something is not part of a set we use the symbol ∉. So to continue our example we know that 11 ∉ S.
Let's take a look at a practice problem.
Problem. For each of the following sets decide if 7 is in the set:
A={1, 2, 3, 5}
The set O of odd numbers.
C={1, 2, 3, 4, 5, …, 100}
The set P of all prime numbers.
Answer.
We can see by inspection that 7 ∉ A.
7 ∈ O because 7 is an odd number.
7 ∈ C. This is a bit of a trick question. While it hasn't been explicitly said it yet, when listing elements people sometimes use … to mean keep "following the pattern". And in the rest of mathematics … keeps getting used in this way, so it is time to get used to it. Here the pattern is we start with 1 and keep making the numbers 1 bigger, so yes, if we keep going we will reach 7.
7 ∈ P. This problem requires that we try to factor 7, or have previously known it was a prime number. Since 2, 3, 4, 5, 6 all leave a remainder when we divide 7 by them, we see that 7 is a prime number.
We now introduce the basic ideas that come up when you have two or more sets.
Sometimes every element in one set is contained in another set. For example, let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and T = {2, 4, 6, 8, 10}. Clearly T is just the even numbers between 1 and 10, and every number in T is already in S. In this example we would say that T is a subset of S. Instead of saying which set is the smaller one, we could instead say which set is the bigger one by calling it a superset. That is, we could say S is a superset of T. As one deals with sets more and more, it becomes increasingly tempting to say that T is smaller thanS (or maybe less thanS). In fact, we already have! Because the relationship of one set being contained in another is so similar to relationship of one number being less than another it is natural to introduce a symbol very similar to inequality symbol <. To avoid confusing sets with numbers we don't want to use exactly the same symbol, so we will round out the point a bit and use the symbol ⊂. So instead of writing out in words "T is a subset of S", we could write T ⊆ S. Just like inequality we can flip the symbol around. We just have to make sure the rounded side points to the smaller thing. That is, for our example we could write S ⊃ T.
What if two sets have exactly the same elements? In this case we say that the two sets are equal. So if someone else came along and said let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then we say that S and U are equal. This time we don't worry about confusing sets and numbers and we will stick with the symbol = to denote when two sets are equal. So we can write S = T. Are there relationships corresponding to ≤ and ≥? Yes, the are ⊆ and ⊇ and they work like you might expect. Here is a table that explains how each of these symbols work.
Expression
Meaning
T ⊂ S
Means T is a subset of S. That is every element of T is an element S, but there is some element of S that is not in T.
T ⊃ S
Means S is a subset of T. That is, every element of S is an element T, but there is some element of T that is not in S.
S = T
Means that S and T have exactly the same elements. That is, use a few more words every element of S is an element of Tand every element of T is an element of S.
T ⊆ S
Means every element of T is an element of S. (Notice that saying it this way allows for the possibility that S = T)
There are two other things to do with sets. Given sets in S and T we may want to talk about all the elements that are in either S or T. Since a set is just a collection of things, and "the elements that are in either S or T" is a collection, you can think of this as defining a new set called the union of S and T. We write the union of S and T by S ∪ T. We use the symbol ∪ because it looks like a u, and u is the first letter in word "union". Let's do an example.
An important thing to notice in this example is that S ∪ T doesn't contain two 6's. The union contains all the elements in either set, but 6 is still just one thing that happens to be in both sets.
Give sets S and T, instead of thinking about things that are in either set, it is sometimes handy to think about things that are in both sets. Again we think of the collection of "the elements in both S and T, this set is called the intersection of S and T. We write the intersection of S and T by S ∩ T. We don't use the symbol ∩ because it looks like an i. It doesn't. Somehow, an i between two symbols just wouldn't look as good, so we want to pick something else. This symbol is just the upside down symbol of the symbol for union. Let's consider what the intersection looks like in the problem above.
In this section, we give names to some of the important classes of number.
The first important set of numbers is probably the first set of numbers we are really introduced to, namely the Natural Numbers, which we will call . The natural numbers are:
.
The next set is just a little bigger, and includes the next number we usually learn in elementary school. The natural numbers, together with the number 0 will be called the Whole Numbers, and denoted by . The whole numbers are:
.
Of course we are missing the negative numbers. The set of whole numbers together with all of the negative numbers is called the Integers denoted by . (You might ask why a letter that looks like Z. The reason is because it comes from the German word for number, Zahlen. English speakers are not the only ones to make important contributions to mathematics! Today, Z is the letter used almost universally.) The integers are:
Next, as you might guess we need a set of numbers that includes fractions. The set of all numbers that can be written as a fraction is written is called the Rational Numbers and is denoted by . You might ask why a letter that looks like Q? Well, first mathematicians save the letter R for real numbers (described below) and F for a general number field (a concept a bit beyond this book). But since a quotient is another word for fraction, and we are not using Q for anything else, it seems the sensible choice. The rational numbers are:
What about just the decimal numbers, we spent a long time working with them. As mentioned in the section on variables the set of all numbers decimal numbers (including those whose that continue indefinitely after the decimal point) is known as the Real Numbers and is written with the symbol . In this case we will not attempt to give a formula that describes the set, and instead just rely on its English description. But we should give a few examples of real numbers.
It may seem difficult to believe, but not every number can be written has a fraction. As we will see later, one such number is , but this is far from the only example. Indeed every integer can be written as a decimal just by adding a decimal point and infinitely many zeros to it. For example, 0 = 0.000… and -3 = -3.000… and we have made those to integers into decimal numbers. What about fractions? Yes every fraction can be written as a decimal simply using long division. We can also add, subtract, multiply and divide any two real numbers to get another real number (as long as we don't divide by 0). Unfortunately it gets to be very difficult to describe why this is. The algorithms we learn in school for adding, subtracting, multiplying and dividing real numbers all being with the decimal place furthest to the right. If the decimal goes on forever, it is awfully hard to find the decimal furthest to the right. So what do we do? For the moment the answer has to be "not worry about it too much". Later, after we have mastered a few more mathematics courses we will be ready to tackle the task of making sense of the arithmetic of real numbers. In the mean time your intuition about decimal numbers will probably not lead you astray. And where ever possible we will stick to fractions, or expressions like , rather than having to deal with infinitely long decimals.
Notice, the above list of numbers is increasing. That is, .
There is one last set of numbers it we should name. The Irrational Numbers is the set of all real numbers which are not rational numbers, we shall denote this set by , though other books may choose other names. To give a formula, we write:
We have never quite given the definition of a set this way. We added emphasis that the numbers x needed to come from the set of real numbers. This is also a common way to denote a set, though we may not use it much. We should point out there are numbers that are irrational. The most famous example is , but there are many many more. In fact the square root of any number which is not a perfect square will be an irrational number. As will the cube root of anything which isn't a perfect cube, etc.
We should point out that
Why? Well because of the definition of . Maybe your thinking "wait, what does this mean again?" Remember that two sets are equal if they have the same elements. So we really should explain why everything in is in and why everything in is in . But we will leave this as cn exercise to the ambitious reader.
The number 0 is a whole number, that is it is in . It cannot be in any smaller set that we named because it is not in the natural numbers.
The number 21 is not a pefect square, so is in .
The decimal expansion ends, meaning it can be represented as fraction, specifically 110,211/10,000,000, so it is a rational number. That is 0.0110211 ∈ of two integers - a rational number thus also a real number.
The square root of 49 is 7, which is a natural number.
Since -32 is a negative number the smallest set it could be in is it is an integer, the integers. |
Constructing Versatile
Mathematical Conceptions
Mike Thomas
The University of Auckland
Overview
Define versatile thinking in mathematics
Consider some examples and problems
Versatile thinking in mathematics
First…
process/object versatility—the ability to
switch at will in any given representational
system between a perception of symbols as
a process or an object
Not just procepts, which are
arithmetic/algebraic
Lack of process-object versatility
(Thomas, 1988; 2008)
Visuo/analytic versatility
Visuo/analytic versatility—the ability to
exploit the power of visual schemas by
linking them to relevant logico/analytic
schemas
A Model of Cognitive Integration
Higher level sc hemas
conscious
Directed
C–links and
A–links
unconscious
Low er level sc hemas
Representational Versatility
Thirdly…
representational versatility—the ability to
work seamlessly within and between
representations, and to engage in procedural
and conceptual interactions with
representations
Treatment and conversion
Duval, 2006, p. 3
Icon to symbol requires interpretation through
appropriate mathematical schema to ascertain
properties
External
world
external sign Interact
interpret
with/act on
translation or ‗appropriate' Internal
conversion schema world
Example
This may be an icon,
a 'hill', say
We may look
'deeper' and see a
parabola using a
quadratic function
schema
This schema may
allow us to
convert to algebra
A possible problem
The opportunity to acquire knowledge in a
variety of forms, and to establish connections
between different forms of knowledge are apt
to contribute to the flexibility of students'
thinking (Dreyfus and Eisenberg, 1996). The
same variety, however, also tends to blur
students' appreciation of the difference in
status which different means of establishing
mathematical knowledge bestow upon that
knowledge.
Dreyfus, 1999
Algebraic symbols: Equals schema
• Pick out those statements that are equations
from the following list and write down why you
think the statement is an equation:
• a) k = 5
• b) 7w – w
• c) 5t – t = 4t
• d) 5r – 1 = –11
• e) 3w = 7w – 4w
Equation schema: only needs an
operation
Perform an operation and get a result:
Another possible problem
Compartmentalization
―This phenomenon occurs when a person has
two different, potentially conflicting schemes
in his or her cognitive structure. Certain
situations stimulate one scheme, and other
situations stimulate the other…Sometimes, a
given situation does not stimulate the scheme
that is the most relevant to the situation.
Instead, a less relevant scheme is activated‖
Vinner & Dreyfus, 1989, p. 357
A formula
Linking of representation systems
(x, 2x), where x is a real number
Ordered pairs to algebra to graph
Abstraction in context
We also pay careful attention to the
multifaceted context in which processes of
abstraction occur: A process of abstraction is
influenced by the task(s) on which students
work; it may capitalize on tools and other
artifacts; it depends on the personal histories
of students and teacher
Hershkowitz, Schwarz, Dreyfus, 2001
dy
Abstraction of meaning for dx
Expression Rate of Gradient Derivative Term in
change of an
tangent equation
dy 5x
dx 16 6 11 2
2x dy 1 3 0 5 8
dx
dy 4y
dx 7 4 7 1
d( dy ) 1 1 0 3
z dx
dx
dy
Process/object versatility for dx
dy
Seeing solely as a process causes a
dx
dy
problem interpreting d( )
dx
dx
2
d y
and relating it to 2
dx
f ( f ( x))
Student: that does imply the second
derivative…it is the derived function of the
second derived function
b b k
a
g(x)dx
a k
g(x k)dx
To see this relationship ―one needs to deal
with the function g as an object that is
operated on in two ways‖
Dreyfus (1991, p. 29)
Proceptual versatile thinking
3
If ,
f (t)dt 8.6
1
then write down the value of
4
2
f (t 1)dt
Versatile thinking–change of
representation system
nb The representation does not correspond; an
exemplar y=x2 is used
Newton-Raphson versatility
Many students can use the formula
below to calculate a better
approximation of the root, but are
unable to explain why it works
f (x1 )
x2 x1
f (x1 )
Newton-Raphson
f (x1 )
f (x1 )
x1 x2
Newton-Raphson
When is x1 a
suitable first
approximation for
the root a of
f(x) = 0?
Student V1: It is very important
that the approximation is close
enough the root and not on a
turning point. Otherwise you
might be finding the wrong root.
Student knowledge construction
Learning may take place in a single representation system, so
inter-representational links are not made
Avoid activity comprising surface interactions with a
representation, not leading to the concept
The same representations may mean different things to students
due to their contextual schema construction (abstraction)
Use multiple contexts for representations
Reference
Thomas, M. O. J. (2008). Developing
versatility in mathematical thinking.
Mediterranean Journal for Research in
Mathematics Education, 7(2), 67-87.
From: moj.thomas@auckland.ac.nz
Proceptual versatility–
eigenvectors
Ax x
Two different processes
Need to see resulting object
or 'effect' as the same
Work within the representation
system—algebra
Work within the representation
system—algebra
Same process
Conversion
u v
v
u
Student knowledge construction
Learning may take place in a single representation
system, so inter-representational links are not made
Avoid activity comprising surface interactions with a
representation, not leading to the concept
The same representations may mean different things
to students due to their contextual schema
construction (abstraction)
Use multiple contexts for representations
Conversions
Translation between registers
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Duval, 1999, p. 5
Representations
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Semiotic representations systems — Semiotic registers
QuickTime™ and a
TIFF (LZW) decompressor
are neede d to see this picture.
Processing — transformation within a register
Duval, 1999, p. 4
Epistemic actions are mental actions by
means of which knowledge is used or
constructed
Representations and
mathematics
Much of mathematics is about what we can
learn about concepts through their
representations (or signs)
Examples include: natural language,
algebras, graphs, diagrams, pictures, sets,
ordered pairs, tables, presentations,
matrices, etc. (nb icons, indices and symbols
here)
Some of the things we learn are
representation dependant; others
representation independent
Representation dependant ideas...
"…much of the actual work of mathematics is to
determine exactly what structure is preserved
in that representation.‖
J. Kaput
Is 12 even or odd?
Numbers ending in a multiple of 2 are even. True
or False?
123?
123, 345, 569 are all odd numbers
113, 346, 537, 469 are all even numbers
Representational interactions
We can interact with a representation by:
Observation—surface or deep (property)
Performing an action—procedural or
conceptual
Thomas, 2001
Representational interactions
We can interact with a representation by:
Observation—surface or deep (property)
Performing an action—procedural or conceptual
Thomas, 2001
Procedure versus concept
Let x2
f (x)
x 2 1
For what values of x is f(x) increasing?
Some could answer this using algebra
and f (x) 0 but…
Procedure versus concept
4. 00
3. 00
2. 00
1. 00
- 2. 50 - 2. 00 - 1. 50 - 1. 00 - 0. 50 0. 50 1. 00 1. 50 2. 00 2. 50 3. 00
- 1. 00
- 2. 00
- 3. 00
For what values of x is this function increasing?
Why it may fail
We should not think that the three parts
of versatile thinking are independent
Neither should we think that a given
sign has a single interpretation — it is
influenced by the context
Icon, index, symbol
A B
D C
ABCD — symbol
Icon to symbol
―
‖
Moving from seeing a drawing (icon) to seeing a
figure (symbol) requires interpretation; use of an
overlay of an appropriate mathematical schema to
Function
Sign Into your Account
What's Docstoc
Docstoc is the premier online destination to start and grow small businesses.
It hosts the best quality and widest selection of professional documents (over 20 million)
and resources including expert videos, articles and productivity tools to make every small business better. |
is an excellent book for people who are new to Mathematica. It gets right to the point by providing hundreds of examples showing how to use Mathematica to solve problems ranging from basic arithmetic through calculus, ordinary differential equations and linear algebra. It also clearly shows how to use Mathematica to draw 2D and 3D graphs. This book is an excellent supplement to the Mathematica Book, which is included on the Mathematica CD.
This is an excellent introduction to Mathematica. I have suggested that they include this with each copy of the software and offer it (free?) to prospective buyers. A great review of the basic aspects of a very extensive software application.
This is an excellent introductory book on Mathematica. If you need to learn some basic features of this language, it is the best tool.
The author did not spend too much time on theory. The rules and syntax are explained very clearly with illustrative examples. Of course, the theory may first sound very dry and dull, but once you follow the examples, you will see that things will settle in your mind easily. Clear, concise and effective discussion of the topics makes your job easy and enjoyable.
The best way to learn and explore various features is to try to modify the parameters in the examples. Make use of your own creativity to discover new features. In many examples, the same outcome can be obtained by several different ways.
This book, in general, covers the fundamentals of the language, but it is sufficient to use differentiation, integration, 2D and 3D graphics, differential equations, linear algebra, etc. Mathematica, certainly, is quite a sophisticated language, there are many more intricate features.
For more advanced features, please try "Mastering Mathematica" by John W. Gray.
I bought this product hoping to get a quick but good introduction about Mathematica. This book has definitely given me that. This book does not cater to advanced level Mathematica programming issues.
One of the main reasons why bought this book is to look at many examples on individual topics, and I am certainly not disappointed.
I strongly recommend this book to anybody interested in getting a quick but reasonably good intro into Mathematica. Advanced programming and deeper level understanding of Mathematica can be taken up using other specialized books.
I agree, this book should be included with Mathematica. It's concise, to the point. Builds topics in logical steps with multiple examples and worked through problems.
Most of the stuff that applies works in later versions of Mathematica but it's a simple matter within Mathematica to find out how the dated procedures can be done in the newer versions. I have the latest version and have no problem using this book.
The price alone makes it well worth it. If you want to get started quickly and progress rapidly you definitely should buy this book!
although there remains useful information in this work, it is largely outdated, and offers techniques and methods incompatible with more recent incarnations of mathematica.
it can still offer effective instruction if you are willing to tread the interminable help pages in mathematica for the appropriate syntax and parameters, but to a new user even that is a forbidding task. strongly recommend you look elsewhere, until the current edition is updated.
If you're never used Mathematica before, this book is a good starting point. The Mathematica book (which is the same as the electronic help files from Mathematica) is certainly a better reference as far as depth and content is concerned. Therefore, if you're looking for a reference that will give you information on the tools from the software, you're better off with the help files. If you want to learn how to use the software itself and make some sense out of the otherwise akward user interface, then this book is definitely for you.
This is an excellent book. The authors cover everything from basic algebra and equations through calculus and differential equations, all using mathematica 4. It doesn't really explain why the program works the way it does, but it doesn't have any need to, since it is an introductory book. This book is basically formatted so you can learn by example. The author explains that he sees this as the best way to learn the program, and so you get 750 examples and solved problems.
Most useful chapters: 2D graphics 3D graphics Multivariable calculus
All the chapters are good, those just stand out. The learn-by-example setup works very well here, and there is an index so you can go to whatever topic you need. True, it may not have the level of depth some people desire, but that is what Trott's book is for.
All in all, this is the best place to start with mathematica (worth $12 and then some). I recommend Trott's programming guidebook when you finish this book. Even though I have Trott, I still reference this ALL the time, because it's easier to find what you need for less complex topics. Great book!! |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.