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The site is part of the learning area of BBC Scotland that includes a variety of K-12 subjects. This area focuses on the... see more The site is part of the learning area of BBC Scotland that includes a variety of K-12 subjects. This area focuses on the material for the Scottish "Higher" exams in mathematics. These national exams are normally taken by students age 15 or 16, and play a significant role in university admission. The main subject groups for the material in this mathematics area site are algebra, calculus, geometry and trigonometry. The material is simple but well executed. The goal of this site is to visualize the mathematical structure behind M.C. Escher's picture called "Print Gallery" (1956).... see more The goal of this site is to visualize the mathematical structure behind M.C. Escher's picture called "Print Gallery" (1956). The visualization itself is largely non-mathematical and is accomplished through many still images and animations. The actual mathematics, involving conformal mappings of the complex plane, is contained in a pdf copy of the original AMS publication. The Droste Effect refers to any image that contains itself on a smaller scale. The Graphical representation of complex eigenvectors simulation aims to help students make connections between graphical and... see more The Graphical representation of complex eigenvectors simulation aims to help students make connections between graphical and mathematical representations of complex eigenvectors and eigenvalues. The simulation depicts two components of a complex vector in the complex plane, and the same vector under several transformations that can be chosen by the user. A slider allows students to change the second component of the initial vector. The simulation shows whether or not the vector is an eigenvector, and if so displays the associated eigenvalue. The simulation includes a small challenge in asking the student to find the elements of one of the transformation matricesThe Graphical representation of eigenvectors simulation aims to help students make connections between graphical and... see more The Graphical representation of eigenvectors simulation aims to help students make connections between graphical and mathematical representations of eigenvectors and eigenvalues. The simulation depicts the two components of a unit vector in the xy-plane, and the same vector under several different transformations that can be chosen by the user. A slider allows students to change the orientation of the initial vector. The simulation shows whether or not the vector is an eigenvector, and if so displays the associated eigenvalue. The simulation includes a small challenge in asking students to find the elements of one of the transformation matrices 4This website is a great guide for teachers. It has several interactive links to choose from that cover multiple standards in... see more
Modify Your Results "The mathematical methods that physical scientists need for solving substantial problems in their fields of study are set out clearly and simply in this tutorial-style textbook. Students will develop problem-solving skills through hundreds of worked examples, self-test questions and homework problems. Each chapter concludes with a summary of the main procedures and results and all assumed prior knowledge is summarized in one of the appendices. Over 300 worked examples show how to use the techniques and around 100 self-test questions in the footnotes act as checkpoints to build student confidence. Nearly 400 end-of-chapter problems combine ideas from the chapter to reinforce the concepts. Hints and outline answers to the odd-numbered problems are given at the end of each chapter, with fully-worked solutions to these problems given in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www. cambridge. org/essential"-- This tutorial-style textbook develops the basic mathematical tools needed by first and second year undergraduates to solve problems in the physical sciences. Students gain hands-on experience through hundreds of worked examples, self-test questions and homework problems. Each chapter includes a summary of the main results, definitions and formulae. Over 270 worked examples show how to put the tools into practice. Around 170 self-test questions in the footnotes and 300 end-of-section exercises give students an instant check of their understanding. More than 450 end-of-chapter problems allow students to put what they have just learned into practice. Hints and outline answers to the odd-numbered problems are given at the end of each chapter. Complete solutions to these problems can be found in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www. cambridge. org/foundation. spacetime. Gravitation as spacetime curvature is introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle www. cambridge. org/9780521679718
Web Resources Podcasts Title: Logarithmic and Exponential Functions TV Description: This website has different videos explaining how to graph logarithmic and exponential functions. Standard(s): MI1 (9-12) 11: Describe the development of mathematical tools and their applications. (Alabama) [MA2013] PRE (9-12) 18: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7] Interactives/Games Title: Karl's Function Plotter Description: Standard(s): [MA2013] AL1 (9-12) 22: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). [A-REI10 24: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. [A-REI12Karl's Function Plotter tablesLearning Activities
Algebra: Word Problems Help and Practice Problems Find study help on linear applications for algebra. Use the links below to select the specific area of linear applications you're looking for help with. Each guide comes complete with an explanation, example problems, and practice problems with solutions to help you learn linear applications for algebra. Study Guides Distance, Rate, and Time Another common word problem type is the distance problem, sometimes called the uniform rate problem. The underlying formula is d = rt (distance equals rate times time). From d = rt , ... Introduction to Formulas in Word Problems - Cost and Profit For some word problems, nothing more will be required of you than to substitute a given value into a formula, which is either given to you or is readily available. The most difficult ... Word Problems About Grades Finding the Grade Needed to get a Specific Average Grade Grade computation problems are probably the most useful to students. In these problems, the formula for the course grade and ... Introduction to Money Word Problems Coin problems are also common algebra applications. Usually the total number of coins is given as well as the total dollar value. The question is normally "How many of each coin is there?" ... Introduction to Mixture Word Problems Mixture problems involve mixing two different concentrations to obtain some concentration in between. Often these problems are stated as alcohol or acid solutions, but there are many more types. For ...
Discrete Mathematics: Gateway to Advanced Mathematics There is a world of mathematics beyond what students learn in high-school algebra, geometry, and calculus courses. This seminar serves as an introduction to this realm of elegant mathematical ideas. With an explicit goal of improving students' mathematical reasoning and problem-solving skills, this seminar provides the ultimate intellectual workout. Five important themes are interwoven in the course: logic, the nature of proof, combinatorial analysis, discrete structures, and mathematical philosophy. For conference work, students may choose to undertake a deeper investigation of a single topic or application of discrete mathematics or to conduct a study in some other branch of mathematics. This seminar is a must for students interested in advanced mathematical study and highly recommended for students with an interest in computer science, law, or philosophy. Some prior study of calculus is required.
St. Francis High School Classes COURSE DESCRIPTION: This course is designed to introduce students to the major concepts and tools for collecting, analyzing and drawing conclusions from data. Students will be exposed to four broad conceptual themes: 1) Exploring data, 2) Planning a study, 3) Anticipating patterns, and 4) Statistical inference. COURSE DESCRIPTION: This course is designed for those with a good background in mathematics but who do not wish to continue in the precalculus track by taking trigonometry.. Topics covered will include linear models and linear inequalities, matrices, linear programming, determinants, combinatorics and probability. Graph theory will be visited as time allows.
Almost all adults suffer a little math anxiety, especially when it comes to everyday problems they think they should be able to figure out in their heads. Want to figure the six percent sales tax on a $34.50 item? A 15 percent tip for a $13.75 check? The carpeting needed for a 12˝-by-17-foot room? No one learns how to do these mental calculations... more... The reader is introduced to higher mathematics in an experimental way. He works with numerous interactive Java- simulations treating mathematical topics from number theory to infinitesimal calculus and partial differential equations. On the way he playfully learns the EJS simulation technique. Beyond the mathematics simulations the data pool contains... more... Sales Handle A no-nonsense practical guide to trigonometry, providing concise summaries, clear model examples, and plenty of practice, making this workbook the ideal complement to class study or self-study, preparation for exams or a brush-up on rusty skills. About the Book Established as a successful practical workbook series with over 30... more... Despite what we may sometimes imagine, popular mathematics writing didn't begin with Martin Gardner. In fact, it has a rich tradition stretching back hundreds of years. This entertaining and enlightening anthology--the first of its kind--gathers nearly one hundred fascinating selections from the past 500 years of popular math writing, bringing to... more... The easy way to brush up on the math skills you need in real life Not everyone retains the math they learned in school. Like any skill, your ability to speak "math" can deteriorate if left unused. From adding and subtracting money in a bank account to figuring out the number of shingles to put on a roof, math in all of its forms factors into daily... more...
A Problem Solving Approach to Mathematics for Elementary School Teachers 9780321570550 ISBN: 0321570553 Edition: 10 Pub Date: 2009 Publisher: Addison-Wesley Summary: Billstein, Rick is the author of A Problem Solving Approach to Mathematics for Elementary School Teachers, published 2009 under ISBN 9780321570550 and 0321570553. One hundred ninety nine A Problem Solving Approach to Mathematics for Elementary School Teachers textbooks are available for sale on ValoreBooks.com, one hundred ninety seven used from the cheapest price of $2.85, or buy new starting at $62.25 Hardcover. TEXTBOOK ONLY 10th Edition May contain highlighting/underlining/notes/etc. May have used stickers on cover. Ships same or next day. Expedited shipping ta [more] Used-Good
More About This Textbook Overview An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory. A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography. Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to be introduced to some of the main themes in number theory. Editorial Reviews From the Publisher From the reviews: "This number theory text is somewhat different than traditional number theory texts. The authors' guiding principle is unique factorization and its consequences. … This is not a traditional number theory text, but one that tries to guide the reader through the beginnings of the subject towards the modern frontiers. This is helped along by a good sized bibliography plus many problems … . it might provide an interesting experience when used at the graduate level." (Don Redmond, Mathematical Reviews, Issue 2006 j) "The book under review contains several topics which are usually not brought together in an introductory text. The book is meant to give a broad introduction to advanced undergraduate students … of number theory. … Each chapter contains many exercises and historical notes. … In my opinion, because so many topics are treated in an accessible way, the book is very well suited for an introductory course in number theory." (Jan-Hendrik Evertse, Zentralblatt MATH, Vol. 1089 (15), 2006) "In An Introduction to Number Theory, the authors strive to have the best of all worlds: they cover a broad range of topics … . This book could be used for a number of different courses. … The full book would be appropriate for a first-year graduate course. It's also a nice introduction to the subject for established mathematicians form other fields. … its extensive bibliography, tasteful collection of topics, and clear presentation make it a pleasant reference even for working number theorists." (Rob Benedetto, MathDL, January
Mathematics Upon graduation, a student will be able to: Major-Specific Areas of Knowledge Solve mathematical problems. Learn new mathematics independently. Evaluate mathematical arguments. Have depth of study in at least one area of mathematics. Have a basic understanding of several branches of mathematics. Communication Communicate mathematical ideas effectively both orally and in writing.
97801302275gebra: A Combined Approach This CD allows students the opportunity to practice exercises that correlate to the exercises at the end of the sections in the textbook. They are algorithmically generated so students can retry an exercise as many times as they would like with new values each time. Every exercise is accompanied by an example and an interactive guided solution that gives students helpful feedback if they enter an incorrect answer. Selected exercises also include a video clip so students can have help visualizing concepts
More About This Textbook Overview A National Science Foundation (NSF) funded high school series for all students Contemporary Mathematics in Context engages students in investigation-based, multi-day lessons organized around big ideas. Important mathematical concepts are developed in relevant contexts by students in ways that make sense to them. Courses 1, along with Courses 2 and 3, comprise a core curriculum that upgrades the mathematics experience for all your students. Course 4 is designed for all college-bound students. Developed with funding from the National Science Foundation, each course is the product of a four-year research, development, and evaluation process involving thousands of students
Summary: CONTEMPORARY MATHEMATICS FOR BUSINESS AND CONSUMERS, BRIEF is a 14-chapter educational adventure into today's business world and its associated mathematical procedures. The book is designed to provide solid mathematical preparation and foundation for students going on to business courses and careers. It begins with a business-oriented review of the basic operations, including whole numbers, fractions, and decimals. Once students have mastered these operations, they a...show morere introduced to the concept of basic equations and how they are used to solve business problems. From that point, each chapter presents a business math topic that utilizes the student's knowledge of these basic operations and equations. In keeping with the philosophy of "practice makes perfect," the text contains over 2,000 realistic business math exercises--many with multiple steps and answers designed to prepare students to use math to make business decisions and develop critical-thinking and problem-solving skills. Many of the exercises in each chapter are written in a "you are the manager" format, to enhance student involvement. The exercises cover a full range of difficulty levels, from those designed for beginners to those requiring moderate to challenge-level skillsBetter World Books Mishawaka, IN With CD! Shows definite wear, and perhaps considerable marking on inside. Find out why millions of customers rave about Better World Books. Experience the best customer care and a 100% satisfaction gu...show morearantee. ...show less $1.99 +$3.99 s/h Good Better World Books Mishawaka, IN Sorry, CD missing
Basic Multivariable Calculus - 93 edition Summary: Basic Multivariable Calculus helps students make the difficult transition to advanced calculus by focusing exclusively on topics traditionally covered in the third-semester course in the calculus of functions of several variables. The concepts of vector calculus are clearly and accurately explained, with an emphasis on developing students' intuitive understanding and computational technique. Only first year calculus required--all necessary linear al...show moregebra is explained Incorporates wide range of physical applications, dozens of graphics, and a large number of exercises Volume and Cavalieri's Principle The Double Integral over a Rectangle The Double Integral over Regions The Triple Integral Change of a Variable, Cylindrical and Spherical Coordinates Applications of Multiple Integrals 6. Integrals over Curves and Surfaces Line Integrals Parametrized Surfaces Area of a Surface Surface Integrals 071672443X Used texts may NOT contain supplemental materials such as CD. $4.50 +$3.99 s/h Good Nettextstore Lincoln, NE 1993One Stop Text Books Store Sherman Oaks, CA 1993-03-15
Algebra : Introductory and Intermediate - 4th edition Summary: With all the support of the renowned Aufmann approach, this popular combination text helps your students prepare to master college algebra and to apply algebra in the real world. New! Bulleted annotations have been added to the solution steps of Examples and to the You Try It solutions in the appendix, further enhancing the Aufmann Interactive Method. New! Examples have been clearly labeled How To, ...show moreallowing students to more easily refer back to solution steps when completing corresponding exercises. Updated! The Chapter Summary has been reformatted to include an example column, offering students the additional support of an algebraic representation of concepts, rules and definitions. Updated! In response to instructor feedback, the number of Chapter Review Exercises and Cumulative Review Exercises has increased. Updated! More operation application problems integrated into the Applying the Concepts exercises encourage students to judge which operation (adding, subtracting, multiplying, dividing) is needed to solve a word problem. New! Integrating Technology (formerly Calculator Notes) margin notes provide suggestions for using a calculator in certain situations. For added support and quick reference, a scientific calculator screen is displayed on the inside back cover of the text. New! Objective-based Worksheets accompany every section in the book for extra classroom practice or homework. These worksheets are found on the ClassPrep CD and Online Teaching Center. Aufmann Interactive Method (AIM) encourages students to try the math as it is presented. Every section objective contains one or more sets of matched-pair examples. The first example is completely worked out; the second example, called 'You Try It,' is for the student to work. Complete worked-out solutions to these examples in an appendix enable students to check their solutions and obtain immediate reinforcement of the concept. Integrated, easy-to-navigate learning system organized by objectives guides students with a consistent, predictable framework. Each chapter opens with a list of learning objectives, which are woven throughout the text and integrated with the print and multimedia ancillaries. The AIM for Success Student Preface guides students in making the most of the text's features. Study Tip margin notes throughout the text refer students back to the Student Preface for advice. Prep Tests at the beginning of each chapter help students prepare for the upcoming material by testing them on prerequisite material learned in preceding chapters. The answers to these questions can be found in the Answer Appendix, along with a reference to the objective from which the question was taken. The Go Figure problem that follows the Prep Test is a challenge problem for interested students. Extensive use of applications that use real source data shows students the value of mathematics as a real-life tool. Focus on Problem Solving section at the end of each chapter introduces students to various problem-solving strategies. Students are encouraged to write their own strategies and draw diagrams in order to find solutions. Unique Verbal/Mathematical connection simultaneously introduces a verbal phrase with a mathematical operation, followed by exercises that require students to make a connection between a phrase and a mathematical process. Projects and Group Activities at the end of each chapter offer ideas for cooperative learning. Unique Instructor's Annotated Edition features a format rich with new instructor support materials, which are provided at point-of-use in the margins surrounding reduced student pages60953995 +$3.99 s/h Good BookCellar-NH Nashua, NH 0618609539 Has heavy shelf wear, but still a good reading copy. We are a tested and proven company with over 900,000 satisfied customers since 1997. Choose expedited shipping (if available) for much f...show moreaster delivery. Delivery confirmation on all US orders. ...show less $74.81 addition
Elementary Algebra Book Description: Algebra can be like a foreign language. But one text delivers an interpretation you can fully understand. Building a conceptual foundation in the "language of algebra," ELEMENTARY ALGEBRA, 4e provides an integrated learning process that helps you expand your reasoning abilities as it teaches you how to read, write, and think mathematically. Packed with real-life applications of math, it blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. The authors' five-step problem-solving approach makes learning easy. More student-friendly than ever, the text offers a rich collection of student learning tools, including Enhanced WebAssign online learning system. With ELEMENTARY ALGEBRA, 4e, algebra makes sense
Algebra: A Self-Teaching Guide With a "learn-by-doing" approach, it reviews and teaches elementary and some intermediate algebra. While rigorous enough to be used as a college or ...Show synopsisWith a "learn-by-doing" approach, it reviews and teaches elementary and some intermediate algebra. While rigorous enough to be used as a college or high school text, the format is reader friendly, particularly in this Second Edition, and clear enough to be used for self-study in a non-classroom environment. "Pre-test" material enables readers to target problem areas quickly and skip areas that are already well understood. Some new material has been added to the Second Edition and redundant or confusing material omitted. The first chapter has undergone major revision. Chapters feature "post-tests" for self-evaluation. Thousands of practice problems, questions and answers make this algebra review a unique and practical text.Hide synopsis Description:New. Practical Algebra If you studied algebra years ago and now...New. Practical Algebra If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you're a student who needs an introduction to the subject, here's the perfect book for you. Practical Algebra is
Are you looking for rigour or a more intuitive understanding of mathematical analysis? If you are looking for a more theoretical approach to analysis, then Apostol's Calculus Vol 1, 2 are probably the best place to start, assuming that as a third grader you have the right background for it. – RankeyaFeb 29 '12 at 1:27 The author is in the "third grade of high school", i.e. the OP is probably in the 8 th or 9 th grade of school.I am currently studying calculus from Apostol and I really like his gentle and yet rigorous style. – EisenFeb 29 '12 at 6:12 @SabyasachiMukherjee When I hear "third grade of high school" I can't think of anything other than 11th grade. – Alex Becker♦Feb 29 '12 at 7:35 3 Answers As I said, for a rigorous and theoretical approach to calculus, Apostol's Calculus Vols. $1$ and $2$ are very good. Depending on your background, for multivariable calculus, Spivak's Calculus on manifolds is also good. Spivak's Calculus (which does single variable calculus) is also one of my favorites. I think I first learned calculus from Richard Courant's Introduction to Calculus and Analysis. I think Courant's and Robbin's What is Mathematics? also has good intuitive explanations of differentiation and integration. For a book more intuitive, and perhaps something that a third grader would have background for, try Silvanus Thompson's and Martin Gardner's Calculus Made Easy. Of course, a book that worked for me might not work for you. I would suggest that you go to a library and browse through a number of different calculus books (there are a lot of them out there), till you find the one that appeals to you the most. If you really are in the third grade, then I would assume there is no real hurry for you to master calculus, and if there is, then the books above are a good place to start. I think one's first exposure to calculus-no matter how gifted or ambitious the student is-should be a physically and geometrically motivated approach that illustrates most of important applications of calculus. Sadly,many people think that means a "pencil-pushing" or "cookbook" approach where things are done sloppily and with no careful explanation of underlying theory. That's simply not true. You can certainly do calculus non-rigorously while still doing it carefully enough to give students the broad picture of the underlying theory. The best example of this kind of book,to me, is Gilbert Strang's Calculus. Strang's emphasis is clearly on applications and it has more applications then just about any other calculus text-including many kinds of differential equations in physics(mechanics),chemistry( first and second order kinetics),biology (modeling heart rythum) and economics and a basic introduction to probability.But Strang doesn't avoid a proof when it's called for and the book has many pictures to soften the blows of these careful proofs. This would be my first choice for a high school student just starting out with calculus.
map to 8th Grade Math, Virginia Edition If Students Need to Know It, It's in This Book This book develops the mathematics skills of eighth-graders. It builds skills that will help them ...Show synopsisIf Students Need to Know It, It's in This Book This book develops the mathematics skills of eighth-graders. It builds skills that will help them succeed in school and on the Virginia Standards of Learning Assessments. Why The Princeton Review? We have more than twenty 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 Virginia Standards of Learning (SOL) Assessments Our experts at The Princeton Review have analyzed the Virginia SOL Math Assessment, and this book provides the most up-to-date, thoroughly researched practice possible. 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 based on Virginia standards and objectives -detailed lessons, complete with skill-specific activities -two complete practice Virginia SOL Math Assessments For more information about our other test-preparation products for school and home, call "1-800-REVIEW-2 "or visit "k12.princetonreview.com
Start to Learn Calculus the Intuitive Way The purpose of this site is to make calculus fun and exciting to learn for you. Is this possible? I think so, because there is nothing more fun than learning something new. And I mean really learn. You know that feeling when something just clicks in your head, right? That's what I'm talking about... Whether you already have some experience or you are just starting out you can tremendously benefit from this site. Learning calculus is hard work, I won't lie to you. However, hard work doesn't need to be tedious or boring. When I say hard work I mean that you will need to put on the time and dedication. In this site you will learn calculus the intuitive way, that is, forming an intuitive understanding of the concepts. With this understanding, the rest will be easier for you. What I mean by this? Let's say, what do you think you will remember better. Something like this? Or something like this: The graph is easier to remember for most people. Let me tell you a secret... The difference between students that get A's and the rest is in how they approach learning. It's not intelligence! "A" students form an intuition of the concepts, using a graph like this one, for example. The rest just nervously copy the formulas the teacher writes on the board, then they try to memorize. Of course, after you formed an intuition, writing the equation will be easy for you with some practice. Where to Begin? This site covers all the topics in a first calculus course. If you are already taking a course in school, you can immediatelly visit the topics you didn't really understand. Even if you think you understood a topic well, I recommend you to visit the pages about them. You may have new insights... Be sure to bookmark this site, so you can return to it again and again. The sequence you may follow to learn calculus is: Functions: This topic isn't usually covered in a calculus course. It is essential for a real understanding of calculus, though. That's why I created a series of pages about it. This is the place to start if you never saw calculus before... Limits: This is the topic most calculus courses start with. The idea of limit is the central idea of all calculus. In this pages you will learn the basics of them... Derivatives: The derivative is one of the two most important concept of calculus, the other being the integral. The derivative can be seen as a generalized idea of slope or rate of change, and it is essential for all sciences... Integrals: The integral is a generalized idea of area. As the derivative, it is essential for all sciences. In all these pages you will learn about these concepts by first forming an intuition, then solving common problems that apply these concepts. Who am I? My name is Pablo Antuna and I'm just a guy passionate about math and teaching. I decided to create this site so I can teach calculus the way I wish I had been taught it. You can learn more about me in the about page. Intuitive Calculus Insights is a free e-zine that provides you with unique ideas and tips for succeeding in calculus. I send it every two weeks, and you'll only receive valuable and unique information. Subscribe now, and you'll also receive Free and Unlimited access to the two resources only available to subscribers: The special report Top Ten Tips for Succeeding in Calculus: In this report you'll learn the things I learned the hard way, and can make the difference between success and failure in calculus. The Intuitive Calculus Tables: These tables have everything you need regarding formulas in calculus. Everything from limits to integrals. Ready for print. No need to go for your big textbook every time you're seeking a formulaTopics Keep in Touch Resources Search this Site: Free Report: Top Ten Tips for Succeeding in Calculus Sign up below to get access to the two resources only available to subscribers: Free Report: Top Ten Tips for Succeeding in Calculus: everything I learned the hard way. The Intuitive Calculus Tables: all the formulas you'll need just in one place, ready for printPlus: Intuitive Calculus Insights. A free subscription to my newsletter, which I send once every two weeks. You'll receive study tips and other valuable things that'll help you succeed in calculus. Learn more here...
More About This Textbook Overview Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving Mathematical Problems includes numerous exercises and model solutions throughout. Assuming only a basic level of mathematics, the text is ideal for students of 14 years and above in pure mathematics. Editorial Reviews From the Publisher "There are a handful of really wonderful books that can introduce a young high-school student to the beauty of mathematics. This is definitely one of them. Besides,this book is probably going to be known as the first book written by one of the best mathematicians of the twenty-first century."-- TheMathematical Asociation of America Related Subjects Meet the Author Terence Tao was born in Adelaide, Australia, in 1975. In 1987, 1988, and 1989 he competed in the International Mathematical Olympiad for the Australian team, winning a bronze, silver, and gold medal respectively, and being the youngest competitor ever to win a gold medal at this event. Since 2000, Terence has been a full professor of mathematics at the University of California, Los Angeles. He now
Course Description: This standards-based seventh grade course emphasizes the development and understanding of the real number system and algebraic thinking. Students acquire skills in adding, subtracting, multiplying, and dividing signed numbers including integers. Students solve one step equations involving real numbers. Problem solving in the course includes consumer applications of ratio, proportion, and percent. It continues to develop other important mathematics topics including patterns, functions, Math 8 or Algebra I in the eighth grade:
School Teachers Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers" leave the course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual skills upon which those ideas are built
Mathematics WMI is an open web-based eLearning system in mathematics for grammar school and/or university students, including basic mathematical functions (equation solving, function plotting, symbolic differentation and antiderivation) and thematic modules.
Mathematics Education resources At the Times Higher Awards ceremony on 24th November 2011, it was announced that Loughborough and Coventry Universities had won the award for Outstanding Support for Students, in recognition of the work of sigma, Centre for Excellence in University-wide mathematics and statistics support. Whilst sigma at Coventry and Loughborough Universities received the award, the real winner was mathematics and statistics support across the country. In this booklet, we outline how sigma's work has contributed to the growing recognition of the importance of mathematics and statistics support and to the development of a national and international community of practitioners. Authors : Ciaran Mac an Bhaird and Duncan Lawson This guide edited by Michael Grove and Tina Overton has been developed for those looking to begin pedagogic research within the science, technology, engineering and mathematics (STEM) disciplines. Its purpose is to provide an accessible introduction to pedagogic research along with a practical guide containing hints and tips on how to get started. The guide was produced following a series of national workshops and conferences that were started in 2011 by the National HE STEM Programme and continued in 2012 with the support of the Higher Education Academy. Recruitment to post-graduate mathematics programmes and to lecturer positions in mathematics departments in UK universities has become dominated by international students and staff. Although mathematics is generally regarded as 'the universal language', the reality is that different countries have very different cultures when it comes to the teaching and learning of mathematics. There are significant variations in the pre-university mathematical experience, in terms of the curriculum content, learning styles, levels of abstraction, and assessment methods. Even within the UK, a considerable number of pre-higher education mathematics qualifications are available and, it is not always clear what mathematics can be expected when students commence their degree programmes. With increasing numbers of international students and academic staff in UK HE, the scene is becoming more complicated. Students enter degree courses with a wide range of backgrounds and bring with them very different experiences. At the same time, academic staff, having experienced different education systems, may have some unrealistic expectations from their students. With an HEA Teaching Development Grant (Individual Scheme 2012 -2013), this research by Aiping Xu, Coventry University has investigated the mathematical cultures of a range of the main international suppliers (of students and staff) to UK HE mathematics departments. Using semi-structured interviews and online questionnaires, personal experiences of academic staff who have studied or taught more than two educational systems have been drawn upon. Some examinations have also been studied in detail.
Student learning outcomes: Mathematics, Secondary develop both increased depth and breadth of mathematical content knowledge. Students will be able to communicate mathematics both orally and in writing. Students will demonstrate the ability to solve challenging problems from the mathematical sciences and write mathematical proofs. Students will demonstrate the ability to solve problems in a variety of areas of mathematical sciences including statistics, algebra, calculus and geometry. Students will experience and learn an assortment of pedagogical tools and practices that can be adapted to a variety of school environments and learners. Students will demonstrate that they can identify and utilize the appropriate strategies and tools (including the use of technology) to solve mathematics problems. Students will transition their thinking about learning mathematics from a student's perspective to a teacher's perspective. Students will demonstrate that they can apply various strategies and tools (including the use of technology) to teaching mathematical concepts, thinking, and content appropriate for secondary students. Students will produce evidence of insights gained from interactions with in-service teachers, results of working collaboratively with peers to solve mathematics problems, and evidence of reflection on experiences in public school mathematics classrooms.
BUSINESS MATHEMATICS BRIEF, 12/e teaches business...Very good. BUSINESS MATHEMATICS BRIEF, 12/e teaches business math with a strong focus on current issues, real companies, and realistic business scenarios. It places essential business math concepts in context
'An abstract concept image featuring: a variety of mathematical equations, ranging from derivatives, to trig laws to graphing; a pencil writing an equation ending in 2 pi x; a graphics calculator showing three functions; a student working out a vector problem; a 3D graph; and the famous e^pi.i + 1 = 0 equation.' #179292286 #179292286 24029056 #87717694 #87717694 Textbooks and an apple #78469558 #78469558 Portrait of a young smiling teacher #453522691 #453522691 male teacher writing various high school maths and science formula on whiteboard #452193677
3.2" backlit color display TI-Nspire CAS (computer algebra system) student software for home kit Enables you to continue or complete assigned work outside of the classroom. Calculator function Lets you enter and view expressions, equations and formulas, select syntax, symbols and variables from a template that supports standard mathematical notation and scroll through previous entries to explore your outcomes. Graphing function Allows you to utilize images and overlay with graphical elements on the screen. A geometry function makes it easy to create and explore geometric shapes for interactive learning. Lists and spreadsheets function Enables you to capture and track the values of a graph and collected data to observe numerical patterns, organize the results of statistical analysis, label columns, insert formulas into cells and more. Data and statistics function Helps you summarize statistical data using different graphical methods and perform a variety of descriptive and inferential statistics calculations on real data sets.
More About This Textbook Overview In this book the renowned Russian mathematician Georgi E. Shilov brings his unique perspective to real and complex analysis, an area of perennial interest in mathematics. Although there are many books available on the topic, the present work is specially designed for undergraduates in mathematics, science and engineering. A high level of mathematical sophistication is not required. The book begins with a systematic study of real numbers, understood to be a set of objects satisfying certain definite axioms. The concepts of a mathematical structure and an isomorphism are introduced in Chapter 2, after a brief digression on set theory, and a proof of the uniqueness of the structure of real numbers is given as an illustration. Two other structures are then introduced, namely n-dimensional space and the field of complex numbers. After a detailed treatment of metric spaces in Chapter 3, a general theory of limits is developed in Chapter 4. Chapter 5 treats some theorems on continuous numerical functions on the real line, and then considers the use of functional equations to introduce the logarithm and the trigonometric functions. Chapter 6 is on infinite series, dealing not only with numerical series but also with series whose terms are vectors and functions (including power series). Chapters 7 and 8 treat differential calculus proper, with Taylor's series leading to a natural extension of real analysis into the complex domain. Chapter 9 presents the general theory of Riemann integration, together with a number of its applications. Analytic functions are covered in Chapter 10, while Chapter 11 is devoted to improper integrals, and makes full use of the technique of analytic functions. Each chapter includes a set of problems, with selected hints and answers at the end of the book. A wealth of examples and applications can be found throughout the text. Over 340 theorems are fully proved 8, 2005 getting started in math analysis This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time. Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book. Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin. There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Find a Sugar Hill, GA AlgebraFear not, your life skills with my experience will give you the confidence you need. Elementary math is just that -- elementary -- once you learn the connections/relationship between one concept and another. For example, to find an answer for a division problem, look for the multiple of the numbers. ...This algebra deals mostly with linear functions. Algebra 2 is a more advanced, more complex version of algebra 1. Here we get more involved with non-linear functions as well as imaginary and complex numbers
Foundations of Discrete Mathematics 9780470211526 ISBN: 0470211520 Publisher: Wiley & Sons, Incorporated, John Summary: Introduces the process of abstraction, studies two elementary structures on sets, and covers the necessary generalities concerning algebraic structures. Presents powerful abstract mathematical concepts from algebra and combinatorics, supported by concrete applications. All background material is provided, including elements of logic, set theory, abstract algebra, linear algebra, and graph theory. Each chapter develop...s a new mathematical concept, then shows how to apply it. Includes numerous end-of-chapter problems and exercises. Joshi, K. D. is the author of Foundations of Discrete Mathematics, published under ISBN 9780470211526 and 0470211520. Ten Foundations of Discrete Mathematics textbooks are available for sale on ValoreBooks.com, five used from the cheapest price of $172.35, or buy new starting at $189.42.[read more]
From the Publisher reinforce what you've learned. In addition, the book includes many real-world examples that show you how mathematics is used to model in fields like engineering, business, physics, chemistry, and biology. Description: Cynthia Young''s 3rd Edition of "College Algebra "focuses on revisions and additions including hundreds of new exercises, more opportunities to use technology, and themed modeling projects that help connect content to real world issues. The text builds upon the ... Description: This text bridges the gap between traditional and reform approaches to algebra encouraging students to see mathematics in context. It presents fewer topics in greater depth, prioritizing data analysis as a foundation for mathematical modeling, and emphasizing the verbal, ...
EDCU11029 Procedural Mathematics 1 Course details Mathematics as a science of number and pattern has rules, procedures and concepts that are foundational to higher mathematics courses. This enabling course in procedural mathematics identifies, expands upon and reinforces the specific content areas of arithmetic, algebra and geometry at a level beyond the expectation of secondary education and at a level pre-emptive of the requirements of further tertiary mathematics programs. In addition to this identification, recognition and use of mathematical algorithms is the conceptual knowledge of the mathematics content and the development of maturity in using the procedural mathematics in a variety of discipline areas. As a mathematical science enabling course this course focuses on working mathematically.
Calculus Problems p. 1 / 4 Embed or link this publication Description Calculus has widespread application in areas like Engineering and Science. Since this study is pivotal in branching out to other fields, it is important to get the best help right from the formative years itself. Solving Calculus problems is not easy b Popular Pages p. 1 calculus problems calculus has widespread application in areas like engineering and science since this study is pivotal in branching out to other fields it is important to get the best help right from the formative years itself solving calculus problems is not easy but with the help of tutorvista s online tutors this will become much easy and simple our online tutors will help you out to solve your calculus problems and understand the concepts with a better hand get your help from our tutors and ensure yourself quality learning on the subject topics covered in calculus given below are the topics covered by our online calculus help program learn more about equation of a line p. 2 functions limits and continuity differentiation differential equations indefinite integrals definite integrals application of derivatives exponential and logarithmic series understand all these topics with personalized attention and gain quality help online get calculus homework help students can get all the calculus homework help needed from the expert tutors all the help required with solving and understanding problems for homework and examinations can be got online as the calculus help is provided online students can get help immediately and accurately with the understanding of the concept tutorvista s online help isn t just about working out a few problems and logging off features like regular homework help and exam prep enable a student to get all the help he/she needs read more on graphing linear equations p. 3 there is also an extensive library of e-learning material like free question banks simulations and animations available to help the student ace the subject calculus is basically a study of higher grades and tutorvista has special tutors for higher grade students who teach the subject with an expertise grade 7 grade 10 grade 8 grade 11 grade 9 grade 12 online tutoring with an expert get online calculus tutorials from tutorvista our online tutors will solve all your problems they are available 24x7 so that you can connect with them just when you need help our calculus tutors will help you with your homework and assignments and also give you step-by-step explanations for all problems making even complex problems simple and easy.
The Learning Guide for Introductory and Intermediate Learning Guide helps students learn how to make the most of their textbook and its companion learning tools, including MyMathLab. Organized by the textbook's learning objectives, this workbook provides additional practice for each section and guidance for test preparation. Published in an unbound, binder-ready format, the Learning Guide can serve as the foundation as the student's course notebook.
New AQA GCSE Mathematics Unit 2 Foundation Teacher's Book for an Amazon.co.uk gift card of up to £14.00, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more Book Description Covering AQA GCSE Foundation Unit 2, our teacher books are written for the AQA GCSE Mathematics A modular specification and offer teaching ideas, lesson help and study tips for your students to achieve their target grade. Covering AQA GCSE Foundation Unit 2, our teacher books are written for the AQA GCSE Mathematics A modular specification and offer teaching ideas, lesson help and study tips for your students to achieve their target grade43.99,"ASIN":"1408506270","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":43.98,"ASIN":"1408506238","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":9.87,"ASIN":"1408506254","isPreorder":0}],"shippingId":"1408506270::ZoVzbc9FmjU4ktKEJG2qIcQBsl4azAZfXmiojrTb5LC%2BHBNXb7zRn2LrVWijggwz7B1yHqIbkFfz%2F43GA3T8Whx5ve9nPzPw,1408506238::5CKM6EV2SOuQKGIKUziLQqDo0wqZHsjiIHj2NiL39L8pL9y%2F90ZNZdQ9AtS%2F%2FmgGbjcNtQULjdguKAXiHDrdQmqTGAXiKIic,1408506254::BzFNj3tZ7DFA6%2FPh8Q4AfsmprB4BR%2BJbZP7VhqzQZ4fxWo1kgdcIUzkS881Ro1LKLDJ3MijPZm%2F1cYUvTMkv4hetLkgzMdz
Product description The Algebra 1 Teacher's Edition (3rd ed.) contains the student pages with overprint answers as well as solutions for exercises and additional math problems. Teachers will find the one-on-one activities, presentation suggestions, and common student errors features to be helpful resources. This edition also gives suggested assignments, and schedules are given for three tracks-minimum, standard, and extended. The CD that accompanies the Teacher's Edition has Projection Ready Answers, Mathardy (Review Game) and Visuals. Type: Spiral Bound (Teacher's Guide)Category: > Home SchoolingISBN / UPC: 9781606820469/160682046XPublish Date: 3/1/2012Item No: 295785Vendor: Bob Jones University Press
Summary: Helping students grasp the "why" of algebra through patient explanations, Hirsch and Goodman gradually build students' confidence without sacrificing rigor. To help students definit...show moreions through a wealth of illustrative examples-both numerical and algebraic-helps students compare and contrast related ideas and understand the sometimes-subtle distinctions among a wide variety of situations. This author team carefully prepares students to succeed in higher level mathematics
gives students a solid foundation in algebra and prepares them for success in their next algebra course. McKeague's depth of experience, his passion for teaching, and his connection with students at this level are apparent throughout the book. His incremental approach carefully guides students to a self-assured competence with the topics they are learning. NOTICE: if you require online access to homework, worksheets, or answer sets, you MUST purchase this book through the XYZ Textbook Amazon store. The publishers cannot guarantee these online resources for books purchased through other third parties.
More than 40 million students have trusted Schaum?s Outlines for their expert knowledge and helpful solved problems. Written by renowned experts in their respective fields, Schaum?s Outlines cover everything from math to science, nursing to language. The main feature for all these books is the solved problems. Step-by-step, authors walk readers through coming up with solutions to exercises in their topic of choice. Description: Study faster, learn better, and get top gradesA clear review of standard college course of Mathematics for Elementary School Teachers, this book will be designed to improve your basic knowledge of math content required for this level, engage you ...
complements and extends the Edexcel GCSE Modular Mathematics Examples and Practice book for Foundation Stage 2.This book complements and extends the Edexcel GCSE Modular Mathematics Examples and Practice book for Foundation Stage
Excursions in Modern Mathematics - 5th edition Summary: For undergraduate courses in Liberal Arts Mathematics, Quantitative Literacy, and General Education. This collection of "excursions" into modern mathematics is organized into four independent parts, each consisting of four chapters--1) Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics. The book is written in an informal, very readable style, with pedagogical features that make the material both interesting and clear....show more Coverage centers on an assortment of real-world examples and applications, demonstrating the usefulness, relevance, and attractiveness of liberal arts mathematics. Features NEW--"Profiles" --Incorporates biographical profiles of featured mathematicians at the end of each chapter. Humanizes the materials in each chapter and introduces some of the key mathematicians in these fields. NEW--"Projects and Papers" --Found at the end of chapter problem sets, offers topics for projects and papers appealing to instructors who want their students to do cooperative learning or research papers. Provides a great vehicle for explorations and class discussions that students and professors can pursue together. NEW--Updated problems and examples. Many of the problems and examples have been updated and improved based on reviewer/user feedback. Provides updated examples and additional exercises. Carefully chosen topics--Meet the following criteria: Accessibility--the material does not require a heavy mathematical infrastructure. Applicability--the presentation connects the mathematics presented and the real-life problems that motivate it. Currency--much of the material dates within the last 100 years, and some--fractals for instance--within the last 15. Aesthetics--develops an appreciation for mathematics by combining its elegance with its simplicity. Numerous exercises--Over 1500 in total. Exercises are divided into 3 levels of difficulty: Walking: straight forward applications of the concepts discussed in the chapter. Jogging: exercises that require extra effort and/or insight on the part of the student. Running: exercises that really challenge the students' ability and understandingCentral Kentucky Book Supply Nicholasville, KY 0131001914 A good value based purchase. Acceptable condition. Used texts may NOT contain supplemental materials such as CD27 +$3.99 s/h Acceptable goodwill_of_oc Santa Ana, CA Book has light external/internal wear. It may have creases on the cover and some folded pages.This is a USED book. $5
This lesson helps students further their understanding of linear functions by applying the material to a real-world example. The class will use data on an airline flight including travel time, ground speed, time... This lesson uses the example of successive discounts at a retail store to demonstrate numeric, algebraic, and graphical representations of compositions of mathematical functions. Students will get the opportunity to... This lesson from Illuminations helps illustrate recursive sequences. The interdisciplinary lesson uses elements from the short story The Devil and Daniel Webster by Stephen Vincent Benet. A mathematical game has been... This site from Northern Illinois University provides online notes for students using the Abstract Algebra textbook (which is also available online). The materials cover the topics of integers, functions, groups,... This learning unit from the Center for Cultural Design provides a background on Pre-Columbian pyramids. The lesson is used to demonstrate the concept of symmetry. Both two and three dimensional examples of this type of...
Geometries This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern ...Show synopsisThis This edition reflects the recommendations of the COMAP proceedings on Geometry's Future, the NCTM standards, and the Professional Standards for Teaching Mathematics. References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems34351885-5 Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780534351885-4
The Eighth Edition of this highly dependable book retains its best features--accuracy, precision, depth, and abundant exercise sets--while substantially updating its content and pedagogy. Striving to teach mathematics as a way of life, Sullivan provides understandable, realistic applications that are consistent with the abilities of most readers. Chapter topics include Graphs; Polynomial and Rational Functions; Conics; Systems of Equations and Inequalities; Exponential and Logarithmic Functions; Counting and Probability; and more. For individuals with an interest in learning algebra as it applies to their everyday lives. Mike Sullivan's time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. In the Ninth Edition, Precalculus has evolved to meet today's course needs, building on these hallmarks by integrating projects and other interactive learning tools for use in the classroom or online. These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus:Enhanced with Graphing Utilities gives students a model for success in mathematics. A proven motivator for readers of diverse mathematical backgrounds, this book explores mathematics within the context of real life using understandable, realistic applications consistent with the abilities of any reader. Graphing techniques are emphasized, including a thorough discussion of polynomial, rational, exponential, and logarithmic functions and conics. Includes Case Studies; New design that utilizes multiple colors to enhance accessibility; Multiple source applications; Numerous graduated examples and exercises; Discussion, writing, and research problems; Important formulas, theorems, definitions, and objectives; and more. For anyone interested in trigonometry. "The Sullivan Enhanced with Graphing Utilities" series fully integrates the graphing calculator. These widely adopted books are known for their precise careful presentation of mathematics. This precision permeates the book and is particularly evident in the examples, pedagogy and exercises. This book includes coverage of trigonometric functions and their applications, analytic trigonometry, polar coordinates and vectors, and exponential and logarithmic functions. For anyone who needs to brush up on everyday or business-related mathematics. This textbook helps students build the knowledge and skills they need to be successful in trigonometry. It contains a number of theorems, definitions, procedures, and equations. The lessons include Graphs and Functions, Trigonometric Functions, Analytic Trigonometry, Applications of Trigonometric Functions, Polar Coordinates Vectors, Analytic Geometry, Exponential and Logarithmic Functions
Discrete Mathematics and its Applications, Seventh Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering. This renowned best-selling text, which has been used at over 500 institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications…from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fieldsChalk Dust Math saves the day! Everything you need to succeed in Basic College Mathematics is here: 12 chapters with reknowned teacher Dana Mosely, text, and complete solutions guide. The DVDs teach every single lesson! It's like having your own private math tutor.
Boca Raton ChemistryAlgebra 2 is a subject which is critical that a student do well in. The ability to master this subject will greatly affect the students perfromance in all subsequent math classes. The concepts in this course build upon one anotherInductive and deductive thinking skills will be used in problem solving situations, and applications to the real world. Emphasizing on proofs to solve (prove) properties of geometric figures. I will cover topics in Algebra ranging from polynomial, rational, and exponential functions to conic sections.
Pre-Algebra The Pre-Algebra curriculum consists of over 130 lessons offering more than 45 hours of instructional video. This curriculum will provide a solid foundation in the skills necessary to move on to Algebra. Pre-Algebra also offers basic geometry instruction as well as an introduction to more advanced topics like Probability and Statistics. Sample Pre-Algebra topics Include: Numbers and Operations Mathematical Reasoning Algebraic Expressions and Equations Integers, Decimals, Fractions and Percents Ratios and Proportions Square Roots Measurement Basic Geometry Graphs and Functions In addition, printed textbooks are available to supplement the electronic courseware. Along with their Summary and Example sections, these textbooks offer Independent Practice problems that match the courseware for students to work on their own.
Companion Products Product Description In the "Getting Ready to Learn" portion of the textbook lesson, "Check your readiness" exercises help students see where they might need to review before the lesson. "Check skills you'll need" list out the skills used in the lesson, and new vocabulary is listed before it's introduced. Sidebar helps tell students where to go for help in the textbook if they need to review, or note when an online tutor video is available. The lesson itself includes "quick check" problems for students to see if they understand the concept just introduced; "key concepts" boxes that summarize definitions, formulas, & properties, online activities for review and practice; vocabulary sidebars and features that help focus on the language of math; and multiple types of practice activities that feature new material, integrate older material, and provide challenges. A homework video tutor for every lesson is provided online. Designed to especially help students prepare for high-stakes tests like the SAT and ACT, as well as standardized tests, test-taking strategies are included in each chapter. Skills handbook, Spanish/English glossary, instant check answers, and selected answers are included in the student textbook. The teacher's guide includes full-color student pages with the correct answers overlaid in pink ink. The margins contain teacher notes, including page numbers for intervention students to review, an overview, questions to activate prior knowledge, objectives, math background summary, supporting resource list, guided teaching instruction, differentiated instruction for special needs, below level, advanced learners, and ELL students, additional examples, alternative methods, assignments, homework checks, error prevention tips, additional resources, and other helpful information. A skills flow, correlations chart, pacing charts, and other teacher tools are provided. Additional answers with some worked solutions are provided in the back of the book. The workbook provides complete daily support for the lesson, and includes a daily notetaking guide, guided problem solving exercises, and additional practice for every lesson. For each chapter vocabulary and study skills are emphasized. The Daily Notebooking Guide is designed to be used while working through the text; each section corresponds to a section in the text, with objectives clearly laid out. Perforated, newsprint-like pages, softcover. The workbook teacher's guide includes the answers to the workbook with reduced pages & overlaid answers for the daily notebooking questions, and line-listed answers for the practice and guided problem solving questions. It also contains the answers to the vocabulary and study skills for each chapter. Perforated, three-hole-punched pages, softcover.
books.google.com - This... and Combinatorial Mathematics Discrete and Combinatorial Mathematics: An Applied Introduction This were added, creating a greater variety of level in problem sets, which allows students to perfect skills as they practice. This new edition continues to feature numerous computer science applications-making this the ideal text for preparing students for advanced study. User ratings The book is very well-written, clear and precise. I have read a couple of other books for discrete math courses, but found nothing exciting there; mostly frustrating errors, bad examples and confusion ...Read full review
Using the Scientific Calculator Step-by-Step Now that you have had an opportunity to review the basic keys and functions that students will need for the GED Mathematics Test, it is time for you to learn the basic step-by-step operation of the Casio fx-260 calculator. Teaching Tip: When working with students on positive and negative integers, remind them that they must first enter the number before they can change the number from a positive to a negative. The calculator cannot perform a function on a number that is not there.
Higher Education Solutions The U.S. Department of Education reported Basic Algebra to have the highest failure and withdrawal rates among post-secondary math courses. Intermediate Algebra and Developmental Math followed closely behind. More than half of the mathematics courses offered at two-year colleges and 10-20 percent of the mathematics courses offered at four-year colleges are developmental math courses. Seventy-five percent of new students enrolling in two-year colleges must take one or more developmental mathematics courses, and on average, half of those student do not pass. All of your students can achieve proficiency in mathematics with the Carnegie Learning Developmental Math Software Solution. Our solution provides students of all ages and skill levels with highly differentiated and self-paced instruction that meets their exact needs to improve their developmental math skills. That means your developmental math courses will be able to retain more students and quickly promote them into their degree programs. Be sure to read about the effect that Carnegie Learning Developmental Math Solution has on retention rates in the Success Stories tab on the left. For detailed information about the program components, visit the Higher Education Curricula page. Read more about our Developmental Math offerings by visiting the following pages: Many students want to prepare for an upcoming test or refresh their knowledge of prior course content before advancing onto their next math course. These students will greatly benefit from the Cognitive Tutor software in their Learning Support Center. Webinars I have successfully used Carnegie Learning's K-12 solution with adult learners, and I like what I see in the new solution specifically built for the needs of the post-secondary market. This is a fresh and exciting approach to remediating college level math students. It offers instructors the flexibility to select specific units for a custom curriculum. It also provides the student individualized step-by-step instruction for each lesson, the ability to go back and review concepts and practice skills from any point in the software and, in general, personalizes the learning path so that the curricula are efficient for college use.
Description Professors teaching this course have two concurrent goals: to teach the microeconomic tools students need for future coursework and careers, and to encourage students to work problems. In Microeconomics: Theory and Applications with Calculus, Perloff brings his hallmark pedagogy to the calculus-based course by integrating Solved Problems and real, data-driven applications in every chapter. This new text offers a serious presentation of calculus-based microeconomic theory and offers a suite of carefully crafted, calculus-based problem sets at the end of each chapter. Microeconomics: Theory and Applications with Calculus covers basic and modern theories first, and then offers advanced theory chapters at the end. By providing graphical representation and real-world applications to illustrate the theory, Perloff demonstrates how individuals, policy makers, and firms use microeconomic tools to analyze and resolve problems. Preview online! An interactive tour of Microeconomics: Theory and Applications with Calculus is available here . Perloff's algebra-based Microeconomics , now in its Fourth Edition, has become a market leader because it clearly introduces theory and helps students develop problem-solving skills through its Solved Problem featureMicroeconomics: Theory and Applications with CalculusMicroeconomics: Theory and Applications with Calculus
Pre-Algebra - 06 edition Summary: A new way of thinking about Algebra readiness! Focused, organized, and easy to follow, Glencoe Pre-Algebra shows your students how to read, write, and understand the unique language of mathematics, so they'll be prepared for every type of problem-solving and assessment situation.58.69 +$3.99 s/h VeryGood AlphaBookWorks Alpharetta, GA 007870424397.30
MATH TREK Algebra 1 04/01/04 For curriculum-based algebra instruction, teachers and students can use MATH TREK Algebra 1. The multimedia program includes tutorials, assessments and student tracking. Students can use the program's scientific calculator, glossary and journal to help them complete the various exercises and activities. The assessment and student-tracking features provide immediate feedback to students so that they can stay on top of their progress. This engaging program, complete with sound, animation and graphics, can be used on stand-alone computers or a network. NECTAR Foundation, (613) 224-3031
One of the most valuable tools acquired in a university education, college algebra is essential for courses from the sciences to computing, engineering to mathematics It can help you do better on placement exams, even before college, and it's useful in solving the computations of daily life Now anyone with an interest in college algebra can master it InCollege Algebra Demystified,entertaining author and experienced teacher Rhonda Huettenmueller breaks college algebra down into manageable bites with practical examples, real data, and a new approach that banishes algebra's mystery.WithCollege Algebra Demystified,you master the subject one simple step at a time-at your own speed Unlike most books on college algebra, general concepts are presented first-and the details follow In order to make the process as clear and simple as possible, long computations are presented in a logical, layered progression with just one execution per step.This fast and entertaining self-teaching course will help you Perform better on placement exams Avoid confusion with detailed examples and solutions that help you every step of the way Get comfortable with functions, graphs of functions, logarithms, exponents, and more Master aspects of algebra that will help you with calculus, geometry, trigonometry, physics, chemistry, computing, and engineering Reinforce learning and pinpoint weaknesses with questions at the end of every chapter, and a final at the end of the book
simple integrals by guessing the antiderivative Compute the area under a given curve using Riemann sums Compute integrals using the power rule, the sum and difference rules, and simple substitution. State and utilize the Fundamental Theorem of Calculus to compute definite integrals. Late Submission of Course Materials: Assignments should be turned in on the specified due date. Classroom Rules of Conduct: Students are expected to participate fully in class learning activities. Phones and beepers are to be placed on "vibrate" or turned off. Students are required to exercise courteous behavior between themselves and with the
Math made easier: advice from experts Many students struggle with various kinds of math, including positive and negative number signs, fractions, factoring, graphing and word problems, instructors in the department of mathematics and statistics said. In fall 2011, the success rate for college algebra, a core math course, was 59 percent, said Mellisa Hardeman, senior instructor in the department. The success rate dropped anther percentage point the following year, she said. In fall 2012, 50 to 60 percent of pre-core math students had difficulties solving math problems, said Denise LeGrand, director of the Mac I math lab. Ike McPhearson, math tutor, explained why students may have trouble comprehending math. One reason is that students may come from a home where education is not valued, he said. A bad experience with an instructor can also change students' attitudes about math. "You can't take yourself too seriously as a teacher," said Hardeman. Instructors can never give a student too much help passing math, she said. Students who took a math course in high school before going to college are less likely to struggle with math, Hardeman said. Some students go to college years after graduating high school, however, and may forget everything they learned in their math classes. Fortunately, there are a number of strategies that can help students overcome these challenges and develop a better understanding of math. "In order to make math easy for students, show different ways of how to understand it," said McPherson, who has tutored high school and college students. Another way of making math fun for students is to create different games, he said. According to LeGrand, the most important way to become better at math is to practice math exercises for 20 to 30 minutes. "They won't see the results right away," said LeGrand, " but if they go to class and focus on work required, they will be successful and they will build confidence." In addition, students can get help from tutors at the math lab. Each semester, the lab hires 12 tutors, LeGrand said. For the math-impaired, there is a new math course called Quantitative and Mathematical Reasoning. The course was designed for students who are not science, technology, engineering or mathematics majors. It focuses on practical math, for example, currency exchange rates. The course fulfills the core math requirement, in place of college algebra. Pre-core math courses, developmental math courses students take if they do not have the prerequisites for college math classes, are becoming more successful, said Tracy Watson, coordinator for pre-core math. The success rate for those courses rose to 77 percent in fall 2012, she said. Previously, the success rate was 37 percent for a 4-year period, she said. This semester, there are 80 math majors at the university. "We all like how math works because it all fits together," Watson said. "Students who major in math develop a sense of thinking and solving problems," said Thomas McMillan, department chair. Once students better understand math, they will have the confidence to solve not only math problems, but problems in everyday life as well
Algebra is mainly only necessary if you may have a career involving math (like being an accountant or actuary), but otherwise it's not that useful but regardless you have to learn it in school, like it or not...I'm good at algebra though and if you need help in math I could help you, feel free to ask me for help :) Solving quadratic equations by factoring are important to you in life because they can help you better understand situations in your life. The same goes for all other mathematical principles since maths affects almost every aspect of our lives.
Book summary Here's a complete, easy-to-grasp course in trigonometry that takes the form of a fantasy novel. The King of Carmorra and his subjects have many practical problems to solve, and their answers can be found by applying principles of trigonometry. Readers follow along and learn to solve many different problems that can be reduced to triangular diagrams. They learn the laws of sine and cosine, trigonometric functions and inverse functions, waves, conic sections, polynomial approximation, and much more. The book is filled with instructive exercises and their solutions, plus illustrative drawings, graphs, and diagrams. This new edition contains updated coverage on using graphing calculators and computer spreadsheets for solving trigonometric problems. [via]
AMATYC Student Math League Math Prize Competition [This page was last updated on June 16, 2010 .] Background The American Mathematical Association of Two-Year Colleges (AMATYC) has an annual mathematics competition for two-year college students. The AMATYC website has links with competition questions and answers from past exams, past results, and also more details about the eligibility rules. Participants are eligible not only for national awards, but also regional and Pierce College awards. The competition consists of two rounds. Each round is a one hour, 20-question test, and students may participate in either or both rounds. Although no calculus is required for any of the problems, the exams are challenging. Each correct answer earns two points and each incorrect answer (to a multiple-choice question) earns a penalty of 1/2 point deduction. Most calculators, including graphing calculators such as the TI-89 and the HP-48G, are allowed. But cell phones, PDA's, and calculators with a "qwerty" keyboard or a removable disk drive are forbidden. Participants need to bring a Pierce ID. Awards are given to the top 3 Pierce students at the graduation breakfast. Practice Questions and answers from past competitions are available (in pdf) from links in the table below. (These and other past exams and answers are posted on the AMATYC website.) The Math Club also practices for the contest on Thursdays 12:30 - 1:30 in 1403. Further information Click here for the Handout on "Counting Formulas". The top Pierce students in each round are eligible for prizes to be awarded at the graduation breakfast.
more details Matrix transforms are ubiquitous within the world of computer graphics, where they have become an invaluable tool in a programmer's toolkit for solving everything from 2D image scaling to 3D rotation about an arbitrary axis. Virtually every software system and hardware graphics processor uses matrices to undertake operations such as scaling, translation, reflection and rotation. Nevertheless, for some newcomers to the world of computer games and animation, matrix notation can appear obscure and challenging.Matrices and determinants were originally used to solve groups of simultaneous linear equations, and were subsequently embraced by the computer graphics community to describe the geometric operations for manipulating two- and three-dimensional structures. Consequently, to place matrix notation within an historical context, the author provides readers with some useful background to their development, alongside determinants.Although it is assumed that the reader is familiar with everyday algebra and the solution of simultaneous linear equations, Matrix Transforms for Computer Games and Animation does not expect any prior knowledge of matrix notation. It includes chapters on matrix notation, determinants, matrices, 2D transforms, 3D transforms and quaternions, and includes many worked examples to illustrate their practical use.
... a brief description of MathStudiestopics, ... The Internal Assessment (IA) portion of the IBMathStudiesSL curriculum is the MathStudies project. ... Course: IBMathStudiesSL Due date: February 2013 Do not turn this in! ... Mathematical Studies (SL), Mathematics (SL) and ... Topics The course consists of seven topics taken over 2 years (150 hours): Algebra Functions and Equations ... The exploration (IA) will also contribute to the final IB grade. ... Mathematical Studies (SL), Mathematics (SL) and ... The exploration (IA) will also contribute to the final IB grade. IB Assessments: ... Written work that involves exploration into a math related topic chosen by the student Cheating: IBMathStudies is a Survey course of a wide range of topics. ... Mathematical StudiesSL Total 150 hours ... Failure to do a mathstudies project will result in a forfeiture of the IB diploma. 130 hours for Topics and 20 hours for Project equal 150 hours for the course. MathStudiesSL (Discrete) George Magliaro ... (IA); these are activities assessed by the ... principles, practices and skills using actual case studies. Content Topics explored in IB Business and Management provide a broader understanding of IBMathStudiesSL no required reading assignments ; Students coming from IBSL or IBHL: 15-20 ... essays taken from the topics studied in the current year that students have one hour and a half to ... IA ; 2 Lab Reports per quarter- IA . Group 4 Project . Also utilize the International Baccalaureate website ... IB Precalculus SL Mathematics Year 1 To the Parent or Guardian: ... Even if you are unable to help your student with their mathstudies, ... IBMathStudiesSLIB English II HL AP U.S ... Latin SL, Topics, Chemistry SL, Physics HL and SL, MathStudies, Music SL and Art HL were higher than the World Wide average in 2013. 585 students received a ... Virgin ia Department of Education and Southern Association of Colleges and ... The International Baccalaureate® (IB) is an educational ... IB Diploma Programme students study six subjects at higher or standard level. ... grade, and if you do not complete your IA tasks you shall not be awarded the diploma. 20th Century TopicsSLIB History of Europe HL IB Psychology SLIB Economics SLIB ... Precalculus Advanced prepares students for the IBMathSL, Calculus AB AP, or Statistics AP. 23 ... This IB Standard Level course is for students who expect to go on to study subjects with a Mathematical StudiesSL is a two year math course intended for ... demonstrates their understanding of the topics being studied along with two IB external assessment papers. ... IB Mathematical StudiesSL is a 2-year course. The IA portion will consist of laboratory investigations. ... Mathematical StudiesSL is a two year math course intended for ... demonstrates their understanding of the topics being studied along with two IB external assessment papers.
A systematic, research-based introduction to the principles and practice of teaching mathematics at primary school level. This second edition includes new material on middle years, and on numeracy, early numbers and fractions.Your hands-on guide to real-world applications of linear algebra Does linear algebra leave you feeling lost? No worries —this easy-to-follow guide explains the how and the why of solving linear algebra problems in plain English. From matrices to vector spaces to linear transformations, you'll understand the key concepts and see how they relate...
Customer Reviews for Weimar Institute Math-It Guide Book on CD-ROM Students from kindergarten to grade eight can benefit from the math teaching method of Elmer W. Brooks as presented in this CD-ROM. All basic math skills from number recognition to statistics and transversals are covered, and is addressed in clear, concise terms. This CD-ROM be used as a basic math skills tool, explaining each term and concept for all covered grades. Now in CD-ROM format! Customer Reviews for Math-It Guide Book on CD-ROM This product has not yet been reviewed. Click here to continue to the product details page.
This review is passed along from the ENC Online newsletter. --------- She Does Math! Marla Parker, editor Reviewed by Judy Spicer, ENC Mathematics Education Abstractor She Does Math! is written for high school and college students who enjoy math and want to know more about potential career paths. It can also answer the question often asked by other students: When am I ever going to use this? The 253-page book presents personal career histories written by 38 professional women. The reader learns how mathematics plays a critical role in careers ranging from environmental psychology to dietetics to engineering. Each history describes how much math the author took in high school and college; how she chose her field of study; and how she ended up in her current job. Each woman presents several mathematics problems typical of those she has to solve on the job. She Does Math! encourages students to take math every year in high school and to encourage high school and college students, especially women and minorities, to consider technical fields when planning their careers. For example, an ophthalmologist explains that in her school days she liked to read mysteries and was fascinated by the precision in physics and calculus. These interests lead her to a career in medicine and to researching how the eyes are used together to achieve three-dimensional vision. The related problems use mathematics to investigate double vision, determining when reading glasses are needed, and the use of eye drops. Solutions to all problems are given. The equity@nici-mc2.org listserv provides a forum for sharing information about issues, strategies and resources regarding educational equity. The Equity Forum is sponsored by the National Institute for Community Innovations ( and is moderated by Joy Wallace (joy@col-ed.org).
You are here MATH FOR ELEM SCHL TEACHERS I First of two courses designed for prospective elementary teachers. Emphasizes the real number system, arithmetic operations, and algebra. Explorations focus on representing, analyzing, generalizing, formalizing, and communicating patterns and structures. Content is presented using problem solving and exploration. Prerequisite: ACT math score of 23 or above, SAT math score of 530 or above
Trigonometry - 7th edition Summary: Gain a solid understanding of the principles of trigonometry and how these concepts apply to real life with McKeague/Turner's TRIGONOMETRY. This book's proven approach presents contemporary concepts in brief, manageable sections using current, detailed examples and interesting applications. Captivating illustrations drawn from Lance Armstrong's cycling success, the Ferris wheel, and even the human cannonball show trigonometry in action. Unique Historical Vignettes offer a fascinating...show more glimpse at how many of the central ideas in trigonometry began. ...show less Charles P. McKeague Charles Mark Turner Mark D. Turner earned his B.A. in Mathematics from California State University, Fullerton. Professor Turner worked in the aerospace industry for two years with the Systems Modeling and Analysis group at The Aerospace Corporation before completing his graduate work at California Polytechnic State University, where he earned his M.S. in Mathematics and Secondary Teaching Credential. Turner is a full-time instructor at Cuesta College in San Luis Obispo, California. He has been a leading influence in the use of graphing calculator and multimedia technology in the classroom, as well as a leading innovator in instructional website design at his institution. Mark has also created educational materials through his own company, Turner Educational Publishing, including a series of Web-based tutorials on the use of the TI-83 graphing calculator. He is a member of the American Mathematics Association for Two-Year Colleges and the California Mathematics Council for Community Colleges, and is a frequent speaker at annual conferences. Professor Turner has received the CMC3 Award for Teaching Excellence26854 used book - free tracking number with every order. ?book may have some writing or highlighting, or used book stickers on front or back $117.55 +$3.99 s/h Good newrecycleabook centerville, OH 1111826854 Some used book stickers on cover - used text may NOT contain supplemental materials such as CD's, info-trac, access codes, etc... - all books hand packed by me personally in bubble wrap mai...show moreler to protect book during shipping - buy with confidence - established seller ...show less $119.95 +$3.99 s/h VeryGood AlphaBookWorks Alpharetta, GA 11118268119$121.95 +$3.99 s/h VeryGood Bookbyte-OR Salem, OR Has minor wear and/or markings. SKU:9781111826857-3-0 $126.45 +$3.99 s/h Acceptable textbooknook Knoxville, TN Used book stickers all over the cover, but the inside is flawless!!! Great Text!!! THIS IS THE TEXT ONLY! $128.05 +$3.99 s/h LikeNew Bookbyte-OR Salem, OR Almost new condition. SKU:9781111826857-2-0 $145148.23 +$3.99 s/h VeryGood textbooknook Knoxville, TN Used book stickers all over the cover, but the inside is flawless!!! Great Text!!! THIS IS THE TEXT ONLY
... read more Differential Manifolds by Antoni A. Kosinski Introductory text for advanced undergraduates and graduate students presents systematic study of the topological structure of smooth manifolds, starting with elements of theory and concluding with method of surgery. 1993 edition. Differential Forms by Henri Cartan The famous mathematician addresses both pure and applied branches of mathematics in a book equally essential as a text, reference, or a brilliant mathematical exercise. "Superb." — Mathematical Review. 1971 edition. Elements of Pure and Applied Mathematics by Harry Lass This completely self-contained survey explores important topics in pure and applied mathematics. Each chapter can be read independently, and all are unified by cross-references to the complete work. 1957 edition.Ordinary Differential Equations by Edward L. Ince Among the topics covered in this classic treatment are linear differential equations; solution in an infinite form; solution by definite integrals; algebraic theory; Sturmian theory and its later developments; much more. "Highly recommended" — Electronics Industries. Product Description: text introduces the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Requiring familiarity with several variable calculus and some knowledge of linear algebra and set theory, it is directed primarily to engineers and physical scientists, but it has also been used successfully to introduce modern differential geometry to students in mathematics. Chapter I introduces exterior differential forms and their comparisons with tensors. The next three chapters take up exterior algebra, the exterior derivative and their applications. Chapter V discusses manifolds and integration, and Chapter VI covers applications in Euclidean space. The last three chapters explore applications to differential equations, differential geometry, and group theory. "The book is very readable, indeed, enjoyable — and, although addressed to engineers and scientists, should be not at all inaccessible to or inappropriate for ... first year graduate students and bright undergraduates." — F. E. J. Linton, Wesleyan University, American Mathematical
Summary: Written by a team of nationally known mathematics educators and mathematicians who have experience in teaching at the public school and university levels, this text builds on the recommendations of NCTM's Principles and Standards. It provides guidance in teaching content and developing processes in students from the middle school through high school. Its emphasis on mathematical modeling as an approach focuses on the development of the competencies needed to teach co...show morentemporary curricular materials in today's schools
Algebra Help In this section you'll find study materials for algebra help. Use the links below to find the area of algebra 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 algebra. Algebra Final Practice Test 1 . The grade in a psychology class is determined by three tests and a final exam. The final exam counts twice as much as a test. A student's three test grades are 78, 82, and 100. What ... Adding/Subtracting Fractions with Variables and Exponents Practice Problems Set 1: Adding/Subtracting Fractions with Variables and Exponents To review addition and subtraction of fractions with variables in one or more of the denominators, go to
Mathematics - AS/A2 Levels 1-2 Years Entry Requirements Five GCSEs at Grade C or above including English Language and Mathematics at Grade B, preferably A. Course Content You can choose from an extensive range of A Levels. We offer around 20 AS Levels, most of which can be taken through to A2 Level, giving you a real opportunity to progress on to Higher Education or into work. We will help you choose your A Level subjects based on your existing qualifications, subjects you enjoy or a career in which you're interested. Course running at Northampton (AS Level (First Year): Northampton: LALEVELFT FBAS; A Level (Second Year): Northampton: LALEVELFT FBA2* *Only use this code if you will have completed your AS year and want direct entry into 2nd year) From our YouTube channel Success stories Lucie Glennon Lucie was so pleased to have achieved the grades she needed to get into The University of Glamorgan. "I am going to study Film Studies there which is what I have always been interested in. I will miss college though – it was such fun and I made some amazing friends. The best thing about college is that the teachers have all worked in industry so you are taught by experts in the field. As I want to get into films that was important. I want to be taught by someone who knows what it's really like. I got two Bs in Film and Media and a C in English. I am so excited about the future."
Prealgebra - 5th edition Summary: Prealgebra, 5/e, is a consumable worktext that helps students make the transition from the concrete world of arithmetic to the symbolic world of algebra. The Aufmann team achieves this by introducing variables in Chapter 1 and integrating them throughout the text. This text's strength lies in the Aufmann Interactive Method, which enables students to work with math concepts as they're being introduced. Each set of matched-pair examples is organized around an objective...show more and includes a worked example and a You Try It example for students. In addition, the program emphasizes AMATYC standards, with a special focus on real-sourced data. The Fifth Edition incorporates the hallmarks that make Aufmann developmental texts ideal for students and instructors: an interactive approach in an objective-based framework; a clear writing style; and an emphasis on problem solving strategies, offering guided learning for both lecture-based and self-paced courses. The authors introduce two new exercises designed to foster conceptual understanding: Interactive Exercises and Think About It
The site is part of the learning area of BBC Scotland that includes a variety of K-12 subjects. This area focuses on the... see more The site is part of the learning area of BBC Scotland that includes a variety of K-12 subjects. This area focuses on the material for the Scottish "Higher" exams in mathematics. These national exams are normally taken by students age 15 or 16, and play a significant role in university admission. The main subject groups for the material in this mathematics area site are algebra, calculus, geometry and trigonometry. The material is simple but well executed The Graphical representation of complex eigenvectors simulation aims to help students make connections between graphical and... see more The Graphical representation of complex eigenvectors simulation aims to help students make connections between graphical and mathematical representations of complex eigenvectors and eigenvalues. The simulation depicts two components of a complex vector in the complex plane, and the same vector under several transformations that can be chosen by the user. A slider allows students to change the second component of the initial vector. The simulation shows whether or not the vector is an eigenvector, and if so displays the associated eigenvalue. The simulation includes a small challenge in asking the student to find the elements of one of the transformation matricesThe Graphical representation of eigenvectors simulation aims to help students make connections between graphical and... see more The Graphical representation of eigenvectors simulation aims to help students make connections between graphical and mathematical representations of eigenvectors and eigenvalues. The simulation depicts the two components of a unit vector in the xy-plane, and the same vector under several different transformations that can be chosen by the user. A slider allows students to change the orientation of the initial vector. The simulation shows whether or not the vector is an eigenvector, and if so displays the associated eigenvalue. The simulation includes a small challenge in asking students to find the elements of one of the transformation matrices 4This website is a great guide for teachers. It has several interactive links to choose from that cover multiple standards in... see more This website is a great guide for teachers. It has several interactive links to choose from that cover multiple standards in regards to math and literacy, for grade levels K-12. The contents are broken down into specific concepts, i.e. shapes, counting to 20, addition, etc. each topic has a variety sites to choose from. The sites are interactive, fun, an easy to navigate. The interactive sites are easy enough for students to complete the activities on their own. This site also offers assessment assistance, and on-line practice modules. Quoted from the site: This applet explores fitting a polynomial p(x) of degree n to a given set of data points. It computes... see more Quoted from the site: This applet explores fitting a polynomial p(x) of degree n to a given set of data points. It computes the best least squares approximation to the data, "best" in the sense that SUM (p(xi) - yi)^2 is minimized. The applet provides controls for choosing the degree n, setting the precision of all displayed numbers, selecting and deselecting data points, and editing the data list. It also allows for entry of your own custom polynomial and computing its least squares error.
From Albert Einstein to C. R. Wylie, we are proud to offer low-priced editions of groundbreaking works by the most respected thinkers in the fields of math and science. Here are a few of our most important authors. To visit our main Math and Science Shop, please click here. And be sure to join our Math and Science Club for a 20% everyday discount, free newsletter, and other exclusive benefits. Recommendations...Einstein's Essays in Science by Albert Einstein, Alan Harris Speeches and essays in accessible, everyday language profile influential physicists such as Niels Bohr and Isaac Newton. They also explore areas of physics to which the author made major contributions. Mathematics and the Physical World by Morris Kline Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non-Euclidean geometries. Also describes how math is used in optics, astronomy, and other phenomena. Foundations of Geometry by C. R. Wylie, Jr. Geared toward students preparing to teach high school mathematics, this text explores the principles of Euclidean and non-Euclidean geometry and covers both generalities and specifics of the axiomatic method. 1964 edition.
books.google.com - Designed as a text for a first course in the college mathematics curriculum that focuses on the formal development of mathematics, this book explains how to read and understand mathematical definitions and proofs, and how to construct and write mathematical proofs. Emphasis is on writing mathematical... Reasoning Mathematical Reasoning: Writing and Proof Designed as a text for a first course in the college mathematics curriculum that focuses on the formal development of mathematics, this book explains how to read and understand mathematical definitions and proofs, and how to construct and write mathematical proofs. Emphasis is on writing mathematical exposition, with guidelines for writing proofs incorporated throughout the text. Learning features include preview activities that prepare students to participate in classroom discussion and activities for in-class group work. Coverage encompasses logical reasoning, constructing and writing proofs, set theoy, mathematical induction, functions, and topics in number theory and set theory. Globalbook For professionals in the sciences or engineering who need to brush up on their advanced mathematics skills. Mathematical Reasoning: Writing and Proof, 2/E ... detalles.php?ISBN=0131877186
The Algebra 2 Tutor DVD Series teaches students the core topics of Algebra 2 and bridges the gap between Algebra 1 and Trigonometry, providing students with essential skills for understanding advanced mathematics. This lesson teaches students how to graph inequalities. In order to do this, students are taught how to use the equation of the inequality to draw a boundary line on the coordinate plane. Students are then instructed how to properly shade the graph so that the inequality is properly reflected. Grades 8-12. 21 minutes on DVD. Customer Reviews for Algebra 2 Tutor: Graphing Inequalities DVD This product has not yet been reviewed. Click here to continue to the product details page.
vision, the geometric laws that relate different views of a scene. Geometry, one of the oldest branches of ... multipleviews of a scene from the perspective of various types of geometries. A key feature is that it ... role incomputer communications. Producers and users of images, in particular three-dimensional images, ... numerous computervision algorithms included in the OpenCV library. You will learn how to read, write, ... a variety of computervision algorithms and be exposed to important concepts in image analysis that will ... mathematical morphology and image filtering. The detection and use of interest points incomputervision is ... filtering. The detection and use of interest points incomputervision is presented with applications for ... Exploit the image geometryin order to match different views of a pictured scene Calibrate the camera from ... programming. It can be used as a companion book in university-level computervision courses. It constitutes an ...
... More About This Book totally rewritten chapter on problem-solving offers students a core set of strategies that apply to a variety of curriculum-related exercises. In addition to subject review, demonstration examples, and practice exercises with answers, the book includes several complete recent Math A Regents exams 2003 Best Book Ever I Bought This book explains every topic in easy to follow topics. This book also gives step by step demonstration examples. I highly recommend this book to anyone who wants to prepare for the Math A exam. This book is also good for the 9th grade SHSAT OR 9th grade SPECIALIZED SCIENCE EXAM because it has many good topics that might be on the exam. It is worth every pennyAmazing This wonderful book is an excellent book to use to prepare for an excellent grade on the Math A regents. To do good, you need to be patient and devote your time to reading and fully comprehending this. I had bought this in Sep., reviewed through out the year, and in June, I got a 100% on the Math A final. It describes in EXCELLENT detail EACH and EVERY topic required to know for the Math A course. It is highyly recommended for students who wish to be successful at Math A. It also provides about 10-20 questions for every topic so you get practice. I recommend you get this book and, for more practice get the Barron's Regents Exams and Answers: Math A. The barron's collection is a great series and provides excellent review. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted April 3, 2006 Excellent Book I really like this book. I didn't understand my teacher very well when she was teaching, so i decided to buy this book and it helped boast my grades and better understand math my way. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted December 18, 2003 Great... This book was very good.. except that I HATE MATH!!!!! wtvr, this book will help for the regents Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted November 19, 2002 Great Book!!!!! This a book that you should get. It really helps with your math skills. A special tool aor need with this book is the Texas Instrument calculator TI-83 and TI-83 Silver Edition. The book shows you how you can use the calculators with simple quesions on the regents. It also come in handy with the Barron's Math B book. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Recommended by Dr. Jay Wile of Apologia Science, VideoText Interactive Homeschool Math is an educational materials company specializing in multi-sensory programs for middle and high school mathematics (pre-algebra, algebra 1 & 2, and geometry/trigonometry). Developed by author and educator Tom Clark, the focus is on the "why" of math, teaching mathematical concepts, not shortcuts and tricks. The instructional strategy includes brief video lessons, using computer graphics, and is supported by course notes, student texts, detailed solutions manuals, progress tests, and instructor's guides. It also includes a toll-free helpline. New online versions of the VideoText curriculum are now available, providing families with even more efficient, economical options for mastering upper-level math.
Matrix Methods: Applied Linear Algebra, 3e, as a textbook, provides a unique and comprehensive balance between the theory and computation of matrices. The application of matrices is not just for mathematicians. The use by other disciplines has grown dramatically over the years in response to the rapid changes in technology. Matrix methods is the essence of linear algebra and is what is used to help physical scientists; chemists, physicists, engineers, statisticians, and economists solve real world problems
Linear Algebra and Its Applications ma...more mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.(less) Hardcover, Fourth Edition, 496 pages Published July 19th 2005 by Cengage Learning (first published April 1st 1976) Community Reviews ...more pull this off, and it's just too Mr.-Rogers-y for my taste. Fortunately, Strang overcomes his cutesy voice by having a really very solid presentation of the material: much better than that of my professor at an unnamed University currently charging extortionist fees for the privilege of learning math from a professor very good at research who hasn't actually spoken to another human being outside of peer-reviewed journals since that summer of 1961 when a girl mistook him for someone else at a coffee shop. I also would complain, briefly, about Strang's notation, which struck me as about the same level as his authorial voice; which is to say, somewhat limited in rigor for the sake of plainness. I guess this is better than the opposite problem but, really, who uses a superscript H for the conjugate transpose of a matrix? An asterisk (*) is used in every single other text I've come across. But complaints aside, Strang is to be commended for focusing on the conceptual framework of linear algebra, with matrices as general linear operators, and eigenvalues/eigenvectors of matrix as fundamental properties of systems' dynamical evolution, instead of just throwing all the math at us and hoping that we can sort it out. (less) ti...more tiny little journey at the end of which something new is discovered. Linear Algebra is a crucial topic for understanding probability, quantum mechanics, and a whole slew of other topics in engineering and mathematics. If you're interested in any of these areas, preparation with this book would not only be helpful but quite fun along the way. ver...more very applied background.(less) An excellent and accessible intro to Linear Algebra. As I've gone further, I've found many (even only medium-complex) topics that Strang omits, but I still find myself coming back here when I really want to understand what's going on in some Linear Algebra context.
10 Steps to Ace Calculus Dried Beans…CALCULUS. Woo…Well, its basically pretty conceptual & if you follow the ten step that follow, you'll be in a better position to ace calculus! Do remember that these are steps and not ways to ace calculus. So you'll need to go step-by-step. You can move on to the next step only after the execution of the first(mathematicians are programmers). 1. Ace PreCalculus Your precalculus must be very strong if you want to move on to Calculus. If you're planning to take up multivariable calculus as well, you also need to build up concepts of: Vectors Parametric coordinates & equations Matrices as a part of precalculus itself. Now, once you are able to see functions all around you…you're ready to go! 2. Functions are Calculus The first course I took up for Calculus was a 6 month course. My instructor took 4 months just to teach functions & their graphs! That was scary, I had a competitive exam & I had to complete the rest of it in 2 months. Surprisingly, we could do it in half the time! When the functions are with you, Calculus is with you. The rest is just application of functions. But, you need to be very good with them & especially with their graphs which help a lot in giving you an insight to the problem. You can visualize the problem& what you're being asked for & how to get itif you take a look at the graph of the functions involved in the problem. To be a master of functions, know their nature. Their likes, dislikes & hobbies. Okay, technically you need to know their domain(they want these for the x), therange(they give these y's) & theirbasic properties(the graph tells it all). 3.Understand the Basic Principles Yes, its now time to know what you are trying to ace…Calculus. To understand the basic principles of Calculus, you jest need to ask yourself three questions: What. Who. Why.…And the answer is the same… Mr. Newton & Mr. Leibniz gave us: On your left is what we call the derivative(represented by d/dx or y' or f') which gives the slope of the tangent to the function at a point. Say you have the function f(x), then the slope of f(x) at x=a would be f'(a). On the right is the integral of the function between two points(represented by the two lines without arrow heads) which gives the area under the curve of the function between these two points. 4. Learn to Apply The Basics One you understand the above basic physical significance of calculus(integral & differential), you should learn ow to use it to solve problems. In any calculus problem, You'll be asked to find one of these… m(slope) or A(area) directly or inderectly. Once you know the significance of the integral & derivative, solution comes to you. 5. Understanding The First principles Lets now take up the derivative. We talked about its physical significance, but how do we compute the derivative? Well, we use something called the first principle. It directly gives us the derivative of a function & we can plug in different values of x into this derivative to get slopes of tangents at varius points. And this is what we use to obtan the basic derivatives… You must derive all of the basic derivatives of common functions(the ones you find in any genuine basic derivatives table). And yes, you may also derive the basic rules of differentiation(sum/product/quotient) using the first principles. You must practice a lot of problems on differnetiation to learn the derivatives of these functions & then extend the to any possible function using the rules of differentiation. 6. Apply those Derivatives Once you are able to differentiate any function that you come accros, you would want to apply these derivatives in real time. So what can they be used for? A)You can use these for finding slopes-something you're already familiar with. The derivative itself is the slope of the tangent at a point. What next? You can then find the equations of tangents & normals using their slopes. B)Derivatives can be used for finding maxima & minima of functions-derivatives can be used to find extremum by examining the slopes of tangents. Take a look at this curve below, you'll find that the slope of the tangent vanishes(tangent becomes parallel to the x-axis) at points of minima & maxima. All the x's corresponding to these points(slope or derivative=0) are the points of extrema & the corresponding y's, the maximum & minimum values of the function. C) The third thing derivatives help us with is determining the nature of a function or curve. We can find the intervals in which the function increases or decreses. Take a look at the image below. You'll observe that the slope of the tangent is positive whenever the function increases while the slope is negative when the function decreases. Yeah, I know exactly how you feel! Now the tangents 1 & 2 have a positive slope(observe that the angle they make with the x-axis < 90 degrees) & the function is increasing. Also, the tangent 3 has a negative slope(angle < 90 degrees) & the function is decreasing. Apart from the above, you can also use derivatives to sketch curves, in approximations & finding physical rates of change(which is what differential caluclus is all about). 7. Reverse the Process of Differentiation Next, we need to reverse the differentiation process to obtain what is called the anti-derivative or indefinite integral. Indefinite, because of lack of the physical dimension to it. If the derivaive of a function f(x) is F(x), then the integral of F(x) is f(x). Thats all it means. You'll be able to get a deeper insight if you look at a genuine table of integrals. Knowing some of the basic integrals, you can use various methods of indefinite integration to find integrals of almost all functions. Yes…there is a reson why I used any function in case of differnetiation & alomost any function in case of anti-differentiation. The reson is simple. You can differentiate all functions(not considering specific values) but you cannot integrate all functions. There are a few non-integrable functions. 8.Add Sense to the Anti-Derivative Its now time to go definite! This is easy, just take the anti-derivative of the function and plug in the values of the points which enclose the area. As you see here, you may find area under a curve or the area between two curves(subtract the two shaded areas in the image). 9. Put Everything Together Put the two of them together. What do you get? Rates of Change & Areas under curves going hand in hand? Weird? But thats what calculus is all about. Thats the whole beauty of calculus. We come across a lot of weird & beautiful relation in math & Calculus gives another one. You might now wish to do some basic differential calculus to see how exactly the two process of differentiation & integration are related to each other. When you mix em' up together, you'll get what is called a differential equation.
Description This text is appropriate for undergraduate courses on numerical methods and numerical analysis found in engineering, mathematics & computer science departments. Each chapter uses introductory problems from specific applications. These easy-to-understand problems clarify for the reader the need for a particular mathematical technique. Numerical techniques are explained with an emphasis on why they work. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
UbD Elementary Algebra Unit Unit Cover Page Unit Title: Operations With Polynomials Grade Level: 10 Subject/Topic: Math Tech 2 Key Words: Monomial Binomial Trinomial Polynomial Standard form Degree of a monomial Degree of a polynomial FOIL pattern Designed by: Sheila Pelton, Dell Miller Time Frame: Approx. 9 days(blocks) School District: Newberry School: MCHS & NHS Brief Summary of Unit (including curricular context and unit goals): This is an introduction so it is essential to emphasize a concrete, descriptive approach. This unit is designed to give students an understanding of operations on polynomials. Unit Design Status: In Progress Completed template pages – Stages 1, 2, and 3 Completed blueprint for each performance task Completed rubrics Directions to students Materials and resources listed Suggested accommodations Suggested Extensions Status: Initial Draft (date: ) Revised draft (date: ) Peer Reviewed Content reviewed Field Tested Validated Anchored EA-1 Stage 1- Identify Desired Results Established Goals: Standard EA-2: The student will demonstrate through the mathematical processes an understanding of the real number system and operations involving exponents, matrices, and algebraic expressions. EA-2.7 Carry out a procedure (including addition, subtraction, multiplication, and division by a monomial) to simplify polynomial expressions. What essential questions will be considered? What understandings are desired? Students will know… Students will understand that…  Why is it necessary or desirable to simplify the objective of this indicator is to carry out a procedure polynomial expressions? (EA-2.7) (including addition, subtraction, multiplication, and division by a monomial) to simplify polynomial expressions. Therefore, the primary focus of the assessment should be for students to carry out such procedures. What key knowledge and skills will students acquire as a result of this unit? Students will know…. Students will be able to… EA-2.7  key terms: coefficient, constant, variable,  carry out a procedure to add polynomials. exponent/power, distributive property, like  carry out a procedure to subtract terms, integer. polynomials.  how to apply the distributive property.  carry out a procedure to multiply  how to identify like terms. polynomials.  laws of exponents needed when  carry out a procedure to multiply two multiplying or dividing. binomials using FOIL.  carry out a procedure to divide a polynomial by a monomial. EA-2 Stage 2- Determine Acceptable Evidence What evidence will show that students understand? Performance Tasks:  Create a poster demonstrating the FOIL process. Students will need to indicate on the poster a clear understanding of what each letter of FOIL represents. Students will be given a scoring rubric for this activity.  Students will create a review booklet demonstrating the following: classifying polynomials; adding, subtracting, and multiplying polynomials; and dividing polynomials by a monomial. What other evidence needs to be collected in light of Stage 1 Desired Results? Other Evidence (e.g., tests, quizzes, prompts, work samples, observations)  Check homework daily  Mini quizzes – operations with polynomials  Teacher observation  Informal Q&A  Test  Jeopardy review game Student Self-Assessment and Reflection:  Self assessment of poster on FOIL  Self assessment of review booklet  Daily journal entry during the last five minutes of class. EA-3 Stage 2- Determine Acceptable Evidence (continued) Assessment Task Blueprint What understandings or goals will be assessed through this task?  Students will add, subtract, and multiply polynomials.  Students will divide a polynomial by a monomial. What criteria are implied in the standards and What qualities must student work demonstrate understandings regardless of the task specifics? to signify that standards were met?  Identification of like terms  Laws of exponents needed for multiplying and dividing with polynomials Through what authentic performance task will students demonstrate understanding? Task Overview: In creating their posters, students will demonstrate an understanding of the FOIL process used for multiplying two binomials. In creating their review booklets, students will demonstrate an understanding of operations with polynomials. What student products and performances will provide evidence of desired understandings?  Posters will represent an understanding of Final performance task (test) will show that students the FOIL process. understand operations with polynomials.  The review booklet will show a clear understanding of the assessment expectations. By what criteria will student products and performances be evaluated? A rubric will be developed for each assignment. Points should be assigned to each of the criteria on the handout to determine how the grade will be calculated. EA-4 Stage 3- Plan Learning Experiences WHERETO What sequence of teaching and learning experiences will equip students to engage with, develop, and demonstrate the desired understandings? Use the following sheet to list the key teaching and learning activities in sequence. Code each entry with the appropriate initials of the WHERETO elements 1. Begin with an entry question (Ask students to identify the number of terms in various given polynomials. Allow them to self-check and compare solutions and discuss why they got the answers they got.) to hook students into the lesson. Allow students to share what they know without passing judgment or providing additional information at this time. W, H, O ( Protection From Adverse Consequences, Novelty and Variety, Organization of Knowledge) 2. Introduce the lesson on addition and subtraction of polynomials. Work examples for the students. Be sure to include key terminology. E1, O (Content and Substance, Organization of Knowledge) 3. Practice working problems at their seats, allowing the teacher to monitor for understanding by observation. R, E2 (Protection From Adverse Consequences for Initial Failure, Affiliation) 4. Give a quiz on addition and subtraction. Review the quiz when it is returned to the students. E2 (Clear and Compelling Product Standards, Product Focus, Authenticity) 5. Review multiplication and the distributive property, along with the laws of exponents needed for multiplying. This would introduce the lesson on multiplying polynomials. Focus on the FOIL process for multiplying two binomials. E1, O (Content and Substance, Organization of Knowledge) 6. Assign the poster activity on the FOIL process. Be sure to closely cover the scoring rubric with the students. A model poster to demonstrate what you're looking for is desirable. R, O, T (Clear and Compelling Product Standards, Product Focus, Affiliation, Choice, Novelty and Variety, Choice) 7. Give a quiz on multiplying polynomials. Review the quiz when it is returned to the students. E2 (Clear and Compelling Product Standards, Product Focus, Authenticity) 8. Review the laws of exponents for dividing. This is the springboard for the lesson on dividing a polynomial by a monomial. E1, O (Content and Substance, Organization of Knowledge) 9. Create the review booklet on operations with polynomials. Be sure all students clearly understand the scoring rubric. R, O, T (Clear and Compelling Product Standards, Product Focus, Authenticity, Choice) 10. Review for a test on operations with polynomials by playing the Jeopardy Review Game. R, T, E2 (Novelty and Variety, Affiliation, Authenticity) 11. Test on operations with polynomials. E2 (Clear and Compelling Product Standards, Content and Substance, Affirmation of Performance) EA-5 Monday Tuesday Wednesday Thursday Friday Day 1 – Day 2- Activities Day 3- Activity5 Day 4 Activity 5 Day 5 – Activities 1-3 3 & 4. E1, O E1, O Activities 5 & 6 W, H, O, E1, R, R, E2 E1, O, R, T E2 Day 6 – Day 7 – Day 8 – Day 9 – Activity Consider the WHERETO elements. Activities 7 & 8 Activities 8 & 9 Activities 9 & 10 11 E2, E1, O E1, O, R, T R, O, T, E2 E2 Stage 3- Plan Learning Experiences (continued) EA
Just the facts (and figures) to understanding algebra. The Complete Idiot's Guide® to Algebra has been updated to include easier-to-read graphs and additional practice problems. It covers variations of standard problems that will assist students with their algebra courses, along with all the basic concepts, including linear equations and inequalities, polynomials, exponents and logarithms, conic sections, discrete math, word problems and more. —Written in an easy-to-comprehend style to make math concepts approachable —Award-winning math teacher and author of The Complete Idiot's Guide® to Calculus and the bestselling advanced placement book in ARCO's "Master" series
How much math is involved in economics? At my school, to fulfill a economics major, you only need to take 1 actual Stats class, and two Calculus classes. I was just wondering outside of those classes, how much does math take part in learning economics? I'm a very "B" student at math, and I was wondering how much you'll see it being used. It all depends. If you want to major in economics with a focus on getting an overview, then basic calculus is fine. But if you are aiming to deeply understand economics to the level where you can go to a good graduate school, it usually requires multivariable calculus, differential equations, linear algebra, real analysis, etc. For alot of the lower level courses I would say it is more a question of are you comfortable with numbers, graphs, etc? Some subjects can get VERY math intensive (game theory) but not usually so for a BA/BS degree. For grad school, the bare minimum is calc through multivariable, linear algebra, stats/prob. I haven't encountered anything too daunting with the math yet in my undergrad, but micro theory had a decent amount of taking derivatives with multiple variables (nothing too hard, though). If you are looking to do grad economics...might as well get an undergrad major in math and just take econ up through intermediate and econometrics. I would say to acquire a deep understanding though, you need to be able to handle math. For instance you might be able to just use a short-hand method on a test (equate slopes) but the most accurate method would be something you learn in multivariable (lagrangians). But to get a BA, calculus 2 and stats should suffice (and doing some reading up on partial derivatives or whatever else you encounter) I'm an Economics major at Berkeley right now (BA). I would say that if you end up in my case, you don't need to love math, "B" student is fine if you're talking Calculus and Statistics, because that's all I needed for my Econ requirements. I don't really like math although I did well in it in high school. Just to give you a little bit of context as to where I'm coming from. I think Economics is really more about theory than it is about math. Sure, there is a little bit of calculus along the way, but you're not really doing calculus the way you do it in a math class. It actually has meaning behind it.. maybe you're calculating the elasticity of a good, for example. I find that this kind of attached meaning makes the math much easier. As far as Econometrics, yes, I had a hard time with it.. but it wasn't impossible, and really it was more about knowing and memorizing than it was about statistics. Plus, Econometrics is only one class out of many that you will take as an Economics major. It'll probably be your hardest class, but after that.. at least, at Berkeley, I get to choose. I can take Economics classes that are more history-based than model-based, and I can take some that are more model-based than history, or I can take some that are more math-based. I feel that whatever math there is in the Economics classes I have taken so far is based on models, and therefore, once you get it, you pretty much have it. It's not like pure math where every time you get a problem it's something new. If you want to have an easier time with it, I would say.. understand the math behind it. You can definitely get by without understanding it completely, but I think it is much more rewarding/interesting to understand why graphs look the way they look, etc, and it makes it much easier to learn because you're not memorizing how things look, you actually know what makes it. Oh also, I should add, if you are planning on doing a graduate degree in Economics, you definitely need to be strong in math. However, I would recommend that you don't go that route unless you love Econ and math/stats. for a BA in economics, depending on the school, you probably just need to take a few statistics courses, a few calculus course (probably at most multi-variable) .... so yeah econ undergrad doesn't require you to be EXCELLENT at math.... BUT if you want to go to grad school for economics.. you have to not only LOVE math, you have to be GOOD at it because it requires A LOT of math. some people even double major in math but imo thats overdoing it. From experience, if you're a math oriented person, then you should do fine, if not great on econ. I think that for someone who is horrible at or totally disinterested in math, econ would likely turn out to be a struggle.
Prealgebra, 5th Edition Prealgebra, 5/e, is a consumable worktext that helps students make the transition from the concrete world of arithmetic to the symbolic world of algebra. The Aufmann team achieves this by introducing variables in Chapter 1 and integrating them throughout the text. This text's strength lies in the Aufmann Interactive Method, which enables students to work with math concepts as they're being introduced. Each set of matched-pair examples is organized around an objective and includes a worked example and a You Try It example for students. In addition, the program emphasizes AMATYC standards, with a special focus on real-sourced data. The Fifth Edition incorporates the hallmarks that make Aufmann developmental texts ideal for students and instructors: an interactive approach in an objective-based framework; a clear writing style; and an emphasis on problem solving strategies, offering guided learning for both lecture-based and self-paced courses. The authors introduce two new exercises designed to foster conceptual understanding: Interactive Exercises and Think About It exercises242.95 Purchase Options Paperback $193.99 $193.99 Save $4848.49 from$48.49 Save up to $194.46! Rent thru 05/06/14 for $48.49 $48.49 Save $194.46! Rent thru 06/05/14 for $54.49 $54.49 Save $188.46! Rent thru 07/15/14 for $60.49 $60.49 Save $182.46! Rent thru 12/02/14 for $69.49 $69.49 Save $173.46! Rent thru 08/29/15 for $75.49 $75.49 Save $167 return60.49 from$60.49 Save up to $182.46! Rent thru 09/03/14 for $60.49 $60.49 Save $182.46! Rent thru 03/02/15 for $69.49 $69.49 Save $173.46! Rent thru 02/25/16 for $75.49 $75.49 Save $167.46! Rent thru 02/14/18 for $84.49 $84.49 Save $158
The Math Help Center has been established to help facilitate student learning in mathematics and statistics at undergraduate level. Tutoring is one of the main components of the Math Help Center. Students are encouraged to utilize this free service to improve their understanding of mathematics and statistics. They will also find a variety of resources created and gathered (check often to see more new materials) in this page to help them find related materials available on the web. Math Help Resources . . . The Algebra Reviewwill be beneficial to anyone who is planning to study mathematics or other most other disciplines at the university level and needs to refresh or learn basic algebra skills. Individual tutoring for many math and statistics courses is available at several locations in the Petty building. The specific locations and times can be found in the Tutoring Schedule. The Algebra Review in TEN Lessonsfrom the University of Akron is a review of Algebra in 10 lessons featuring verbose discussion of topics, typeset quality mathematics, user interactivity in the form of multiple choice quizzes, in-line examples and exercises with complete solutions, and pop-up graphics. Math Abundance is a general math tutorial covering a variety of math topics from algebra and trigonometry to Calculus. ECalculusfrom the University of Akron is a review of Calculus featuring verbose discussion of topics, typeset quality mathematics, user interactivity in the form of multiple choice quizzes, in-line examples and exercises with complete solutions, and pop-up graphics.
mo... read more Customers who bought this book also bought: Our Editors also recommend: An Introduction to the Calculus of Variations by L.A. Pars Clear, rigorous introductory treatment covers applications to geometry, dynamics, and physics. It focuses upon problems with one independent variable, connecting abstract theory with its use in concrete problems. 1962 edition. Calculus of Variations by Lev D. Elsgolc This text offers an introduction to the fundamentals and standard methods of the calculus of variations, covering fixed and movable boundaries, plus solutions of variational problems. 1961 edition. Two-Dimensional Calculus by Robert Osserman Two-dimensional calculus is vital to the mastery of the broader field, and this text presents an extensive treatment. Advantages include the thorough integration of linear algebra and development of geometric intuition. 1986 edition. An Introduction to the Calculus of Variations by Charles Fox Highly regarded text for advanced undergraduate and graduate students explores first and second variations of an integral, generalizations, isoperimetrical problems, least action, special relativity, elasticity, more. 1963 edition. Technical Calculus with Analytic Geometry by Judith L. Gersting Well-conceived text with many special features covers functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, much more. Many examples, exercises, practice problems, with answers. Advanced undergraduate/graduate-level. 1984 edition. Calculus: An Intuitive and Physical Approach (Second Edition) by Morris Kline Application-oriented introduction relates the subject as closely as possible to science with explorations of the derivative; differentiation and integration of the powers of x; theorems on differentiation, antidifferentiation; the chain rule; trigonometric functions; more. Examples. 1967 edition. The Variational Principles of Mechanics by Cornelius Lanczos Philosophic, less formalistic approach to analytical mechanics offers model of clear, scholarly exposition at graduate level with coverage of basics, calculus of variations, principle of virtual work, equations of motion, more. Product Description: modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need only study the first chapter. Students wishing a more extensive treatment, however, will find the first six chapters comprise a complete university-level course in the subject, including the theory of fields and sufficient conditions for weak and strong extrema. Chapter 7 considers the application of variational methods to the study of systems with infinite degrees of freedom, and Chapter 8 deals with direct methods in the calculus of variations. The problems following each chapter were made specially for this English-language edition, and many of them comment further on corresponding parts of the text. Two appendices and suggestions for supplementary reading round out the text. Substantially revised and corrected by the translator, this inexpensive new edition will be welcomed by advanced undergraduate and graduate students of mathematics and physics
Linear Algebra This course is a study of the basic concepts of linear algebra, and includes an emphasis of developing techniques for proving theorems. Students will explore systems of linear equations, matrices, vector spaces, bases, dimension, linear transformation, determinants, eigenvalues, change of basis, and matrix representations of linear transformations.
Studying for Midterms For midterms, we suggest studying by topic. You should have an idea of which topics will be covered on your midterm; clicking on the corresponding link below (under Resources by Topic) will bring you to a list of practice problems related to that topic. If you feel that certain topics are missing, please sign in to the wiki and leave a comment on the discussion tab of this page. To go straight to resources specifically for your course, see the links below. Resources by Topic Caution is needed when working through problems on the same topic from a different course. While problems are often applicable cross-courses, sometimes questions from a particular course require a technique that was not covered in another course, even though both courses covered the same general topic.
Class Schedule MAT130 Section D - 2014 Spring Term MAT130: Introduction to Algebra - 4 Hours This course is designed for students with one year or less of credit in high school algebra or for students needing a review of algebra. Topics to be covered include a review of calculator operations, real number properties and operations, operations with polynomials, linear equations and inequalities, solution of linear systems by elimination and/or substitution, second-degree polynomial factoring, exponents, roots, radicals, and graphing lines.
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Description Multiply your chances of success on the ACT Math Test The ACT Mathematics Test is a 60-question, 60-minute subtest designed to measure the mathematical skills students have typically acquired in courses taken by the end of 11th grade, and is generally considered to be the most challenging section of the ACT. ACT Math For Dummies is an approachable, easy-to-follow study guide specific to the Math section, complete with practice problems and strategies to help you prepare for exam day. Review chapters for algebra, geometry, and trigonometry Three practice tests modeled from questions off the most recent ACT tests Packed with tips, useful information, and strategies ACT Math For Dummies is your one-stop guide to learn, review, and practice for the test Zegarelli is the author of Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies (both from Wiley), and five other books on basic math, calculus, and logic. He holds degrees in both English and math from Rutgers University and is a math tutor and teacher
for Photonics Education has developed program planning and course materials to support education and training for future and current photonics technicians. Classroom materials include Scientific Notification, Unit Conversion, Introductory Algebra, Introductory Geometry, Introductory Trigonometry, Exponents, Logarithms, and Graphing. Visitors can request evaluation copies of any of these materials or purchase them via the contact information given.Mon, 22 Oct 2012 10:56:35 -0500Aviation Maintenance Technician Handbook document from SpaceTEC National Aerospace Technical Education Center is one of three handbooks for people preparing for mechanic certification with airframe or powerplant ratings, or both. This particular volume contains information on mathematics, aircraft drawings, weight and balance, aircraft materials, processes and tools, physics, electricity, inspection, ground operations, and FAA regulations governing the certification and work of maintenance technicians.Wed, 20 Jul 2011 12:22:00 -0500Mathematics Course document from SpaceTEC National Aerospace Technical Education Center presents a core readiness course which will serve to prepare individuals entering the aerospace field. The document is 55 pages and contains materials on both basic and advanced math topics such as whole numbers, fractions, decimals, roots, computing area, measurement systems, and functions of numbers.Mon, 18 Jul 2011 11:33:50 -0500Elementary Algebra by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course "is a study of the basic skills and concepts of elementary algebra, including language and operations on sets, operations on signed numbers, simple linear equations and inequalities in one variable, operations on polynomials (including beginning techniques of factoring), integer exponents, brief introduction to radicals, introduction to graphing, and applications." The course has seven chapters: Basic algebra principles; Linear equations and set theory; Inequalities & absolute values; Graphs of linear equations; Exponents, monomials, and polynomials; Factoring polynomials and solving quadratic equations; and Rational and radical expressions and equations. Each is broken into two or three lessons containing objectives, readings, multimedia components, and sample problems. The Topic View section of the site provides the concepts taught in the course in either alphabetical or sequential order for educators looking for more specific and targeted supporting materials for an introductory algebra classroom.Thu, 17 Jul 2008 03:00:02 -0500Introductory Algebra: Algebra 1B by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course follows up on a previous course, Algebra 1A, which "develops algebraic fluency by providing students with the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. In addition, the course develops proficiency with operations involving monomial and polynomial expressions." Along with providing a syllabus, the Course View section of the site is broken into three units: Exponents, monomials, and polynomials; Relations, functions, & quadratic equations; and Rational & radical expressions & equations. Each unit has five lessons, and each lesson has objectives, readings, multimedia components, assessments, and answers. Also, for instructors looking for more targeted teaching tools, the Topic View of the course presents both a sequential and alphabetical list of individual concepts covered in the course.Wed, 16 Jul 2008 03:00:03 -0500Mathematics and Student Life Skills course, designed for Miami Dade Community College, integrates arithmetic and beginning algebra for the undergraduate student. By applying math to real-life situations most students experience during college, the instructors attempt to make math both fun and applicable. The instructors specifically wish to dissipate the anxiety many college students feel when approaching math at an advanced level. Students can use the information provided on this website to help apply mathematical concepts to their own lives, while instructors can use the assignments, syllabus, and lecture notes to create their own relevant assignments in a mathematics course.Mon, 5 May 2008 03:00:13 -0500BasketMath Interactive by Science Academy Software, this site is a collection of math questions on subjects including basic arithmetic, order of operations, calculating perimeters and distance, exponents, and bar graphs. It is an excellent and fun math review for students of all ages, and a ready-to-use classroom exercise for educators.Wed, 9 Apr 2008 03:00:03 -0500Algebra Review in Ten Lessons University of Akron has created these excellent algebra tutorials that review some of the main topics in the discipline. There are ten lessons, which focus on topics like radicals and exponents, basic algebra, expansion, polynomials, functions, and trig curves. Each lesson has a table of contents and interactive resources like quizzes, in-line examples, and exercises. Words that appear in green or brown are hyperlinks; click on them to learn more about that topic. Tutorials are viewed as a PDF file, and users must have Acrobat Reader 3.0 or greater to access them. This is a perfect resource for anyone who needs to refresh their knowledge of basic algebra concepts, and is also great for those who are just learning about the subject.Wed, 9 Apr 2008 03:00:02 -0500College Algebra Online Tutorials introduction to this site remarks, "If you need help in college algebra, you have come to the right place." Their statement is accurate, as the staff members at the West Texas A&M University's Virtual Math Lab have done a fine job creating a series of online algebra tutorials for students and anyone else who might be returning to the world of algebra. First-time visitors should look at their online guide to the tutorials to learn how their tutorials are organized. After that, they should feel free to browse through any of the 59 tutorials offered here. Each tutorial contains information about learning objectives, full explanations, and numerous examples of how to correctly solve problems.Mon, 10 Dec 2007 03:00:01 -0600Understanding Algebra Algebra is a textbook written by James Brennan of Boise State University. The entire contents of the textbook are located on this site, and a PDF version is also available through the author's Website. Basic topics in algebra are covered in the nine chapters of this book, and examples are dispersed throughout those chapters to facilitate a greater understanding of the concepts. This textbook presents the foundational elements of algebra in a clear, thoughtful manner that everyone will be able to appreciate. Students in introductory algebra courses, or those looking to review basic concepts for a higher-level math course will find this resource extremely useful.Mon, 12 Nov 2007 03:00:03 -0600Intermediate Algebra Handouts by Professor Jody Harris at Broward Community College and created by Professor Levy, these handouts are an excellent resource to print and give to community and technical college students in the algebra classroom. The subjects of the handouts are: rectangular coordinate system, finding the equation of a line, functions, factoring, exponent rules, and quadratic equations. Each PDF explains, in graphic or equation form, the algebraic principle and most contain problems for students to complete to improve their algebra skills.Mon, 29 Oct 2007 03:00:01 -0500College Algebra Handouts by Professor Jody Harris at Broward Community College, these handouts are an excellent resource to print and give to community and technical college students in the algebra classroom. The subjects of the handouts include: quadratic equations; functions, domain, and range; inverse functions; compound interest; and exponential growth and decay. Each PDF explains, in graphic or equation form, the algebraic principle and some contain homework problems for students to complete to improve their algebra skills. [ASC]Mon, 22 Oct 2007 03:00:02 -0500Practical Algebra Lessons to you by Elizabeth Stapel and purplemath.com, this collection of learning modules contains over 100 mathematics modules designed to teach beginning, intermediate, and advanced algebra concepts. Some algebra topics include graphic linear equations; adding, subtracting, multiplying, and dividing polynomials; and solving linear and literal equations. Intermediate algebra topics include domain and range, even and odd functions, factor theorem, and solving systems of non-linear equations. Finally, advanced algebra topics include complex fractions, complex numbers, matrix addition and subtraction, and partial fraction decomposition. This is a great reference and teaching resource for teachers and students of introductory algebra courses. This is an especially good resource for teachers looking for in-class illustrations of fundamental algebra concepts.Fri, 19 Oct 2007 03:00:01 -0500algebasics the motto, "Show me how, now!" algebasics is a fine online mathematics instructional resource that takes young and old alike through the basics of algebra. The breadth of the material is divided into sixteen sections, which begin with, appropriately, "the basics", and proceed all the way to a section on applying algebra to real-world situations. Each section asks users to solve a number of problems so that they will gain mastery of each concept. The interface deployed here is quite user-friendly, as each problem is narrated so that users will better understand the process needed to complete each problem successfully. Overall, it's a well-designed introduction to this area of the mathematical universe, and one that is very easy to use.Sun, 4 Feb 2007 03:00:05 -0600Algebra: In Simplest Terms Annenberg Foundation has been an active part of creating educational and professional development tools and instructional aids for teachers for many years. To reach the broadest audience possible, their Annenberg Media website also contains many of their programs in their entirety. Math educators and aspiring students of algebra will be delighted to learn that their popular series "Algebra: In Simplest Terms" can be found on their site. All told, there are a total of 26 episodes in the series, and they deal with such topics as linear relations, the ellipse, rational functions, and geometric sequences. Narrated by Sol Garfunkel, the series uses practical examples as illustrations of various principles and axioms. It should be noted that to watch the videos, visitors will need to fill out a free online registration form.Tue, 16 Jan 2007 03:00:02 -0600
of a Single Variable This text combines the theoretical instruction of calculus with current best-practise strategies.This text combines the theoretical instruction of calculus with current best-practise strategies
Cambridge University Press Subject: Maths ISBN-10: 0521136377 Educational Level: High School ISBN-13: 9780521136372 Language: English Product Type: Teacher's Edition Publication Year: 2009 Format: Paperback Detailed item info Description Part of a complete course in mathematical methods for senior level students. Key Features Author(s) Josian Astruc, Michael Evans, Neil Cracknell, Peter Jones, Sue Avery Publisher Cambridge University Press Date of Publication 10/11/2009 Language English Format Mixed media product ISBN-10 0521136377 ISBN-13 9780521136372 Subject School Textbooks & Study Guides: Maths, Science & Technical Series Title Essential Mathematics Publication Data Place of Publication Cambridge Country of Publication United Kingdom Imprint Cambridge University Press Out-of-print date 21/02/2013 Dimensions Weight 2000 g Editorial Details Contained items statement Contains Paperback and CD-ROM Edition Statement 3rd Student Manual/Study Guide Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved. Third Edition Looks 98% New Inside still perfect No CD TI-Nspire CAS and Casio ClassPad version
PrefaceThis workbook contains selected mathematical topics that are widely used in ar-chitecture and civil engineering freshmen courses. The topic selection is based onthe "Fuse LAB" project supported by NSF-ATE grant under the direction of PIShelley Smith. Each chapter represents a module to be covered within ARCH 3590Computation Fabrication course and few civil engineering courses. The module alsomake use of computer algebra system Mathematica and provides Mathematica codeswhen necessary. 1 2 Chapter 1Mathematical Modeling andFunctions Module1.1 IntroductionThe process of translating a real-world problem into a usable mathematical equationis called mathematical modeling, and the equation is referred to as a model. Weuse mathematical models to describe numerical data or verbal information.Definition 1.1.1 A function is a rule that takes certain numbers as inputs andassigns to each a definite output number. The set of all input numbers is called thedomain of the function and the set of resulting output numbers is called the rangeof the function.Representation of Functions Functions can be represented by tables, graphs,formulas, and descriptions in words. Example below shows the use of each repre-sentations.Example: A civil engineer is planning to dig a tunnel through a mountain. Thetunnel will begin 575 feet above the sea level and will be constructed with a constantupward slope of 5%; that is, the tunnel will rise vertically 5 feet for every 100 feet ofhorizontal distance. Table 1.1 shows the amount of vertical rise of several horizontaldistances [1].(a) Use the verbal description and the above table to write a model for the elevation of the tunnel in herms of the horizontal distance from where the tunnel begins at the base of the mountain. 3 Horizontal increase (feet) Vertical increase (feet) 100 5 200 10 300 15 400 20 500 25 Table 1.1: Vertical rise vs. horizontal distance(b) Find the elevation of the tunnel at a horizontal distance of 2500 feet from the starting point.(c) If the tunnel exits the mountain at a horizontal distance of 7000 feet from where it began, what is the elevation of the tunnel when it emerges from the mountain?(d) If the tunnel will cost $120 per foot to construct, what will be the cost of building the tunnel?Solution:(a) The elevation of the tunnel starts at 575 feet and rises 5 feet for every 100 horizontal feet. We can write this as Elevation of tunnel = 575 + 5 feet for every 100 horizontal feet Let y be the elevation and x be the horizontal feet in hundreds, then the above statement can be converted into y = 575 + 5x feet above sea level This function is a model for the elevation of the tunnel. Figure 1.1 is the graphical representation of the model.(b) Substitute x = 25 into the equation above, y = 575 + 5 ∗ 25 = 700 feet above sea level.(c) Using the same substitution method in (b), for x = 70 we have y = 575+5∗70 = 925 feet above sea level.(d) Using the Pythagorean theorem for the right triangle in Figure 1.2, we obtain √ d = 70002 + 3502 = 7008.74 ≈ 7009 feet 4 1.2 Library of FunctionsIn this section, we build a collection of widely used functions and study their prop-erties.1.2.1 Linear FunctionsLinear functions have a constant rate of increase or decrease. A function is linearif its slope, or rate of change, is the same everywhere. So, linear functions can beidentified as functions whose outputs are repeated additions. A linear function hasthe form y = f (x) = b + mx.Its graph is a straight line such that m is the slope (rate of change of y with respectto x), b is the vertical intercept. Note that if the slope is zero, m = 0, we have y = b, a horizontal line. Theequation of a line of slope m through a point (x0 , y0 ) is y − y0 = m(x − x0 ).The tunnel problem in section 1.1 is a linear model.1.2.2 Exponential FunctionsExponential function has a constant percentage change, that is to say it is a func-tion whose output is the result of repeated multiplication by a constant at regularintervals. An exponential function has the form y = f (x) = abxwhere a = 0 and b > 0. If b > 1, f (x) is exponential growth and If 0 < b < 1, f (x)is exponential decay as shown in Figure 1.3, which is plotted by computer algebrasystem MathematicaPlot[Exp[x], {x, -3, 3}, PlotRange -> {-1, 8}, AxesLabel -> {"x", "y"},PlotStyle -> Thick]1.2.3 Logarithmic FunctionsA logarithmic (log) function has a vertical asymptote (the line x = 0) and continuesto grow or decline as x becomes large. A log function has the form y = f (x) = a + b ln x 6 y y 8 8 x x y y 6 6 4 4 2 2 x x3 2 1 1 2 3 3 2 1 1 2 3 Figure 1.3: Exponential growth and decay functionswhere b = 0. If b > 0, f (x) is a growth and If b < 0, f (x) is decay function as shownin Figure 1.4.A note on inverse relationship:If we have data whose input/output relationship can be modeled by an exponentialfunction, then the inverse (output/input) relationship can be modeled by a logfunction, and vice versa. This relationship in its simplest form can be stated: If f (x) = ln x and g(x) = ex , then f (g(x)) = ln (ex ) = x and g(f (x)) = eln x =x as long as x is positive.The inverse relationship for the model f (x) = a + b ln x becomes f −1 (x) = AB xwhere A = e−a/b and B = e1/b .1.2.4 Logistic FunctionsSometimes it is unrealistic to believe that exponential growth can continue forever.The constrains that slows down the growth results S-shaped behavior. A mathe-matical function with such an S-shaped curve is called logistic function and it hasthe form L y = f (x) = 1 + Ae−Bxwhere L is the limiting value of the function. If B > 0, f (x) is a growth and IfB < 0, f (x) is decay function as shown in Figure 1.5. Logistic functions have twotypes of curvatures, concave up and concave down. The point at which the graphof f (x) changes its concavity is called an inflection point. 7 y y 5 1 2 y x 4 x 3 2 1 1 2 3 3 1 2 2 1 3 x 4 y x23 2 1 1 2 3 1 5 Figure 1.6: Concave up and concave down quadratic functions 1.2.6 Cubic Functions When the third differences are constant, the data can be modeled by the cubic function and it has the form y = f (x) = ax3 + bx2 + cx + d as long as a = 0. If the scatter plot of a set of data fails to exhibit an inflection point, then it is not appropriate to fit a cubic equation to the data. Figure 1.7 and Figure 1.8 show the cases when the leading coefficient a positive and negative respectively. 1.2.7 Trigonometric Functions Definition 1.2.1 Functions that repeat their values at regular intervals are called periodic. Periodic functions repeat exactly the same cycle forever. Definition 1.2.2 For any periodic function of time: • The amplitude is the half the difference between its maximum and minimum values. • The period is the time for the function to execute one complete cycle. 9 y y 2 2 y=sin(x) y=cos(x) 1 1 x x 5 5 5 5 1 1 2 2 Figure 1.9: Sine and cosine functionsDefinition 1.3.1 Average rate of change of a function f (x) between x = a andx = b is given by ∆y f (b) − f (a) = . ∆x b−aDefinition 1.3.2 The instantaneous rate of change of a function f (x) at pointx = a is defined to be the instantaneous rates of change of f (x) at x = a and denotedby f (a).Visualizing average rate of change and instantaneous rate of change: Fig-ure 1.10 shows the average rate of change of a function represented by the slope ofthe secant line joining points A and B. Instantaneous rate of change is found bytaking the average rate of change over smaller and smaller intervals.Definition 1.3.3 The derivative of a function f (x) at point x = a is defined tobe the limit of the average rates of change of f (x) over shorter and shorter intervalsaround a. Then the instantaneous rate of change of f is the slope of the tangentline at x = a.Some useful derivative rules are given in Table 1.2.Example: One tower is 50 ft high and another tower is 30 ft high. The towers are150 ft apart. A guy wire is to run from Point A to the top of each tower. LocatePoint A so that the total length of guy wire is minimal [2]. 11 Figure 1.10: Average rate of change vs. instantaneous rate of change [2] Figure 1.11: Length of a guy wireSolution: Let x be the distance between point A and 30 ft high tower. Then thelength of guy wire is the sum of the hypothenuses of two triangles in Figure 1.11, √L(x) = 900 + x2 + 2500 + (150 − x)2 for 0 ≤ x ≤ 150. The graph of the thelength function L(x) is given in Figure 1.12. We set L (x) = 0 and solve for x: x (150 − x) L (x) = √ − =0 900 + x2 2500 + (150 − x)2. Let θA be the angle at point A facing to the 30ft high tower and θB be the angle atpoint A facing to the 50ft high tower. Since √900+x2 = cos θA and √ (150−x) 2 = x 2500+(150−x)cos θB , then L (x) = 0 when cos θA = cos θB , or θA = θB . Since two triangles are 13 y 210 200 190 180 170 x 0 20 40 60 80 100 120 140 Figure 1.12: Length of a guy wiresimilar, x 150 − x 225 = =⇒ x = = 56.25 feet. 30 50 41.4 Regressions and InterpolationsIn section 1.2, we discussed several types of models/functions by building a libraryof functions. It is important that we choose the right model/function for our dataset. Here are some guidelines for determining which model to use: Examine the scatter plot of the data 1. If the scatter points appear to lie in a straight line, use linear function. 2. If the scatter points curved with no inflection point, try a quadratic, an expo- nential, or a logarithmic functions. 3. If the scatter plot appears to have an inflection point, try a cubic and/or a logistic function. Check the end behavior of the plot to distinguish cubic from logistic. 4. If the scatter plot appears to be periodic, try trigonometric functions. 5. If above trials fail, try combining two or more functions. 14 the exponential distribution of the loads on the cantilever beam. Below is theMathematica code for the exponential case:expdata = {{0, 12}, {2, 48}, {4, 192}, {6, 768}};exp = FindFit[expdata, a*b^ x, {a, b}, x] Quadratic: The first differences and the second differences are given in Table1.6. The constant second differences imply a quadratic model. It is of the formy = ax2 + bx + c. x y First differences Second differences 1 2 − − 2 3 3−2=1 − 3 6 6−3=3 3−1=2 4 11 11 − 6 = 5 5−3=2 5 18 18 − 11 = 7 7−5=2 Table 1.6: First and second differences We substitute the first three points (1, 2), (2, 3), (3, 6) into the equation y = 2ax + bx + c. We obtain the 3 × 3 linear system. 2 = a+b+c 3 = 4z + 2b + c 6 = 9a + 3b + cFrom the first two, we eliminate c and obtain 3a + b = 1. From the last two, weeliminate c and obtain 5a + b = 3. Now we have a 2 × 2 system to solve: 3a + b = 1 5a + b = 3Solution of the 2 × 2 system is a = 1 and b = −2. Substituting a and b into the firstequation, we receive c = 3. Thus the quadratic model is y = x2 − 2x + 3. Even intho simple example we had to solve a 3 × 3 system which may be tedious. Also useof computer algebra system is recommended for the data does not exactly fit thedata. Below is the Mathematica code for the quadratic case:quaddata = {{1, 2}, {2, 3}, {3, 6}, {4, 11}};parabola = Fit[quaddata, {1, x, x^ 2}, x] Now, one can interpolate, by asking "what is the linear load on the beam atlocation x = 1?" which is not listed in the tabular data. Or we can extrapolate byasking "what is the quadratic load at the location x = 0?" which is out side therange of the data set. These questions can be answered by simply substituting thex values into the corresponding models. 16 1.5 OptimizationMost of the real-world problems are given as data sets as it is presented in the pre-vious section. Then after finding appropriate model for the data (using regression),we use the methods of calculus to determine the largest (maximum) or smallest(minimum) value of the model/function.Definition 1.5.1 Let f be a function defined on an interval I that contains thenumber c. Then • f (c) is the absolute maximum of f on I if f (c) ≥ f (x) for all x in I • f (c) is the absolute minimum of f on I if f (c) ≤ f (x) for all x in IBoth absolute minima and absolute maxima are called absolute extrema .How to find absolute extrema:In order to be able to find the absolute extrema of a continuous functions f on aclosed interval a ≤ x ≤ b: • Find all critical numbers, c, of f (f (c) = 0 or undefined) in the open interval (a, b). • Compute f (x) at the critical numbers found in the previous step. • Compute f (x) at the end points of the interval a and b. • The largest and smallest values found in the previous two steps are the absolute maximum and absolute minimum values of f on the closed interval a ≤ x ≤ b respectively.The second derivative test for absolute extrema:Suppose that f (x) is continuous on an interval I where x = c is the only criticalnumber and that f (c) = 0. Then, • if f (c) > 0, the absolute minimum of f (x) on I is f (c), • if f (c) < 0, the absolute maximum of f (x) on I is f (c). 17 x y Picnic park y Highway Figure 1.13: Rectangular picnic parkExample: The highway department is planning to build a picnic park for motoristalong a major highway. The park is to be rectangular with an area of 5,000 squareyards and is to be fenced off on the three sides not adjacent to the highway shownin Figure 1.13. What is the least amount of fencing required for this job? How longand wide should the park be for the fencing to be minimized [3]?Solution: Since the park is to have area 5,000 square yards, we have xy = 5, 000.The length of the fencing is L = x + 2y, where x > 0 and y > 0 (otherwise we can 5, 000not have a picnic park area). Since xy = 5, 000 or y = , we can eliminate y xfrom the formula for L. Then L will be a function of x: 5, 000 10, 000 L(x) = x + 2y = x + 2 =x+ for x > 0 x xThe derivative of L(x) is 10, 000 L (x) = 1 − . x2 18 angle B and side c faces angle C. First, we review few laws/rules. A b c B C a Figure 1.15: Angles and sides of a triangle • The angles always add to 180◦ . When you know two angles you can find the third. • Law of sines: a b c = = sin A sin B sin C • Low of cosines a2 = b2 + c2 − 2bc cos (A) b2 = a2 + c2 − 2ac cos (B) c2 = a2 + b2 − 2ab cos (C) We have the following possible cases: 1. AAS (Angle, Angle, Side): This means that two angles and one side are known. Note that the known side is not between the angles. We use law of sine to find the other two sides. 2. ASA (Angle, Side, Angle): This means that two angles and one side are known. Note that the known side is between the angles. We use law of sine to find the other two sides. 20 3. SAS (Side, Angle, Side): This means that two sides and one angle are known. Note that the known angle is between the sides. We use law of cosine to find the third side, and law of sine for the other angles. 4. SSA (Side, Side, Angle): This means that two sides and one one are known. Note that the known angle is not between the sides. We use law of sine to find one of the other two angles (therefore the third angle is also known). Then use law of sine again to find the third side. 5. SSS (Side, Side, Side): This means that all three sides are known. We use law of cosine twice to find two of the angles (therefore the third angle is also known).Example: This is an example of SAS case. In this triangle, we know angle A = 49◦and sides b = 5, c = 7 as shown in Figure 1.16. Solve the triangle in Figure 1.16. C 5 a 49 B A 7 Figure 1.16: Angles and sides of a triangleSolution: We use law of cosines first to find side a: a2 = 52 + 72 − 2 · 5 · 7 · cos (49◦ ) a2 = 25 + 49 − 70 cos (49◦ ) a2 = 74 − 45.924 = 28.075 √ a = 28.075 a = 5.298 21 Now we use the the law of sines to find one of the other two angles: sin B sin A = b a sin B sin (49◦ ) = 5 5.298which gives sin B = 0.7122 then B = sin−1 (0.7122) = 45.4◦ . Thus the the thirdangle is C = 180◦ − 49◦ − 45.4◦ = 85.6◦ .1.7 Problems, Projects, and Activities 1. A rectangular-shaped garden has one side along the side of a house. The other three sides are to be enclosed with 60 feet of fencing. What is the largest possible area of such garden [2]? 2. A florist uses wire frames to support flower arrangements displayed at wed- dings. Each frame is constructed from a wire of length 9 feet that is cut into 6 pieces. Vertical edges of the frame consists of four of the pieces of wire that are each 12 inches long. One of the remaining pieces is bent into a square to form the base of the frame; the final piece is bent into a circle to form the top of the frame [2]. (a) How should the florist cut the wire of length 9 feet in order to minimize the combined area of the circular top and the square base of the frame? (b) Verify that the answer to part (a) minimizes the combined area. 3. You need to design a display booth for a company. Because the company generally must pay for the amount of square footage your booth requires, you want to limit the floor size to 300 square feet. The booth is to be 6 feet tall and three-sided, with the back of the booth a display board and the two sides of the booth made of gathered fabric. The display board for the back of the booth costs $30 per square foot. The fabric costs $2 per square foot and needs to be twice the length of the side to allow for gathering. Find the minimum cost of constructing a booth according to these specifications. What should be the dimensions of the booth? 4. (Project) A popular historical site in Missouri is the Gateway Arch. Designed by Eero Saarinen, it is located on the original riverfront town site of St. Louis and symbolizes the city's role as gateway to the West. The stainless steel Gateway Arch (also called the St. Louis Arch) is 630 feet (192 meters) high and has an equal span. 22 In honor of the 200th anniversary of the Louisiana Purchase, which made St.Louis a part of the United States, the city has commissioned an artist to designa work of art at the Jefferson National Expansion Memorial National HistoricSite. The artist plans to construct a hill beneath the Gateway Arch, locatedat the Historic Site, and hang strips of mylar from the arch to the hill so asto completely fill the space. The artist has asked for your help in determiningthe amount of mylar needed [1]. Figure 1.17: The Gateway Arch in St. Louis(a) If the hill is to be 30 feet tall at its highest point, find an equation for the height of the cross-section of the hill at its peak. Refer to Figure 1.17. 23 (b) Estimate the height of the arch in at least ten different places. Use the estimated heights to construct a model for the height of the arch. (c) Estimate the area between the arch and the hill. (d) The artist plans to use strips of mylar 60 inches wide. What is the minimum number of yards of mylar that the artist will need to purchase? (e) Repeat Task 4 for strips 30 inches wide. (f) If the 30-inch strips cost half as much as the 60-inch strips, is there any cost benefit to using one width instead of the other? If so, which width? Explain.5. A square sheet of cardboard 18 inch on a side is made into an open box (i.e., theres no top), by cutting squares of equal size out of each corner (see Figure 1.18) and folding up the sides along the dotted lines. Find the dimensions of the box with the maximum volume [2]. Figure 1.18: The rectangular box6. A cylindrical can is to hold 12 fluid liters (see Figure 1.19). Find the dimensions that will minimize the amount of material used in its construction, assuming that the thickness of the material is uniform [2].7. (Project) The state wants to build a new stretch of highway to link an existing bridge with a turnpike interchange, located 8 miles to the east and 8 miles to the south of the bridge. There is a 5-mile-wide stretch of marsh land adjacent to the bridge that must be crossed (see Figure 1.20). Given that the highway 24 r h Figure 1.19: Open diagram of cylindrical cancosts $10 million per mile to build over the marsh and only $7 million to buildover dry land, how far to the east of the bridge should the highway be whenit crosses out of the marsh [2]? Figure 1.20: Highway design 25 8. A showroom for a department store is to be rectangular with walls on three sides, 6-ft door openings on the two facing sides and a 10-ft door opening on the remaining wall. The showroom is to have 800 square feet of floor space. What dimensions will minimize the length of wall used [2]? 9. (Project) A Norman window has the outline of a semicircle on top of a rect- angle, as shown below. Suppose there is 8 + π feet of wood trim available. Discuss why a window designer might want to maximize the area of the win- dow. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window (see Figure 1.21) [2]. Figure 1.21: Norman window10. (Project) You are designing a cable-stayed bridge, illustrated in Figure 1.22. The objective of this problem is to estimate the optimal height of the tow- ers, using the simple idealization shown in Figure 1.23. Both the cable and column will be idealized as cylinders with uniform cross section. Here is the information you need to make the decision. ( • The force in the cable is Pcable = W/(2 sin θ), where W is the weight of the roadbed. 26 Figure 1.22: Cable-stayed bridge • The force in the column is Pcolumn = −W . • The cable will break if the force per unit area in the cable exceeds σ0cable . • The column will collapse if the magnitude of the force per unit area in the column exceeds σ0column . Figure 1.23: Animation of the cable-stayed bridge(a) Write down an expression for the height of the column in terms of distance d and the angle θ.(b) Write down an expression for the length of the cable in terms of d and the θ.(c) Find the minimum allowable cross sectional area and volume of the cable and column. 27 28 Chapter 2Discrete Mathematics Module2.1 IntroductionDiscrete mathematics is dealing with objects that can assume only distinct, sepa-rated values. The term "discrete mathematics" is used in contrast with "continuousmathematics". The objects that are considered are integers, propositions, sets, andrelations, which are all discrete. The concepts associated with these objects, theirproperties, and relationships among them are the content of this module. SinceDiscrete mathematics encompasses a very wide range of mathematical topics, thismodule presents some selected choice of materials based on the FUSE-Lab Project.2.2 Logic and Sets2.2.1 Logic and Truth TablesLogic is used to establish the validity of arguments. We are not concerned aboutwhat the argument is but interested providing rules so that the general form of theargument can be judged as sound or unsound. A proposition is a declarative statement which is either true or false, but notboth. Here are few examples of propositions: 1. This rose is red. 2. 5 < 21. 3. Pentagons have four vertices. 29 4. 3 + 1 = 5. Exclamations, questions and demands are not propositions since they cannotbe declared true or false. For example "don't be late", "how are you?" etc. Thetruth (T) or falsity (F) of a proposition is called truth value. Let p be a proposition then p (or∼ p or −p or ¬p) symbolizes the negation of ¯p. Table 2.1 shows the relationship between the truth values of p and the negationof p. ¯ p p ¯ T F F T Table 2.1: The truth table for the negation of a proposition A table which summarizes truth values of propositions is called a truth table. There are several alternative ways of stating the negation of a proposition. Ifwe consider the proposition "All roses are red", some examples of its negation are:"It is not the case that all roses are red", "Not all roses are red", "Some roses arenot red". Note that the proposition "No roses are red" is not the negation of "Allroses are red". Remember that to be the negation, the second statement must befalse in all circumstances that the first is true and vice versa. Let p and q be two propositions. The conjunction of p and q, denoted byp ∧ q, is the proposition "p and q". We say that the sentence p ∧ q is true if the twosentences p, q are both true, otherwise conjunction is false. Table 2.2 is the truthtable of conjunction. The sentence "2 + 2 = 4 and 2 + 3 = 5" is true. The sentence "2 + 2 = 4 and π is rational" is false. p q p∧q T T T T F F F T F F F F Table 2.2: The truth table for the conjunction of two propositions Let p and q be two propositions. The disjunction of p and q, denoted by p ∨ q,is the proposition "p or q". We say that the sentence p ∨ q is true when either or 30 both of its components are true and is false otherwise. Table 2.3 is the truth tableof disjunction. The sentence "2 + 2 = 2 or 1 + 3 = 5" is false. The sentence "2 + 2 = 4 or π is rational" is true. p q p∨q T T T T F T F T T F F F Table 2.3: The truth table for the disjunction of two propositions The exclusive disjunction of p and q is denoted by p⊕q. This compound propo-sition is true when exactly one (i.e. one or other, but not both) of its componentsis true. The truth table for p ⊕ q is given by Table 2.4 p q p⊕q T T F T F T F T T F F F Table 2.4: The truth table for the "exclusive or" of two propositions Let p and q be two propositions. The conditional statement p → q is theproposition "if p, then q". The sentence p → q is true if the sentence p is false or ifthe sentence q is true or both, and is false otherwise. It is convenient to realize that the sentence p → q is false precisely when thesentence p is true and the sentence q is false. To understand this, note that if wedraw a false conclusion from a true assumption, then our argument must be faulty.On the other hand, if our assumption is false or if our conclusion is true, then ourargument may still be acceptable. The truth table for p → q is given by Table 2.5 The sentence "if 2 + 2 = 4, then π is rational" is false. The sentence "if π is rational, then 2 + 2 = 4" is true. Let p and q be two propositions. The biconditional statement p ↔ q is theproposition "p if and only if q". The sentence p ↔ q is true if the two sentences p,q are both true or both false, and is false otherwise. The truth table for p ↔ q isgiven by Table 2.6 31 Solution:1. Note that the truth table below is built up in stages. The first two columns give the usual combinations of possible truth values of p and q. The third column gives, for each truth value of p, the truth value of p. When p is true, p is false and vice versa. The last column combines the truth values in columns 3 and 2 using the inclusive disjunction connective. The compound proposition p ∨ q is true when at least one of its two components is true. This is the case in row 1 (where q is true), row 3 ( p and q are both true) and row 4 ( p is true). In the second row, p and q are both false and hence p ∨ qis false. p q p p∨q T T F T T F F F F T T T F F T T2. Here we first obtain truth values for p and q by reversing the corresponding truth values of p and q respectively. Now p ∧ q is only true when both p and q are true, i.e. in row 4. In all other cases p ∧ q is false. p q p q p∧q T T F F F T F F T F F T T F F F F T T T3. The truth table of q → p: p q q q→p T T F T T F T T F T F T F F T F4. The truth table of p ↔ q: 33 p q p q p↔q T T F F T T F F T F F T T F F F F T T T2.3 Boolean Algebra2.3.1 IntroductionA Boolean algebra consists of a set B together with following three operations definedon that set: 1. a binary operation denoted by ⊕ referred to as the sum (or join); 2. a binary operation denoted by ∗ referred to as the product (or meet); 3. for any element b ∈ B, the element ¯ ∈ B where the operation ¯ is called the b b complement of b.The following axioms apply to the set B together with the operations ⊕, ∗, and¯ . A1. Distinct identity elements belonging to B exist for each of the binary opera- tions ⊕ and ∗ and we denote these by 0 and 1 respectively. Thus we have b⊕0 = 0⊕b=b b∗1 = 1∗b=b for all b ∈ B. A2. The operations ⊕ and ∗ are associative, that is (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ B. A3. The operations ⊕ and ∗ are commutative, that is a⊕b = b⊕a a∗b = b∗a for all a, b ∈ B. 34 A4. The operation ⊕ is distributive over ∗ and the operation ∗ is distributive over ⊕, that is a ⊕ (b ∗ c) = (a ⊕ b) ∗ (a ⊕ c) a ∗ (b ⊕ c) = (a ∗ b) ⊕ (a ∗ c) for all a, b, c ∈ B. A5. For all b ∈ B, b ⊕ ¯ = 1 and b ∗ ¯ = 0. b b Note that A5 may lead you to conclude that ¯ is the inverse of b. This is not btrue. Remember that, if b?1 is the inverse of b, then b ∗ b?1 gives the identity withrespect to the operation ∗. However, b ⊕ ¯ gives the identity with respect to ∗ and bb ∗ ¯ gives the identity with respect to ⊕, so that ¯ is not the inverse of b with respect b bto either operation. Note also that note that 0 and 1 are used here as symbols for the two identityelements and not for the numbers which they conventionally symbolize. We musttherefore be careful not to make assumptions which are true for the integers 0 and1 but not necessarily so for identities in general.Example: The simplest Boolean algebra (and also the one of most interest to com-puter scientists) consists of the set B = {0, 1} together with the binary operations⊕ and ∗ and complement operation¯defined by the following tables. ⊕ 0 1 ∗ 0 1 b ¯ b 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 0 (a) Join. (b) Meet. (c) Com- plemet.2.3.2 Properties of Boolean AlgebrasDuality: Given any proposition about a Boolean algebra, we define its dual to bethe proposition obtained by substituting ⊕ for ∗, ∗ for ⊕, 0 for 1, and 1 for 0.The principle of duality is that for any theorem about a Boolean algebra, thedual is also a theorem. Properties of Boolean algebras are listed as theorems below:Theorem 2.3.1 The identity elements 0 and 1 are unique. 35

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