Split (1)

train · ~1.24B rows (showing the first 1.67M)

text stringlengths 8 1.01M
Buy Used Textbook Buy New Textbook eTextbook 180 day subscription $89.99 More New and Used from Private Sellers Starting at $367Algebra for College Students Algebra for College Students Summary Algebra for College Studentsis typically used in a very comprehensive 1-semester Intermediate Algebra course serving as a This Algebra for College students text may also be used in a 1-semester, lower-level College Algebra course as a prerequisite to Precalculus. Author Biography Allen R. Angel received his AAS in Electrical Technology from New York City Community College. He then received his BS in Physics and his MS in Mathematics from SUNY at New Paltz, and he took additional graduate work at Rutgers University. He is Professor Emeritus at Monroe Community College in Rochester, New York where he served for many years as the chair of the Mathematics Department. He also served as the Assistant Director of the National Science Foundation Summer Institutes at Rutgers University from 1967—73. He served as the President of the New York State Mathematics Association of Two Year Colleges (NYSMATYC) and the Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). He is the recipient of many awards including a number of NISOD Excellence in Teaching Awards, NYSMATYC's Outstanding Contributions to Mathematics Education Award, and AMATYC's President Award. Allen enjoy tennis, worldwide travel, and visiting with his children and granddaughter. Table of Contents Preface To the Student Basic Concepts Study Skills for Success in Mathematics, and Using a Calculator Sets and Other Basic Concepts Properties of and Operations with Real Numbers Order of Operations Mid-Chapter Test: Sections 1.1 Exponents Scientific Notation Equations and Inequalities Solving Linear Equations Problem Solving and Using Formulas Applications of Algebra Mid-Chapter Test: Sections 2.1 Additional Application Problems Solving Linear Inequalities Solving Equations and Inequalities Containing Absolute Values Graphs and Functions Graphs Functions Linear Functions: Graphs and Applications The Slope-Intercept Form of a Linear Equation Mid-Chapter Test: Sections 3.1
This book is intended to introduce calculus through examples to prepare the reader for the diverse problems he/she will have to solve using derivatives and integrals. The derivative is introduced right at the beginning of the book as the slope of a tangent. The reader is then exposed to a cross-section of problems that may be solved by knowing how to calculate slopes of tangents. College students have to master techniques of diffrentiation within a semester. The early introduction of derivatives gives them ample time to practice these techniques. The integral is introduced as a limit of sums with its applications to physical problems in mind. A number of problems that may be reduced to limits of sums are then addressed. Techniques of integration are also addressed as are methods of approximating definite integrals. Limits, which are the foundations of calculus are treated rigorously and most of the statements used in the course are proved. Finally, detailed solutions, (not just numerical answers), to about half the problems in the book are given. In particular, the solutions to all "Test - your - self" exercises are given. These are problems the student should attempt to do in the indicated time to test his/her mastery of some fundamental topics. The purpose of this book is to help students in calculus1 get a good practice for the midterms and final exams during their school year in calculus1. All the finals and midterms are real exams (with little changes) from several Universities around America ( USA, Canada, Puerto Rico, Mexico). We believe to get a good grade in the midterms and final, the student should after reviewing his/her homework and notes pick some real midterms and final and do them. The book begins with a discussion of finding the equation of a curve that passes through a fixed number of points. This technique is called curve fitting. This concept is then generalized to finding the equation of a curve that passes through an infinite number of points. This concept is in turn generalized to finding the equation of a curve that passes through a given curve over a specific interval. Such an equation and its curve that can be found to pass through the given curve over a particular interval is called a Fourier series. A Fourier series consists of an infinite number of sine and cosine terms added together in an infinite series whose coefficients must be determined. Such an infinite series of terms can be made to approximate the curve of an elementary function over a particular interval arbitrarily close. Also, since the terms of the Fourier series are Trigonometric terms the curve of their sum namely the Fourier series is periodic with a definite period. This brings us to the concepts of the periodic function. The curve of such a function keeps repeating itself over the same intervals. Numerous examples are provided for finding the Fourier series of various elementary functions over given intervals. The following concepts are also discussed-functions in general form, composite functions, and odd and even functions. The Fourier series of each of these types of functions is also found. Next. The equations for the Fourier series and its coefients are generalized to the complex number system. This allows is to derive the Fourier transform. Everything is logically derived with all of the steps included. The book starts with the definition of the Laplace Transform and uses it to derive the Laplace Transforms of the elementary functions including; constant functions, polynomial functions, exponential functions, trigonometric functions, and hyperbolic functions. All steps in the derivations of the Laplace Transforms for these functions are included. The concept of the Inverse Laplace Transform is then logically developed from the concept of the Laplace Transform. Numerous examples are provided for finding the Laplace Transforms of the various types of elementary functions and finding their corresponding Inverse Laplace Transforms. The Product Rule is derived with all steps included. Also, the Laplace Transform of a derivative is derived with all steps included. Finally, the following types of differential equations and their initial value problems are solved using both conventional methods and the Laplace Transform method: Mathematical Methods for Partial Differential Equations is an introduction in the use of various mathematical methods needed for solving linear partial differential equations. The material is suitable for a two semester course in partial differential equations for mathematicians, engineers, physicists, chemistry and science majors and is suitable for upper level college undergraduates or beginning graduate students. There are four Appendices. The Appendix A contains units of measurements from the Syst&egraveme International d'Unit&egraves along with some selected physical constants. The Appendix B contains solutions to selected exercises. The Appendix C lists mathematicians whose research has contributed to the area of partial differential equations. The Appendix D contains a short listing of integrals. The text has numerous illustrative worked examples and over 340 exercises. The most difficult part of calculus for many students is the study of methods of integration and the solution of indefinite integrals. Differentiation of functions is, on the whole, not difficult. This book contains the worked solutions to 200 integrals and may be used as a supplement to any standard calculus text. An index to the integrands is provided in chapter 3, referring the reader directly to worked solutions in chapter 4. Although it is impossible to cover all cases, the integrals have been selected to cover a range of problems likely to be found in junior college and senior high school examination papers. Definite integrals are not included since the proper solution to a definite integral depends on the limits of integration and the presence of discontinuities. The integrands include algebraic, trigonometric, logarithmic and hyperbolic functions. This book may be used by both teachers and students. Teachers will be able to select examination problems of varying degrees of difficulty. When used by a student, an honest attempt should be made to solve the problem before consulting the index and worked solution. It is hoped the student will develop responsible habits in this regard.
Instructor: Michael Hvidsten Text: Geometry with Geometry Explorer, Phone: 933-7480 by Michael Hvidsten, McGraw-Hill (2005) Office Hours: 2:30-4:30 Mon,Thurs 2:30-4:00 Tues and by appointment Email: hvidsten@gac.edu Class Email Alias: s-mcs-303-001@gustavus.edu The study of geometry is an ancient and noble practice begun by Greek mathematician- philosophers. The most influential of these was Euclid, whose system of axioms and theorems has come to be known as Euclidean Geometry. In this course we will see that Euclid's axiomatic system is not the only logically consistent geometric system. We will investigate non-Euclidean geometries such as finite, hyperbolic, and elliptic geometries, as well as fractal geometry. Of course we will also review Euclidean geometry. Throughout the course we will focus on three different geometric viewpoints: the synthetic, analytic, and transformational. The synthetic perspective is to study geometric figures directly, without any reference to coordinate systems or algebraic descriptions. This is the foundational, axiomatic perspective of Euclid. In analytic geometry we reason from coordinate-based descriptions of geometric objects. This viewpoint was the grand achievement of Descartes and others in the early 1600's and directly led to the invention of calculus in the late 1600's. The transformational view of geometry is the most modern of the three perspectives and arises from the work of Felix Klein in the late 1800's. In transformational geometry we study how geometric objects change (or don't change) under transformations such as rotations, translations, scalings, and other geometric operations. The transformational perspective is used widely today in the area of computer graphics. MCS 303 is a writing course and thus there will be frequent writing assignments during the semester. These will be of three types. First, homework assignments will require written proofs of theorems. Second, computer lab projects will require written reports. Lastly, there will be a final research paper for the course. MCS 303 will have a significant computer laboratory component. We will be using the software package Geometry Explorer to help visualize and understand classroom concepts. All labs will take place in the 3rd floor computer lab. There will be approximately one lab a week. There will also be several in-class mini-projects. These are group projects designed to be done during a class period. Assignments: We will have frequent homework assignments which will normally involve the writing of proofs. Also, there will be weekly lab projects and a final research paper. There will be two hour exams, but no final exam. The final research paper will include an in-class presentation. The grading scale used for the course will be essentially a flat 90-80-70-60 scale. Honor Policy: Students are expected to abide by the college's Academic Homesty Policy. I will typically not proctor exams, but will have students sign an honor pledge for each examination. Students are encouraged to discuss with one another topics from the course. However, the work done in the homework assignments, on lab projects, and on the research paper should be individual work unless otherwise specified. If you are having difficulty with an assignment you may ask fellow students for assistance in understanding the assignment, but not for assistance in doing the assignment. Feel free to ask me for assistance at any time. Disability Services: Gustavus Adolphus College is committed to ensuring the full participation of all students in its programs. If you have a documented disability (or you think you may have a disability of any nature) and, as a result, need reasonable academic accommodation to participate in class, take tests or benefit from the College's services, then you should speak with the Disability Services Coordinator, for a confidential discussion of your needs and appropriate plans. Course requirements cannot be waived, but reasonable accommodations may be provided based on disability documentation and course outcomes. Accommodations cannot be made retroactively; therefore, to maximize your academic success at Gustavus, please contact Disability Services as early as possible. Disability Services is located in the Advising and Counseling Center.
More About This Textbook Overview This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text. Comments on Serge Lang's Algebra: Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books. April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books. The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra. MathSciNet's review of the first edition
Berwyn, IL Trigon Complex Number System Topics may include performing arithmetic operations with complex numbers, representing complex numbers and their operations on the complex plane, and using complex numbers in polynomial identities and equations. Algebra Seeing Structure in Expressions Topics may inclu
Functions and graphs You can adopt these techniques and tasks when planning lessons on functions and graphs. Unless attention is focused on mental processes involved in work on functions and graphs, there is a real risk that pupils will be expected to move rather too quickly from plotting coordinates to tackling challenging generalisations that link algebraic and graphical forms. Pupils require a higher level of thinking to make connections between real-life contexts and the features of a graph. Development of such skills is best supported by collaborative endeavour, allowing pupils the opportunity to share their emerging understanding and to learn from one another. Pupils are better able to tackle challenging problems independently if they have first experienced some success in those areas through interactive group work. Activities relating to functions and graphs can take two main forms: interpreting graphs or generating graphs. These should be developed alongside each other. Interpreting graphs of functions Interpreting pre-drawn graphs provides pupils with opportunities to recognise and generalise the relationship between elements in the function and features of the graph. ICT applications are an ideal medium, both for teacher and pupils, because they provide the means for quickly and accurately testing hypotheses about these links. Progression y=mx y=c+x, y=x+c y=mx+c Recognise these graphs for integer values of m and c Note the relationship between families of graphs as values of m and/or c increase or decrease NoteRecognise these graphs for integer values of a, b and c Note the relationship between families of graphs as values of a and/or b and/or c increase or decrease Generating graphs of functions An important skill is the ability to summarise the key features of a graph through a sketch. This can be developed alongside skills involving graphical calculators or graph-plotting software. In all cases it is crucial to explore problems, discuss results and explain the relationship between the features of a function and the consequent features of the graph. Progression y=mx y=c+x, y=x+c y=mx+c Sketch these graphs for integer values of m and c Explain the relationship between families of graphs as values of m and/or c increase or decrease ExplainSketch these graphs for integer values of a, b and c Explain the relationship between families of graphs as values of a and/or b and/or c increase or decrease Interpreting graphs arising from real-life problems Consider using graphs from other subject areas, such as science or geography, or those that appear in newspapers, other published material or on the internet. Ask pupils to explain what they think the graph might be about. Discussion about the shape of a graph and how it is related to the variables and the context represented, supports pupils' understanding. Progression linear conversion a single straight line, interpreting the meaning of points and sections distance–time linear sections, interpreting the meaning of points and sections temperature change curved sections, interpreting the meaning of points and sections Generating graphs arising from real-life problems Use ICT to generate graphs of real data, including application data from other subject areas. Focus on the degree to which the graph is an accurate interpretation of a real situation (recording temperature change) or part of a mathematical model (distance–time for a cycle journey). Hypothesising about graphs without scales and headings can draw attention to the way in which different scales and starting points can lead to different interpretations. linear conversion a single straight line, interpreting the meaning of points and sections
... More About This Book Math cards review pre-algebra, elementary and intermediate algebra, coordinate and plane geometry, and trigonometry. The Reading cards present strategies to maximize time and determine correct answers. The Science cards cover data representation, research summaries, and conflicting viewpoints. The Writing cards offer tips for creating a strong essay. All cards have corner punch holes that accommodate an enclosed metal key-ring-style card holder. Students can use the ring to arrange flash cards in sequences that best fit their study
Textbooks—published by Key Curriculum Press Each student will receive a hardback text book for use at home and in school. The textbooks have many features to support student learning including: Detailed examples Glossary and index Selected hints and answers Additionally, students receive a code from their teacher to access their textbook online. The online book has interactive features which enable students to link from pages in the book to additional practice problems, dynamic explorations, and calculator notes. Online Resources Electronic resources are available for Algebra, Geometry, Algebra 2, Pre-Calculus and Calculus. You can explore the resources for each book by going to the Kendall Hunt math website. These resources include: A Guide for Parents(in English and Spanish)—A brief summary of each chapter, includes tips for working with students, chapter summary exercises and review exercises with complete solutions. Condensed Lessons(in English and Spanish)—A detailed explanation of each lesson. These can provide extra help for students who have fallen behind or missed class, as well as support for adults who want to understand the details of the mathematics. More Practice Your Skills—a set of additional exercises for each lesson in the book for students who want extra practice. Calculator Notes, Programs and Data—helpful information, programs and tips for using calculators for specific activities. Dynamic Explorations—Structured investigations available online so students and their families can explore mathematics concepts at home.
Product Details Published: 2003 Isbn: 1-885581-45-9 Pages: 193 Math is an important part of everyday life and an integral part of the skills necessary to become certified in the safety profession. Many who pursue certification have long since completed their college math courses and have not actively pursued the math skills they once had. Background Math provides the basics necessary to successfully negotiate the math included on the certification exams, as well as a handy primer for those who already have their credentials. Topics include: Calculator selection and use, including BCSP rules for calculators, strategies for examinations and hierarchy for operations Fractions, reciprocals, proportions, rounding and absolute value Exponents, roots and logarithms and antilogs Systems of measurement, including English, metric, conversions and dimensional analysis Notation, both scientific and engineering Algebraic properties and simple equations, including variables, commutation, associative and distributive properties, order of operations, rules of equations, multiplying polynomials, and solving equations
Search Sponsors App Activity About AppShopper Math Tools Set 2 iOS iPhone This is a set of simple math tools for students of math, engineering, and science. The tools included are: Arithmetic Progression Centroid of a Triangle Complex Number Cubic Equation Distance Between Points Fibonacci Series Geometric Progression Group Work Length of Perpendicular Mid Point Point Slope Form Quadratic Equation Quartic Equation Ratio or Section Slope Intercept Form Two Intercept Form Two Point Form Sum of Consecutive Cubes Sum of Consecutive Squares Sum of Two Cubes Vector Addition Vector Multiplication Vector Subtraction
What is a Clarifying Lesson? A model lesson teachers can implement in their classroom. Clarifying Lessons combine multiple TEKS statements and may use several Clarifying Activities in one lesson. Clarifying Lessons help to answer the question "What does a complete lesson look like that addresses a set of related TEKS statements, and how can these TEKS statements be connected to other parts of the TEKS?" TEKS Addressed in This Lesson Foundations for functions: 2A.1.A, B Quadratic and square root functions: 2A.6.A, B, C; 2A.8.A, B, C, D Materials EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07. Clarifying Lessons Algebra II: Quadratic Functions Lesson Overview Students use quadratic functions to describe the relationship of the height of a football thrown in a parabolic path to its distance from the goal line. Mathematics Overview Students collect and organize data, make a scatterplot, fit a curve to the appropriate parent function, and interpret the results. Students translate among the various representations of the quadratic function, formulate equations, use a variety of methods to solve the quadratic equations, and analyze the solutions in terms of the situation. Set-up (to set the stage and motivate the students to participate) If necessary, have students discuss the layout of a football field before beginning the problem on the worksheet. Have students work in pairs. If appropriate, pair students so that at least one of them has a working knowledge of football. Use the guiding questions to direct students in working through the worksheet. Teacher Notes (to personalize the lesson for your classroom) Guiding Questions (to engage students in mathematical thinking during the lesson) What data are you using to determine the function that best describes the relationship of the height of the ball to its distance from the goal line? (2A.1.B, 2A.6.C, 2A.8.A) How is the graph related to the actual path of the ball? (2A.1.B) What values would not make sense to use for x in this situation? (2A.1.A, 2A.6.A, 2A.8.C) Teacher Notes (to personalize the lesson for your classroom) Summary Questions (to direct students' attention to the key mathematics in the lesson) How can you use the equation to determine the height of the ball at a given position on the field? (2A.1.B) How can you use the graph to determine the height of the ball at a given position on the field? (2A.1.B) How is using the graph like using the equation? How are they different? (2A.8.C) How can you use the equation to determine the distance of the ball from the goal line at a given height? (2A.8.D) How can you use the graph to determine the distance of the ball from the goal line at a given height? (2A.8.B) How is using the graph like using the equation? How are they different? (2A.8.C) Teacher Notes (to personalize the lesson for your classroom) Assessment Task(s) (to identify the mathematics students have learned in the lesson) After discussing the Football Problem in class, give a similar problem for students to complete working independently or working together in small groups. Teacher Notes (to personalize the lesson for your classroom) Extension(s) (to lead students to connect the mathematics learned to other situations, both within and outside the classroom) Have students write their own problems involving situations that can be represented by quadratic functions and share their problems with the class.
Standards in this domain: Understand the concept of a function and use function notation. F-IF.1.-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context. F-IF.4. For.★ F-IF.5. Rel.★ F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ Analyze functions using different representations. F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ a. Graph linear and quadratic functions and show intercepts, maxima, and minima. F-IF.8.Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. F-IF.9. Compare
Squirrel Publications Ltd was founded by husband and wife team Paul and Martine Brasseur. Amazed at the absence of any kind of product that truly filled the gap between private tutor and self study they determined to do what no other publisher had done; namely commission the authoring of a course that would cover all school maths (in the Uk from numbers up to and including A-Level) in such a way that students could learn but also where they needed teaching were actually taught. With no maths background themselves, over the subsequent 15 years they became the bane of the teachers authoring the material. Nothing was to be left out because it was "obvious"…."obvious" to a maths teacher may have involved quite a leap for a student! The premise was that anyone can start the course with the certainty that every concept and tool would be fully explained, placed in context and demonstrated across a range of scenarios before self exploration took place so that they could use the knowledge and skills themselves.. All exploration would have fully worked answers so students could learn from their mistakes and understand their progress. What students find the most helpful is that this breadth and depth is guided by a teacher with over 300 hours of audio. Think about how long a teacher actually talks in a 40 minute maths lesson: say 20 minutes of actual dialogue? That's 900 lessons worth, taking you from "what is a number?'' to ''how do I integrate inverse trig functions in calculus'' and everything in between with a teacher who never tires and goes at the pace just right for you! It works. COMPANY DIRECTORS WHY US? Squirrel Publications believes that for effective learning to take place whoever is studying must be motivated and have access to as many entry points for them to understand, explore and practice so that they understand and develop skills. The author or teacher has to therefore be thorough, inclusive and comprehensive. Short cuts are illusory and mostly driven by commercial factors. We have avoided these to ensure the highest quality standards. If you are an author and believe that education should be accessible as well as comprehensive so that the highest standards of quality are maintained then we're likely to share your vision and passion. Talk to us and let's explore how we can work together.
versión impresa ISSN 0325-8203 Resumen There are three different stages that generally take place while learning mathematics: the first concrete stage is when children learn to add, usually incorporating fingers as the most common object of support; a second pictorial stage or of icons, in which children assimilate a pictorial representation of the concrete object and, finally, an abstract or symbolic stage, in which children handle symbols that represent mathematical quantities. The main goal of this paper is to discuss the construction of a mathematics test for university students administered to a sample of 564 participants. The aim of this study is to evaluate achievements in mathematic abilities of young people that have finished high school and are ready to start university studies. The sample's mean age was 24 years old with a standard deviation of 8.7 years and was 40% male and 60% female. The test has 50 items that measure simple algorithms for arithmetic problems: some items require the use of decimal numbers, some stress the use of proportions or percentages, and a few others are algebraic and geometric questions. All of the items are multiple choice tasks with four options and only one correct answer. There is no time limit to take the test, but its duration is usually not longer than one hour. The mean of the total scores is 25.36 with a standard deviation of 8.06. The exploratory item analysis shows the percentages of correct answers for each item, as well as the values of item-total correlations. Cronbach's alpha reliability index is .936. To study the construct validity we factor analyzed the results and came up with one factor that saturates many items of arithmetic calculation type and three other factors which are not very significant. The test measures, fundamentally, arithmetic calculation, but a lengthy analysis indicates that the items imply other inferential processes. The application of this test indicates that it is very difficult to differentiate mathematical abilities from the aptitude to solve new problems, and that, we are actually evaluating an individual's problem-solving abilities. Such an aptitude improves only with a mathematical instruction centered on the understanding processes, so that if the students are taught to understand the structure and the logic of mathematics, they will have more flexibility and will be more capable of remembering, adapting and organizing data. One of the difficulties observed in the test was that some participants thought that, when multiplying, the values always increase and, when dividing, they always diminish. That is the reason why they struggled so much with decimal exercises. The 50% of the examined participants had difficulties in solving problems with decimals, many had difficulties in finding percentage or interpreting a simple graph of columns. Currently, manual computers are used, but students have difficulties in the interpretation of its results, e.g. when it is presented in mathematical notation. It is very difficult, in this type of test, to differentiate mathematical abilities from mathematical knowledge because it is also important to consider a very strong inferential cognitive aspect.
EXPECTED STUDENT COMPETENCIES TO BE ACQUIRED: The successful student at the end of the course will be able produce well-written correct solutions for problems similar to those assigned for homework in this course. ASSIGNMENTS: Homework will be assigned daily and will occasionally be collected as a check on how you are keeping up. Although most of the homework assignments will not be collected, that doesn't mean you don't have to do it! A major part of learning mathematics involves DOING mathematics! Also, homework is useful in preparing for the type of questions, which may appear on quizzes or exams.Many homework problems will be given on quizzes and some on tests. Evaluations:There will be given two tests and one final exam during this term.There will also be given quizzes twice a week, except on weeks when we have a test, in which case only one quiz will be given.. Tentative Test Dates: Test 1 June 8, 2010 Test 2 June 17, 2010 Final Exam Date June 29, 2010 at 8:00 AM GRADING: Your success in meeting the course objectives will be measured by your scores on homework, quizzes, lab activities, three one-hour exams, and a cumulative final exam. The weights of the various components of your grade in determining your final course grade are shown below, along with the grade scale for the course. WEIGHTS: GRADE SCALE 1. Two exams (100 points each) 90-100 A 70-74 C 2. Quizzes, homework (150 points) 85-89 B+ 65-69 D+ 3. Cumulative Final Exam (150 points) 80-84 B 60-64 D Final average score 75-79 C+ 0-59 F NOTES: One quiz/homework grade will be dropped before determining your final quiz average. There will be no makeup quizzes.There will be no makeup tests, except under exceptional (documented) circumstances.In the case you cannot take an exam at the scheduled time, contact the instructor before the test (or as soon as possible after), to arrange a make up.In no case, will any student be allowed to have more than one make up exam during the term. If you leave right after taking a quiz, your quiz might not be graded NOTES: Please silence your cell phones during class time.No use of cell phones or any electronic device during class times or exam times. SPECIAL NOTES: If you have a physical, psychological, and/or learning disability which might affect your performance in this class, please contact the Office of Disability Services, 126A B&E, (803) 641-3609, and/or see me, as soon as possible. The Disability Services Office will determine appropriate accommodations based on medical documentation. ATTENDANCE POLICY: I may occasionally take attendance. It is highly recommended that the student not miss any class. However, the Attendance Policy established by the Department of Mathematical Sciences states that the maximum number of unexcused absences allowed in this class before a penalty is imposed is four for a regular semester. ACADEMIC CODE OF HONESTY: Please read and review the Academic Code of Conduct relating to Academic Honesty located in the Student Handbook. If you are found to be in violation of this Code of Honesty, a grade of F(0) will be given for the work. Additionally, a grade of F may be assigned for the course and/or further sanctions may
{"itemData":[{"priceBreaksMAP":null,"buyingPrice":11.69,"ASIN":"0486277097","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":3.99,"ASIN":"0486270785","isPreorder":0}],"shippingId":"0486277097::T2kU%2FutiwHe0Kgd9%2BqQwMcxxGzzZTv3BhvpHZFtLGkVF41wy5gt7mVaz1Gu49gET5RoYgncdqwb72D%2FIKvAkykVuAmQZt2SFTVBnpD7NIWk%3D,0486270785::oKWzOfHt82Tii4kBfxDKKlBfLYSXYgds0v4TxCfIZihKnxCT5LCpOcyjMTVGJttPZ6nDFPH8iRDtoTWKwWtNVTStAx1sD%2BJTD60k5ojmK "320 unconventional problems in algebra, arithmetic, elementary number theory and trigonometry." The problems are mathematically accessible to students at the high school level or higher, as they call more upon analytical thinking than upon advanced mathematical techniques. There is a range of difficulties, with harder problems marked with stars in the book. (The hardest problems are marked with double stars.) The problems are divided into twelve sections: "Introductory Problems," "Alterations of Digits in Integers," "The Divisibility of Integers," "Some Problems from Arithmetic," "Equations Having Integer Solutions," "Evaluating Sums and Products," "Miscellaneous Problems from Algebra," "The Algebra of Polynomials," "Complex Numbers," "Some Problems of Number Theory," "Some Distinctive Inequalities," and "Difference Sequences and Sums." Much of the book is devoted to providing hints and solutions, which are both thorough and clear. This is a great resource for preparing for competition, for developing your analytical thinking, or just for having fun (that is, if you are the sort of person who finds solving math problems fun). The reason I am giving 5 stars to this book is for its unique collection of problems. It has been very entertaining reading the book so far (I have not completed the book). There however are a few errors which can be easily figured out by the reader. The treatment of each problem is unique. This is a great resource for challenging math problems. After getting annoyed with newspaper "problem of the week" type books, this was a refreshing find. Don't let the "high school level" disclaimer fool you - there are some seriously difficult problems in here. If you're the type to find logic and math problems fun, I would recommend dropping $15 for this text. It's well worth the time. I found this book very interesting, because it deals with many many problems of algebra and number theory. You can find many interesting and tough theorems (not all of them are widely known nor taught) with their demonstration. I particularily liked the section about "distinctive inequalities": it deals with a great number of inequalities and you can learn some new techniques for solving them. The book lacks of geometry, that's true (only some trig somewhere), but it gives (in my opinion) really a strong preparation on topics concerning algebra... try it! This books is what every book on math olympiads should be, it deals with high level problems in a way that readers can easily follow; I also liked it because there are some problems who have many interesting solutions and generalizations It is a problem and solution type of book. The organization is not ideal. The problems are not ordered by difficulties. There are some very difficult problems. The solutions are quite rigorous and mostly well explained. Probably nice for a teacher to use, not good for self study. Almost all the problems are for algebra, which seems narrow for preparation for high school math Olympiad. From the forward, it is stated that they are for 8th and 9th graders in USSR, which probably explain the lack of Euclidean geometry. If you like challenging math problems then this book is for you. It's ideal if you want to prepare for a national math olympiad or if you just like hard math problems. Get this book and you won't regret it !!!
Student Resources The Mathematics Learning Support Centre (MLSC) has extensive resources to assist students in their learning of mathematics and statistics. There are two locations on campus, one in the West Park and the other in Central Park. More information about these is available in student working areas. There are paper-based resources, for example facts and formulas leaflets and handouts, which are available free of charge. Mathematics and statistics software is available on the computers and help can be obtained from the teaching staff on duty. There are over 300 textbooks, for reference purposes, as well as many workbooks. Students and staff are welcome to come to either the West Park or Central Park locations to see at first hand the resources available. No appointment is needed. Loughborough students can find Department-Specific Support for their studies on the new Mathematics Learning Support Centre Learn Page. You can visit the MLSC Learn page here and go to the section of your department to find support and online resources (iPod videos, leaflets, quick reference sheets) for your studies.
Specification Aims To introduce students to matrix analysis through the development of essential tools such as the Jordan canonical form, Perron-Frobenius theory, the singular value decomposition, and matrix functions. Brief Description of the unit This course unit is an introduction to matrix analysis, covering both classical and more recent results that are useful in applying matrix algebra to practical problems. In particular it treats eigenvalues and singular values, matrix factorizations, function of matrices, and structured matrices. It builds on the first year linear algebra course. Apart from being used in many areas of mathematics, Matrix Analysis has broad applications in fields such as engineering, physics, statistics, econometrics and in modern application areas such as data mining and pattern recognition. Examples from some of these areas will be used to illustrate and motivate some of the theorems developed in the course. Learning Outcomes On successful completion of this course unit students will be familiar enough with matrix analysis and linear algebra that they can effectively use the tools and ideas of these fundamental subjects in a variety of applications,
Continuing their 30-year tradition of excellence, this revision of Lial/Hornsby Intermediate Algebra features the best possible text and supplements package using the most up-to-date strategies to help students succeed. One such strategy, evident in the new table of contents and consistent with current teaching practices, involves the early introduction of graphing lines in a rectangular coordinate system and functions. This organization provides students with increased exposure to basic graphing and function concepts, an integral part of later... MORE mathematics courses, throughout their study of intermediate algebra. It also allows the integration of interesting applications featuring real world data in the form of ordered pairs, tables, graphs, and equations. As a natural follow-up to the treatment of linear equations in Chapter 3, systems of linear equations are now presented in Chapter 4. Also consistent with this approach, graphs of quadratic equations are included earlier in the text when quadratic equations are solved rather than with the material on conic sections as in previous editions. The chapter on exponential and logarithmic functions appears earlier as well. If you choose not to cover graphing linear equations and functions earlier as the new edition suggests, you can defer Chapters 3 and 4 and cover them later after either Chapter 6 or 7. Section 5.3 and the material on graphing and functions in Sections 6.1, 6.4, 6.6, and 7.5 can easily be delayed or omitted. Key Message: The Lial series has helped thousands of readers succeed in developmental mathematics through its approachable writing style, relevant real-world examples, extensive exercise sets, and complete supplements package. Key Topics: Reviewalities, and Functions; Inverse, Exponential, and Logarithmic Functions; Nonlinear Functions, Conic Sections, and Nonlinear Systems; Sequences and Series Market: For all readers interested in Intermediate Algebra.
MAT 295 & 296 Calculus I & II Here's a chance to learn calculus and develop the skills to apply it. In MAT 295, you'll use a graphic calculator to gain hands-on experience in curve sketching and learn increasingly sophisticated techniques for the evaluation and tabulation of functions, as well as the numerical evaluation of limits, derivatives, and roots (the last by Newton's method). In MAT 296, you'll explore exponential logarithms and inverse trigonometric functions and see how they can be applied to a wide range of topics. These courses are foundational for anyone wishing to study science and/or engineering. MAT 397 Calculus III Get a head start on your college requirements! MAT 397 is the third course in a three-semester sequence that dives deeper into calculus. It's a course designed for mathematics, science, and engineering majors, as well as those in other majors, who intend to take advanced courses in mathematics. MAT 414 Intro to Ordinary Differential Equations Interested in taking your math knowledge to the next level? Then MAT 414—a course that focuses on the study of differential equations—is for you! Topics include the analytic and qualitative aspects of first-order differential equations (linear and nonlinear), second order linear equations, Laplace transforms, and systems of first order linear equations.
The purpose of this course is to further develop linear and nonlinear patterns such as quadratic and exponential functions. Students will be able to write expressions, solve linear systems, and graphically represent these functions through modeling. In addition, this course introduces students to the fundamental concepts of data analysis, probability, and geometry. There is major emphasis on investigation and making sense of problem situations in order to generalize and communicate mathematics both orally and in writing. A graphing calculator is required for this course.
CliffsQuickReview course guides cover the essentials of your toughest classes. You're sure to get a firm grip on core concepts and key material and be ready for the test with this guide at your side. Whether you're new to functions, analytic geometry, and matrices or just brushing up on those topics, CliffsQuickReview Precalculus can help. This guide... more... Navigate politics, paperwork, and legal issues Find your instructional style and make learning fun for your students! Gain the upper hand on your first day of school! This friendly guide reveals what they didn't teach you in your education classes, offering practical advice and tons of real-life examples to help you set up and maintain an orderly... more...
Quadratics Self AssessmentReview all major concepts with your students using this wonderfully crafted Quadratic Self-Assessment! Students are given a sample problem and a practice problem to encourage understanding. They assess if they can do a particular topic well or if they need more assistance. Topics covered are: Intro to Quadratic Functions, Intro to Solving Quadratic Equations, Solving by Completing the Square, Factoring, Writing Quadratics in vertex form, using the Quadratic Formula, complex solutions, and real-world problems. Save yourself time and energy with this amazing review product! PDF (Acrobat) Document File Be sure that you have an application to open this file type before downloading and/or purchasing. 335
Book Description Publication Date: May 22 2002 \| Series: Charles River Media Game DevelopmentProduct Description About the Author Eric Lengyel is a Senior Software Engineer at the 3DO Company in Redwood City, CA. He holds an MS in Mathematics from Virginia Tech and has written several articles for industry periodicals including gamasutra.com. He is also the area editor in geometry management for Game Programming Gems 2. Finally, no more searching through all my college math textbooks for the reference I need for real-time 3D software development. The basics of vectors and matrices are of course included, but in much more depth than you got in school, more than likely - and with emphasis on how they are useful in 3D game programming. So many game developers lack an intuitive feel for such basics as transformation matrices, dot products, and cross products and are hobbled by this; just read up to chapter three and the lights will go on, so to speak. The chapter on lighting is particularly, well, enlightening - not only are the various lighting models explained in detail (including some I was unfamiliar with before), but the author provides means for accomplishing them in real-time using texture and vertex shaders. The notation used in the book is modern and consistent, and the code samples clearly written. I believe this is the first volume to combine complete mathematical explanations of essential 3D computer graphics operations with practical advice on how to implement the sometimes complex math efficiently in real-time systems. The chapters on picking and collision detection are also complete and include practical advice on implementation in addition to the theory behind it. This is not a book for most high school math students - the author assumes you've at least been through some higher level math and can talk the basic language of mathematics. However, it does not presuppose that you are familiar with anything but basic calculus, and more importantly, it doesn't assume that you're familiar with some quirky notational system specific to the author. I haven't been in a math class for ten years, but I had no trouble understanding any concepts introduced in this book upon the first read. like others books in charles media , written by someone in the industry , which means value infos will be in the book . this book is good for someone studied linear algebra I,II + calculs courses and want to see the applied math in computer applications such as games . i suggest before reading this book , reading a book about linear algebra + gemetry + calculs so as not to lost in that book because this book not for beginners in math .
I'm getting really bored in my math class. It's algebraic cubes, but we're covering higher grade material. The topics are really complicated and that's why I usually doze off in the class. I like the subject and don't want to fail, but I have a real problem understanding it. Can someone help me? Oh boy! You seem to be one of the best students in your class. Well, use Algebrator to solve those problems. The software will give you a comprehensive step by step solution. You can read the explanation and understand the problems. Hopefully your algebraic cubes class will be the best one. A truly piece of math software is Algebrator. Even I faced similar difficulties while solving relations, triangle similarity and monomials. Just by typing in the problem from homeworkand clicking on Solve – and step by step solution to my math homework would be ready. I have used it through several math classes - Algebra 2, Algebra 2 and College Algebra. I highly recommend the program.
Intermediate Algebra. The engaging Martin-Gay workbook series presents a student-friendly approach to the concepts of basic math and algebra, giving students ample opportunity to practice skills and see how those skills relate to both their lives and the real world. The goals of the worktexts are to build confidence, increase motivation, and encourage mastery of basic skills and concepts. Martin-Gay ensures that students have the most up-to-date, relevant text preparation for their next math course; enhances students' perception ... MOREof math by exposing them to real-life situations through graphs and applications; and ensures that students have an organized, integrated learning system at their fingertips. The integrated learning resources program features text-specific supplements including Martin-Gay's acclaimed tutorial videotapes, CD videos, and MathPro 5. The engaging Martin-Gay workbook series presents a user-friendly approach to the concepts of basic math and algebra, giving readers ample opportunity to practice skills and see how those skills relate to both their lives and the real world. The goals of the workbooks are to build confidence, increase motivation, and encourage mastery of basic skills and concepts. Martin-Gay enhances users' perception of math by exposing them to real-life situations through graphs and applications; and ensures that readers have an organized, integrated learning system at their fingertips. The integrated learning resources program features book-specific supplements including Martin-Gay's acclaimed tutorial videotapes, CD videos, and MathPro 5. This book includes key topics in algebra such as linear equations and inequalities, systems of equations, polynomial functions and equations, quadratic functions and equations, exponential functions and equations, logarithmic functions an equations, and rational and radical expressions. For professionals who wish to brush up on their algebra skills. Presents a user-friendly approach to the concepts of basic math and algebra, giving readers ample opportunity to practice skills and see how those skills relate to both their lives and the real world. For professionals who wish to brush up on their algebra skills. Softcover.
Authors Title Date of Original Version 1993 Type Book Abstract or Table of Contents This book is primarily intended to be a textbook for undergraduate students majoring in mathematics or in physics. Most, if not all, of the existing textbooks on Relativity use verbal descriptions together with what one might call \the mathematics of variables and coordinates". We believe that such an approach makes it difficult to convey a deeper understanding of the subject and an intuition for relativistic phenomena, an ability to "think relativistically" as it were. Such an understanding can be gained, we believe, by employing a more contemporary type of mathematics: the mathematics of sets, mappings, and relations. Typically, physics majors rarely learn about this kind of mathematics, and mathematics majors, when they learn about it, are not told how useful it can be to gain a deeper understanding of the concepts of physics.
Saxon Publishers has designed math programs for primary grades (K-3), middle grades (4-8), and secondary grades (9-12) that produce confident students who are not only able to correctly compute, but also to apply concepts to new situations. These materials gently develop concepts, and the practice of those concepts is extended over a considerable period of time. This is called "incremental development and continual review." Material is introduced in easily understandable pieces (increments), allowing students to grasp one facet of a concept before the next one is introduced. Both facets are then practiced together until another is introduced. This feature is combined with continual review in every lesson throughout the y ... more Calculus is designed for prospective math majors in college as well as students preparing for engineering, physics, business analysis, or the life sciences. The text covers all topics normally found in Advanced Placement AB-level calculus program, as well as many topics from a BC-level program. Problem sets contain multiple-choice and conceptually oriented problems similar to those in Advanced Placement exams. Also included are numerous applications to physics, chemistry, engineering, and business. Includes: With LIFEPAC Pre-calculus, your student is given a comprehensive study of advanced math, trigonometry, and pre-calculus. Topics are: Relations and Functions; Trigonometric Functions; Circular Functions and Graphs; Quadratic Equations; and Probability. Easy-to-understand, personalized instructions ease anxiety and give student's an assertive, positive attitude toward calculus. Packed with valuable information, this course, which covers functions and identities, is a great preparatory course for college math classes. The LIFEPAC Pre-calculus Set contains ten work texts and a teacher's guide that may be purchased individually. the text). Quizzes help students put together material in two sections, tests relate material throughout the chapter, while quarterly exams combine information in three chapters.
Liberal Arts & Science Studies Practical Mathematics Course Outline Upon completion of this course you will be able to answer questions and solve problems in arithmetic, algebra, geometry, trigonometry and probability. The contents of the course will be useful in further mathematics studies as well as technical and business applications. The knowledge covered in this course is used on the job daily by apprentices, sales clerks and technicians. This is a single stream course. The topics covered within this course includes the following: • Introduction to Numbers • Fractions • Decimal Numbers • Ratio and Proportion • Percentages • The Metric System and Other Systems of Measurement • Algebra and Formula • Geometry • Linear Equations and Graphing Sets and Inequalities • Factoring • Quadratic Equations and Graphing • Algebraic Fractions • Trigonometry • Probability and Statistics Duration Approximately 20 hours Text Books The text books required for this course are outlined below and can be purchased through any online book supplier: ISBN#: 0-17-604938-X ISBN#: 0-17-605649-1
GraphSight is a feature-rich comprehensive 2D math graphing utility with easy navigation, perfectly suited for use by high-school an college math students. The program is capable of plotting Cartesian, polar, table defined, as well as specialty graphs. Importantly, it features a simple data and formula input format, making it very practical for solving in-class and homework problems. The program comes with customizable Axis options, too.
Calculus And Its Applications - 9th edition Summary: Calculus and Its Applications has, for years, been a best-selling text for one simple reason: it anticipates, then meets the needs of today's applied calculus student. Knowing that calculus is a course in which students typically struggle--both with algebra skills and visualizing new calculus concepts--Bittinger and Ellenbogen speak to students in a way they understand, taking great pains to provide clear and careful explanations. Since most students taking this course ...show morewill go on to careers in the business world, large quantities of real data, especially as they apply to business, are included as well 92.10 +$3.99 s/h Good LOOK AT A BOOK OH Miamisburg, OH 2007
Math 6: This is a rigorous course that builds a strong foundation for future math courses by emphasizing number sense and problem solving, preparing students to take Algebra as 8th graders. Students will be introduced to linear equations, inequalities, statistics, probability, and geometry. This course includes operations with fractions, decimals, and integers. Students, who successfully complete Math 6, are on target to complete all UC/CSU requirements in high school. The majority of incoming sixth graders will be placed in Math 6. Pre-algebra is considered an advanced placement for 6th graders. Student placement for the 6th grade math course will be primarily determined by a student's spring MAP score. Students who score at least a 248 will be placed in pre-algebra. Pre-Algebra: The topics in pre-algebra include an emphasis on solving multi-step equations including those involving rational numbers, graphing linear equations and inequalities, and problem solving. Students will continue their exploration of statistics, geometry, properties of exponents, and consumer mathematics. Pre-Algebra Support: Pre-Algebra with support is a course designed to enhance a student's number sense skills while continuing to cover the necessary topics in the 7th grade curriculum. The placement for this course will be determined by a student's MAP scores in 6th grade and by teacher recommendation. Algebra: The topics in Algebra include an emphasis on solving systems of equations, solving and graphing quadratics, polynomials, problem solving and rational expressions. Placement in Algebra will be based on MAP scores and teacher recommendation based on performance in Pre-Algebra. Algebra Readiness: The topics in Algebra Readiness include an emphasis on fractions, integers, solving equations and inequalities, graphing linear equations, and problem solving. This course continues the work that was begun in the Pre-Algebra Support class to prepare students for Algebra in high school. Geometry: The topics in Geometry include geometric constructions, proof writing, inductive and deductive reasoning, properties and attributes of polygons and circles, spatial reasoning, and an introduction to trigonometry. The Geometry course also introduces some of the advanced algebraic concepts that students will encounter in Algebra 3-4. This course is for those students who have successfully completed Algebra 1-2 based on test scores and teacher recommendation.
Mathematics Courses The ability to comprehend, develop, and utilize mathematical concepts is invaluable throughout life. All people need some fluency in this area in order to contribute to and to fare well in our contemporary world. The Mathematics Department, therefore, seeks to enhance students' knowledge and understanding of math allowing them to attain success in college and throughout life. The broad purposes and objective are best served by cultivating in students the ability to draw reasonable conclusions from information found in various sources, whether written, spoken, or displayed in tables and to solve problems which they may encounter in life.
Subject Matter Anxieties Most studies on achievement anxiety do not differentiate by subject matter. But some people develop anxiety about performance in specific subject areas or with regard to particular skills. They may be comfortable in most academic contexts, but have great difficulty performing in one domain. Two domains that have been studied are mathematics and writing. Mathematics Anxious Alma is in good company. Mathematics anxiety, or "mathophobia," is widespread. College students report much more anxiety about mathematics than they do about English, social science (Everson, Tobias, Hartman, & Gourgey, 1993), or even writing (Sapp, Farrell, & Durand, 1995). It is estimated that about one-third of college students suffer from some level of mathematics anxiety (Anton & Klisch, 1995; see Mitchell & Collins, 1991). A commonly used measure of mathematics anxiety is the Mathematics Anxiety Scale (MAS). The scale includes 10 items and studies of middle school, high school, and college students suggest that it taps the same two dimensions of anxiety often found in more general test anxiety measures—a general sense of worry about mathematics and negative feelings and emotional reactions (Pajares & Urdan, 1996). Mathematics does not generate as much anxiety in young children as in older children and adults. In Goodlad's (1984) study of over 17,000 young students, mathematics was rated about the same as reading in a list of "liked" subjects (after art and physical education). In the National Assessment of Educational Progress, nine-year-olds ranked mathematics as their best-liked subject; thirteen-year-olds ranked it second best, but in contrast to the younger children, seventeen-year-olds claimed that mathematics was their least liked subject (Carrpenter, Corbitt, Kepner, Lindquist, & Reys, 1981). Significant declines in positive attitudes toward mathematics have also been shown over the adolescent years (Wigfield, Eccles, Mac Iver, Reuman, & Midgley, 1991; Wigfield & Eccles, 1994). Apparently children are not born with mathematics anxiety. Rather, negative attitudes toward mathematics develop over time, especially during adolescence. Why does mathematics, in particular, cause so much anxiety in older students and adults? One can only speculate. Lazarus (1975) points out that mathematics anxiety has a "...peculiar social acceptability. Persons otherwise proud of their educational attainments shamelessly confess to being 'no good at math'" (p. 281; see also Sapp, 1999). The way mathematics is usually taught may also explain why mathematics anxiety is common. Lazarus (1975) suggests that the cumulative nature of mathematics curricula is one explanation; if you fail to understand one operation, you are often unable to learn anything taught beyond that operation. From observations of mathematics and social studies classes, Stodolsky (1985) proposed that mathematics instruction fostered in students the belief that mathematics is something that is learned from an authority, not figured out on one's own. She found that mathematics classes were characterized by (1) a recitation and seatwork pattern of instruction; (2) a reliance on teacher presentation of new concepts or procedures; (3) textbook-centered instruction; (4) textbooks that lacked developmental or instructional material for concept development; (5) a lack of manipulatives; and (6) a lack of social support or small-group work. The instructional format, the types of behavior expected from students, and the materials used were also more similar from day to day in mathematics than in social studies classes. This lack of variety may contribute to anxiety because students who do not do well in the instructional format used in mathematics are not given opportunities to succeed using alternative formats. Later studies by Stodolsky also suggest that mathematics teachers see their subject area as more sequential and static than teachers of other subjects (Stodolsky & Grossman, 1995; see also Wolters & Pintrich, 1998). Sapp (1999) speculates that mathematics teaching often focuses on memorization of procedures, which doesn't prepare students for more conceptual, advanced mathematics. Thus, they feel ill-prepared and become anxious when rote procedures are no longer sufficient. Stodolsky (1985) also suggests that mathematics is an area in which ability, in the sense of a stable trait, is believed to play a dominant role in performance-either one has the ability or one does not. And if one lacks ability in mathematics, nothing can be done about it. By contrast, people generally believe that performance in other subjects, like reading or social studies, can be improved with practice and effort; they hold an "incremental" theory of ability. There is consistent evidence that females suffer more from mathematics anxiety than do males (Hembree, 1990; Pajares & Urdan, 1996; Randhawa, 1994; Wigfield & Meece, 1988). Some researchers have proposed that mathematics anxiety contributes to observed gender differences in mathematics achievement and course enrollment, but the one study that actually assessed anxiety and enrollment plans found no relationship (Meece, Wigfield, & Eccles, 1990). There is little agreement on the reasons for such gender differences. Ability differences, socialization differences, differences in the level of self-confidence, and the number of mathematics courses taken have all been proposed as explanations. Whatever the reasons for the frequency and intensity of mathematics anxiety, particularly among females, it is a problem that warrants special attention by educational researchers and practitioners. The good news is that interventions to reduce math anxiety have been successful. Sgoutas-Emch and Johnson (1998) found that writing in a journal about frustrations and feelings reduced college students' anxiety in a statistics course. Writing Perhaps everyone, at one time in their lives, experiences a certain amount of panic facing a blank piece of paper or computer screen, especially if the due date for a written product—a paper for a class or a report for work—is close at hand. "Writer's block" is so debilitating for some that they avoid courses and professions that require writing. (See Daly & Miller, 1975b; Daly, Vangelisti, & Witte, 1988; Rose, 1985; Selfe, 1985.) Although psychoanalytic explanations have been suggested (Barwick, 1995; Grundy, 1993), the few studies that have been done suggest that writing anxiety reflects some of the same dynamics that explain general achievement anxiety. Writing anxiety, like general achievement anxiety, is associated with relatively low expectations for success as well as lower writing quality (Daly, 1985; Pajares & Valiante, 1997). Rose's (1985) research on writer's block makes it very clear that the causes are usually multifaceted, and that although they may have their roots in early familial experiences, later and current experiences in writing contexts are also important. Researchers have developed a measure of writing anxiety (Daly & Miller, 1975a), which has been shown to be more strongly associated with writing performance than a more general measure of achievement anxiety (Richmond & Dickson-Markman, 1985). Studies using the measure have found some gender differences, with females showing somewhat less writing anxiety than males. People high in writing anxiety were also high on reading anxiety and anxiety about public speaking and interpersonal communication, but relatively low on math anxiety (Daly, 1985). Research has also examined associations between teachers' feelings about writing and their teaching strategies. Studies have found, for example, that highly apprehensive female teachers assign fewer writing assignments and are more likely to be concerned with issues of form and usage and less likely to emphasize personal or creative expression and effort than less apprehensive teachers (see Daly, 1985; Daly et al., 1988). Associations between teachers' own anxiety about writing and their teaching methods were strongest in upper elementary school, when many important writing skills are supposed to be taught. The instructional context can exacerbate concerns about competencies that feed anxiety, and they contribute to trait anxiety both over time and collectively. One study found that high school students who were relatively high in writing anxiety reported having experienced more criticism for their writing and less encouragement and support, and they reported seeking help for writing problems less than students low on writing anxiety (Daly, 1985). Daly (1985) proposes that writing anxiety will be greatest under the following circumstances: evaluation is salient the task is ambiguous the writer feels conspicuous task difficulty is perceived to be high the writer feels lacking in prior experience relevant to the task the task is personally salient the setting or task is novel the writer perceives the audience as uninterested but evaluative Teachers may be able to reduce writing anxiety by minimizing students' concerns about evaluation, making assignments and criteria for grading clear, and making sure that students have the prior experience and familiarity they need to complete the writing task. Writing tasks, like all tasks, should be challenging but not so difficult or different from what students have experienced in the past as to provoke a sense of incompetence or low expectations for success. A genuine and supportive audience (e.g., classmates, parents) might also help.
Stop searching through textbooks, old worksheets, and question databases. Infinite Pre-Algebra helps you create questions with the characteristics you want them to have. It enables you to automatically space... Basic Algebra Shape-Up is interactive software that covers creating formulas; using ratios, proportions, and scale; working with integers, simple and multi-step equations. Students are able to track their... Answers.com is a revolutionary Answer Engine: a one-click reference library on demand. With 1-Click Answers software, just Alt-click on any word in any Windows application (e-mail, browser, Office, and so... Problem Solved! 4.6 is a comprehensive but easy to use help desk program which includes both a windows based application and a mini-http server that you have running in less than an hour. The web interfaceThis bilingual problem-solving mathematics software allows you to work through 5018 Algebra equations with guided solutions, and encourages to learn through in-depth understanding of each solution step and...,... This script defines the Matrix class, an implementation of a linear Algebra matrix. Arithmetic operations, trace, determinant, and minors are defined for it. This is a lightweight alternative to a numerical... Passguide Vmware VCP-510 testing engine simulates a real VCP-510 exam environment. Questions & Answers are compiled by a group of Senior IT Professionals. Questions are updated frequently - you get a free... Algebra Helper shows how signs change when moving a quantity from one side of an equation to the other. Algebra Helper is implemented in JavaScript. Currently works in Firefox and Internet Explorer 8.... Boolean Algebra assistant program is an interactive program extremely easy to use. The program is intended for the developers of small digital devices or simply of radioamateurs, and as for those who is... Interactive College Algebra course designed to ensure engaging, self-paced, and self-controlled e-learning process and help students to excel in their classes. Java- and web-based math course includes... Answers... Mavscript allows the user to do calculations in a text document. Plain text and OpenOffice Writer files (odt,sxw) are supported. The calculation is done by the Algebra system Yacas or by the Java interpreter... FAQ Search allows you to place a searchable and easy-to-update Frequently Asked Questions database on your site in minutes. One script file, and a single database text file can be uploaded in a few minutes... MathProf is an easy to use mathematics program within approximately 180 subroutines. MathProf can display mathematical correlations in a very clear and simple way. The program covers the areas Analysis,... Calculator is a minimalist, easy to use Windows calculator that takes convenient advantage of the number key pad in your keyboard. Calculator can compute any Algebra expression instantly. Calculator runs... It is one of the most powerful math tools there is. It gathers between the simplicity and ease of use of a simple calculator and the ability to solve complex math procedures. Here is a brief description of...
This course, designed to be taken concurrently with COSC-072, covers mathematical tools and principles that are valuable to the computer scientist. Topics include: propositional and predicate logic; mathematical proofs, including induction; counting and basic probability theory; logarithmic and exponential functions; elementary graph theory; and "Big-O" notation and asymptotics. Prerequisite: none. Spring. Other academic years There is information about this course number in other academic years:
A study of the structure of algebraic systems with emphasis on groups, rings, integral domains and fields. Detailed Description of Content of Course The major topics covered in these two courses are those which represent the foundation of the field of modern algebra. It is, for most students, an introduction to the axiomatic method. Topics included in MATH 423 are: Most instructors will present the course material in a lecture format. Students may be required to prepare and present problems for class discussion. Goals and Objectives of the Course In the twentieth century abstract algebra has become one of the three main divisions of mathematics. This course offers an introduction to that area as well as an introduction to the axiomatic method so improtant to modern mathematics.
A Course of Pure Mathematics Centenary edition 1 rating: 5.0 A book by G. H. Hardy "First published in 1908, this classic still gives undergraduate students their first dose of the differential and integral calculus, the properties of infinite series and other notions involving limit. Hardy's nineteenth-century sensibilities … see full wiki A masterpiece of mathematics Although the sequence of the presentation of the fundamentals of mathematics has changed over the last century, the substance has not. There is no greater evidence of this fact than this classic work by Hardy, which could be used without alteration or additional explanation as a text in modern college mathematics courses. Hardy was rightfully known as a bit of an eccentric, yet he was a brilliant pure mathematician and he will always be held in the highest regard for his actions in aiding the Indian prodigy Ramanujan. Less well known but still extremely significant is his expository writing; there are few who wrote as clearly as he did. This book was extremely influential in the teaching of mathematics over the last century. The primary subject matter is: ) Real variables ) Functions of real variables ) Complex numbers ) Limits of functions of a positive integral variable ) Limits of functions of a continuous variable ) Derivatives and integrals ) Theorems on the differential and integral calculus ) Convergence of infinite series and infinite integrals ) Logarithmic, exponential and circular functions of a real variable ) General theory of the logarithmic, exponential and circular functions There are few proofs, but an enormous number of examples. The mathematical influence of G. H. Hardy over mathematical education was and remains strong, as can be seen by reading this masterpiece. Published in Journal of Recreational Mathematics, reprinted with permission
This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning. show more show less List price: $145.00 Edition: 1977 Publisher: Elsevier Science & Technology Books Binding: Trade Cloth Pages: 279 Size: 5.98" wide x 9.02" long x 1.00 of Set Theory - 9780122384400 at TextbooksRus.com.
This course is that portion of Abstract Algebra that studies elementary group theory. It considers the properties of groups, subgroups, and functions; this leads to groups of permutations and groups isomorphic to them. Homomorphisms of groups along with the induced quotient groups culminate in the Fundamental Homomorphism Theorem; this rounds out the course. Either Math 232 or this course fulfills the requirement for Modern Algebra by the Indiana State Department of Education for Secondary Teacher Education students of mathematics. REQUIREMENTS AND GRADING The course will cover sequentially the first sixteen chapters of the text. Understanding rigorous mathematical proofs is an essential part of this course. Your course grade will be based on four criteria: Examinations (two during the semester, plus the final exam), written homework assignments, a class presentation, and attendance. Attendance will be taken each class period and will affect your grade by a maximum of two percentage points in either direction. The criteria will be weighted in the following manner: Examinations: 50% Homework: 30% Presentation: 20% Attendance: +/-2% NOTES ON GRADING CRITERIA There will be an assignment from each chapter covered in the text. I will always announce when an assignment is due, however, in case for any reason you don't hear me give the due date, the game plan will always be as follows: Assignments should be attempted by the next class period (so that questions may be asked in class) and are due at the end of the second class period after the appropriate chapter is finished. E.g., if I complete Chapter 4 on Tuesday September 19, the assignment for Chapter 4 is due at the end of the Tuesday September 26 class. Late assignments will not be accepted except under extreme circumstances. I cannot stress enough the importance that doing your homework has to your success in an advanced math course such as this. Math is not a "spectator sport". Even though class attendance and participation will help you a great deal in keeping up with the material, you cannot adequately learn math without practicing it on your own. Your assignments must be NEATLY DONE, and all work must be shown. Multiple pages must be STAPLED, and each page must have a smooth edge - pages torn out of spiral binders are NOT ACCEPTABLE. The first two exams will be given during the regular class period (approximately September 28 and November 9), and the third will be given in finals week. They will consist mostly of short proofs and definitions, with an occasional computational problem. If you keep up with class notes and do all homework assignments in a timely manner you will be adequately prepared for the exams. Your class presentation can be either a detailed explanation of a proof from the text (or elsewhere) or lecture about some aspect of the history or current state of abstract algebra. The presentations should be at least ten minutes long, and we'll start these sometime in late September - early October.
Resources: Description of the Lake Braddock Middle School MATH program: The LBSS Middle School math program is comprised of six different math courses; Mathematics 7, Mathematics 7 Honors, Algebra I Honors, Mathematics 8, Algebra I, and Geometry Honors. Math 7 and 8 is a two-year program designed to prepare students for Algebra I in the 9th grade. Content is composed of four primary mathematical strands, the Language of Algebra, Proportional Reasoning, Quantitative Literacy and Space and Shapes. For advanced math students Algebra I is offered in the 7th grade for those students who have passed a county defined set of prerequisites. Algebra I is also offered in the 8th grade. For students who took Algebra I in the 7th grade and earned an A or B, Geometry Honors is offered in the 8th grade. Algebra I and Geometry Honors are high school courses and as such, will be placed on a students final high school transcript. For the students not quite ready for Algebra I in the seventh grade, the middle school offers a "Mathematics 7 Honors" course with a strong enrichment-based curriculum branching into upper level mathematics.
RELATED LINKS 4 credits, Spring, 2000 Professor Larry Krajewski Office: Murphy Center 526 Office Phone: 796-3658 Home Phone: 782-1648 [No calls between 10 p.m. and 7 a.m. please] Hours: 3M, 11W, 12F & by appointment E-mail: lkrajewski@centurytel.net llkrajewski@viterbo.edu Prerequisite: C or better in Math 255 Text: Mathematics for Elementary School Teachers by Tom Basserear,Houghton Mifflin, 1997 Final Exam: 1:10 class Wednesday, May 10, 7:40-9:40 a.m. 2:10 class Thursday, May 11, 9:50-11:50 a.m. Description: This course is designed to introduce the preservice K-9 teacher with ideas, techniques and approaches to teaching mathematics. Manipulatives, children's literature, problem solving, diagnosis and remediation, assessment, equity issues, and the uses of the calculator are interwoven throughout the topics presented. The math content areas are rational numbers and geometry. The Viterbo College Teacher Education Program has adopted a Teacher As Reflective Decision Maker Model. Each course is designed to contribute to the development of one or more of the knowledge bases in professional education. This course contributes to the development of the knowledge bases: Knowledge of the Learner, Curriculum Design, Planning and Evaluation, and Instructional and Classroom Management. Goals: To Resources You may qualify for free tutoring in the Learning Center. Methodology: Lecture, class discussion, small group work, student presentations. Todd Wehr Library The following books are on reserve: Solutions Manual Selected Bibliography for Gender Equity in Mathematics and Technology Resources Published in 1990-1996 ,Women & Mathematics Education Objectives Upon successful completion of this course, the student will be able to: 1. ..explore, conjecture, reason logically and use a variety of mathematics methods effectively to solve nonroutine problems. 2. ..establish classroom environments so that his or her students can explore, conjecture, reason logically and use a variety of mathematics methods effectively to solve 3. ..nonroutine problems and develop a lifelong appreciation of math in their lives; 4. ..design and use several forms of assessment, such as portfolios, journals, open-ended problems, tests, and projects 5. ..become familiar with educational research on effective teaching of mathematics. Student Responsibilities As teachers you should appreciate the importance of class participation. Your active participation makes the course go. Math is not a spectator sport. Assigned problems and textbook exercises are ways for you to develop problem solving skills and reflect on your learning. Requirements Six summaries of articles on the following topics (include a copy of the article in your summary; article must be at least two pages long.) Fractions Geometry Equity and mathematics Measurement Technology Assessment The purpose of this assignment is to acquaint you with some resources outside of the textbook and to introduce you to some ideas or activities that you may want to share with the class when we are investigating the appropriate topic. Please follow these guidelines: Include a copy of the article with your summary. Use the reporting form included in your packet. Articles must be at least two pages long in the original citation. Articles taken from the internet must be complete (No missing pictures, diagrams, or equations.) A problem notebook with assigned problems from the text and class. You must work out the solutions. Copying answers from the solutions manual is not appropriate. Completion of a minimum of 12 hours of field experience working with an elementary student on mathematics A journal of your sessions with an elementary student. [NOTE: you MUST MEET WITH YOUR STUDENT AND FULFILL THIS REQUIREMENT IN ORDER TO PASS THIS COURSE.] Two math activities, one on geometry and one on fractions Five investigations Two in-class exams Evaluation Percentage Problem notebook 15% Readings 3% Student journal 15% Investigations (5) 25% Math activities 2% Tests (2) 40% Topics I. Geometry A. Spatial Reasoning B. Van Hiele levels C. Two dimensional geometry D. Three dimensional geometry E. Translations, Reflections and Rotations F. Symmetry G. Similarity II. Measurement A. Length B. Area C. Volume III. Extending the number system A. Integers B. Rational numbers and fractions 1. Models for rational numbers 2. Comparing rational numbers 3. Renaming rational numbers 4. Addition and subtraction 5. Multiplication and division C. Decimals D. Proportions and ratios E. Percents A Note: Some Hopefully you will see mathematics as an open- ended, So I ask your help in establishing a mathematical community where one uses logic and mathematical evidence as verification rather than the teacher, where mathematical reasoning replaces the memorization of procedures, and where conjecturing, inventing, and problem solving are encouraged and supported. persons actively involved in problem solving. To nurture a mathematical idea in the mind of a child might be easier if it first thrived in the mind of the child's teacher. Americans with Disabilities Act If you are a person with a disability and require any auxiliary or other accommodations for this class, please see me and Wayne Wojciechowski, the Americans With Disabilities Act Coordinator (MC 320 - 796- 3085 ) within ten days to discuss your accommodation needs. BIBLIOGRAPHY
easy-to-follow approach to intermediate algebra. The book features realistic, relevant application problems, non-routine problems drawn from everyday life, and carefully chosen examples and exercises. It also includes open-ended problems that invites exploration. Includes new applications, written to incorporate more interesting, real-world data in order to increase interest and to help apply the math learned. Table of Contents Basic Concepts The Real Number System Operations with Real Numbers Powers, Square Roots, and the Order of Operations Integer Exponents and Scientific Notation Operations with Variables and Grouping Symbols Evaluating Variable Expressions and Formulas Linear Equations and Inequalities First-Degree Equations with One Unknown Literal Equations and Formulas Absolute Value Equations Using Equations to Solve Word Problems Solving More Involved Word Problems Linear Inequalities Compound Inequalities Absolute Value Inequalities Equations and Inequalities in Two Variables and Functions Graphing Linear Equations with Two Unknowns Slope of a Line Graphs and the Equations of a Line Linear Inequalities in Two Variables Concept of a Function Graphing Functions from Equations and Tables of Data Systems of Linear Equations and Inequalities Systems of Equations in Two Variables Systems of Equations in Three Variables Applications of Systems of Linear Equations Systems of Linear Inequalities Polynomials Introduction to Polynomials and Polynomial Functions: Addition, Subtraction and Multiplication
Algebra 2 Help.pdf... [More] that you understand the subject and score well in it.
N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. N-Q Quantities N-Q Reason quantitatively and use units to solve problems. N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-CN.1 Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. N-CN.2 Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N-CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. N-CN Represent complex numbers and their operations on the complex plane. N-CN.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. N-CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. N-CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. N-CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. N-VM Vector and Matrix Quantities N-VM Represent and model with vector quantities. N-VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, \|v\|, \|\|v\|\|, v). N-VM.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. N-VM.3 Solve problems involving velocity and other quantities that can be represented by vectors. N-VM Perform operations on vectors. N-VM.4 Add and subtract vectors. N-VM.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. N-VM.4.c Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. N-VM.5.b Compute the magnitude of a scalar multiple cv using \|\|cv\|\| = \|c\|v. Compute the direction of cv knowing that when \|c\|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). N-VM Perform operations on matrices and use matrices in applications. N-VM.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N-VM.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. N-VM.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. N-VM.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. N-VM.12 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. A Algebra A-SSE Seeing Structure in Expressions A-SSE Interpret the structure of expressions A-SSE.1 Interpret expressions that represent a quantity in terms of its context. A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients. A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-APR.5 Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle. A-APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-REI Understand solving equations as a process of reasoning and explain the reasoning A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. A-REI.4.b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. A-REI.11 Explain. A-REI.12 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. F-IF.1 Understand Interpret functions that arise in applications in terms of the context F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F-BF.3 Ident F-BF.4 Find inverse functions. F-BF.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. F-BF.4.b Verify by composition that one function is the inverse of another. F-BF.4.c Read values of an inverse function from a graph or a table, given that the function has an inverse. F-BF.4.d Produce an invertible function from a non-invertible function by restricting the domain. F-BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F-TF Extend the domain of trigonometric functions using the unit circle F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. F-TF.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-x in terms of their values for x, where x is any real number. F-TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F-TF.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. F-TF.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. F-TF Prove and apply trigonometric identities F-TF.8 Prove the Pythagorean identity sin2(theta) + cos2(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle. F-TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. G Geometry G-CO Congruence G-CO Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G-CO.2G-CO.5G-CO.6 G-SRT.2 Given G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S-ID Summarize, represent, and interpret data on two categorical and quantitative variables S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. S-ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals. S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. S-IC.6 Evaluate reports based on data. S-CP Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data S-CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not"). S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S-CP.3 Understand S-CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. S-CP Use the rules of probability to compute probabilities of compound events in a uniform probability model S-CP.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. S-MD.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. S-MD.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. S-MD.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. S-MD.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. S-MD Use probability to evaluate outcomes of decisions S-MD.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. S-MD.5.a Find the expected payoff for a game of chance. S-MD.5.b Evaluate and compare strategies on the basis of expected values. S-MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S-MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
Certain contemporary mathematical problems are of particular interest to teachers and students because their origin lies in mathematics covered in the elementary school curriculum and their development can be traced through high school, college, and university level mathematics. This book is intended to provide a source for the mathematics (from beginning to advanced) needed to understand the emergence and evolution of five of these problems: The Four Numbers Problem, Rational Right Triangles, Lattice Point Geometry, Rational Approximation, and Dissection. Each chapter begins with the elementary geometry and number theory at the source of the problem, and proceeds (with the exception of the first problem) to a discussion of important results in current research. The introduction to each chapter summarizes the contents of its various sections, as well as the background required. The book is intended for students and teachers of mathematics from high school through graduate school. It should also be of interest to working mathematicians who are curious about mathematical results in fields other than their own. It can be used by teachers at all of the above mentioned levels for the enhancement of standard curriculum materials or extra-curricular projects. Readership High school students, undergraduate and graduate students, and teachers of all levels interested in mathematics. Reviews "I am a huge fan of this book! ... "Roots to Research" is very accessible, supported throughout with insightful examples and exercises that motivate both the ideas and the formal notation. I recommend this book to future and current math teachers, math majors, and working mathematicians who are interested in reading about cool math. ... the Sallys have done the mathematical community a service by writing a book that illustrates an approach that more of us should take when teaching upper-level undergraduate and graduate math courses." -- MAA Monthly "Many references are given but the book is largely self-contained. The authors have done a remarkable job of giving a seamless presentation of material at very different levels of difficulty. Teachers and students will appreciate this book both for the information it contains and as a model of expository writing." -- Mathematical Reviews "The book gives a very good introduction in how to solve mathematical problems and it is well suited as a basis for a beginner's seminar at universities."
Search Course Communities: Course Communities Lesson 13: Completing the Square Course Topic(s): Developmental Math \| Quadratics This lesson introduces completing the square as a means of expanding the set of quadratic equations that may be solved beyond the extraction of roots and factoring. Simpler cases are first presented and then towards the end of the lesson a procedure for completing the square of (ax^2 + bx + c = 0) is given.
Description By purchasing a value pack, you will save compared to purchasing these two books separately. Target audience Suitable for Year 10 students. Series overview We're proud to introduce the only series has been structured according to the latest research on how students learn mathematics and on how to avoid common misunderstandings, making it easy for you to provide an innovative and effective education to your students with Pearson Mathematics. Built from the ground up for the Australian Curriculum, we've been able to base the series on the latest pedagogical research on how students learn best. We've combined carefully selected grading with thoughtful open-ended questions at the end of every exercise, based on research conducted by the lead writer of the Australian Mathematics Curriculum, Peter Sullivan. Through careful integration of all the sub-headings of the proficiency strand (fluency, understanding, reasoning, problem solving and open-ended questions) a full coverage of the curriculum has been achieved. Equipped as well with the 5e+ format (the engage, explore, elaborate, evaluate and extend model), Pearson Mathematics helps you to provide the right balance of scaffolding and openness for inquiry-based investigations.
COURSE STRUCTURE CLASS IX One Paper Time:Three Hours Marks : 80 Unit Marks Number systcm 05 Algebra 20 Commercial Mathematics 12 Gcometry 18 Trigonomatry 07 Mensuration 08 Statistics 10 Total 80 UNIT I NUMBER SYSTEM (10 Periods) Introduction to Irrational Numbers Irrational numbers as non-terminating and non-repeating decimals. Real numbers and the real number line. Surds and Rationalization of surds. (Irrational numbers may be introduced by recalling rational numbers as terminating or non-terminating recurring decimals. ) Problems of proving a number to be irrational number should be avoided. Representing an inoational number on the number line should be avoided for numbers other that √2 ,√3 and √5 . Rationalization of only those surds are to be Iincluded which involve square roots and are binomials or trino1milials) UNIT II ALGEBRA (42 Periods) Factorisation of Polynomials Review of factorisation of algebraic expressions done in earlier classes. Factorisation of polynomials ", of the form ax2+bx+c, a =/=0, by splitting the middle term, where a, band care real numbers. Factorisation of algebraic expressions of the type X3+y3, x3_y3, X3+y3tz3_-3xyz. I{emainder theorem, factor theorem and factorisation of polynomials of dcgree not excccding three. (While f"actorisation of the polynomial ax2+bx+c, a =/=0, a,b,c, should be rationals or square root of rationals only. (1) If a+b 1-c=o" then a3+b3+c3=3abc, Questions involving above concept will be included. (2) Simple expressions reducible to the form a3+b3+c3-3abc may be included). Ratio and Proportion I{ccall or the concepts of ratio and proportion. Continued proportion, invertendo" alterncndo, componcndo and dividendo. Linear Equations in Two Variablcs Review or 14inear equations in one variable. Co-ordinates of a point and plotting of points with given integral co-ordinates in cartesian plane. Introduction to liner equations in two variables. Graph ofa linear eqLlation in one/two variables in cartesian plane. (For the graph of linear equation in two variables, equation sIlould be so choscn so as to get as far as possible integral valued coordinates.) Internal Assessment 20 Marks Evaluation of Activities 10 marks Project work 05 marks Continuous evaluation 05 marks CLASS X One Paper Time: Three Hours Marks:80 Unit Marks Algebra 20 Commercial Mathematics 10 Geometry 18 Trigonometry 08 Mensuration 08 Statistics 10 Coordinate Geometry 06 Total: 80 UNIT I ALGEBRA (55 Periods) Linear Equations in Two Variables System of' linear equations in two variables, Solution of the system of linear equations (i) Graphically. (ii) By algebraic methods: (a) Elimination by substitution (b) Elimination by equating the co-effcients. ( c) Cross multiplication. Applications of Linear equations in two variables in solving simple problems for different areas. (Restricted upto two equations with integral values as a point of solution. Problems related to life to be incorporated). Polynomials HCF and LCF polynomials by factorization Rational Expressions Meaning of raltional expressions. Reduction of rational exprcssions to lowest terms using factorisation. Four fundamcntal opcrations on rational expressions. (Properties like commutativity, assotciativity)', distribution law etc. not be discussed. Cause involving factor theorem may also be given), Quadratic Equations Standard form of a quadratic equation ax2+bx+c='0, (a=/= 0). Solution of ax2+bx+c=0 by (i) factorisation (ii) quadratic fonnula. Application of quadratic equations in solving word -problems from different areas. (Roots should be real) (Problems related to day-to-day activities to be incorporated). Arithmetic Progression (AP) Introduction to AP by pattern of number. General term of an AP, Sum to n-terms of an AP. Simple problems. (Common difference should not be irrational number). UNIT II COMMEI{CIAL MATHEMATICS (15 Periods) Instalments Instalment payments and instalment buying (Number of instalments should not be more than 2 in case of buying). ( Only equal instalments should be taken. In case of payments through equal instalments, not more than three instalments should be taken. UNIT VI STATISTICS (15 PERIODS) Mean Mean of grouped data. (Calculation by assuming assumed mean should also be discussed). Probability Elementary idea of probability as a measure of uncertainty (for single event only). Pictorial Representation of Data Reading and construction of pie chart. [(i) Sub parts of a pie chart should not exceed five). (ii) Central angles should be in multiples of 5 degrees.] UNIT VII COORDINATE GEOMETRY (15 PERIODS) Distance between two points. Section formula. (internal division only.) Internal Assessment 20 Marks Evaluation of Activities 10 Marks Project Work 05 Marks Continuous Evaluation 05 Marks Prescribed Books Mathematics for Class IX, NCERT PUBLICATION Mathematics for Class X, NCERT PUBLICATION Guidelines for mathematics laboratory in schools class IX – CBSE Publication
Euclid's Elements Description Euclid's Elements is a remarkable geometry course centered around the propositions of Euclid's Elements. In the first half of each class students will present prepared talks to explain and demonstrate a geometric proof to the group. Students will learn to make presentations, critically think, and build their confidence as they go through this yearlong course. Students become the teachers of other students! The second half of each class will have students work through QED™ designed activities to Euclid's Elements was the first logical presentation of mathematics as designed from a set of first principles. Until publication of modern text books in the 20th century, the Elements was the standard geometry text for all advancedmathematics students. Our course reintroduces this tradition striving to foster excellence in geometry, logic, critical thinking, and public speaking. Sequence The Fundamentals An exciting introduction to Euclid's classic "The Elements". Students will learn logical flow, presentation skills, and the fundamentals of geometry from Euclid's first and second book. science: students will work with a range of physical problems to determine ideas of measurement, accuracy, and precision. Information Each course in Euclid's Elements is a 30-hour workshop for a total of 90 hours of in-class material. Academic year sequences are held in 10 week classes each 3 hours long. Summer camps are held in a single week and cover 30 hours of material. Each student will receive a full copy of Euclid's Elements. Students are expected to be prepared each week to make a presentation to their classmates. Classes consist of hands-on activities, lecture, and practice problems. Academic year students are required to complete a minimal amount of practice problems. Prerequisites Students should have completed Algebra and should be reading at an 8th grade level.
Evaluation and Observation Content Quality Rating: Strengths: This is a collection of 40 animations, some with limited interactivity. Included are animations that demonstrate concepts from Euclidean geometry and the first three semesters of calculus. Many of the animations come from topics with which students have a low comfort level. The three dimensional renderings are each meant to point out one specific concept that can best be understood with a computer visualization. The student can grab the object with a mouse and rotate and move it to get another perspective. Some of the objectives include interactivity where the student can grab a point and move it to watch the associated geometric objects change. For example there is an activity where the student moves a tile through a grid and the corresponding tile in polar coordinates also move. The main page includes a key to instructors on the most effective, somewhat effective, and barely usable animations. It is well formatted, containing both titles and small graphics that indicate what the animation will show. Concerns: Although a few of the animations are accompanied by an explanation of the topic, most are not supported in this way. Potential Effectiveness as a Teaching Tool Rating: Strengths: This collection will serve as a powerful demonstration of the concepts by the instructor or an exploration session where the student can work in a computer lab. Many of these can be an effective part of an explanation on theory. For example when an instructor is discussing volumes by washers, the activity can be pulled up and the student can not only see the washer, but can also see the radii changing as the center moves along the x-axis. Concerns: Occasionally and not consistently, one reviewer's computer failed to open some of the objects. Ease of Use for Both Students and Faculty Rating: Strengths: There are instructions on how to use each of the objects. Navigation was simple and when a point or other construct can be manipulated, it is clear how to accomplish the task. Students will have no trouble working with the animations. Concerns: Occasionally and not consistently, one reviewer's computer failed to open some of the objects.
Programme for International Student Assessment (PISA) Mathematics Introduction The Programme for International Student Assessment (PISA) is governed and administered by the Organisation for Economic Co-operation and Development (OECD), an entity comprised of thirty countries. The PISA exams cover scientific, reading, and mathematical literacy, and have been administered every three years since 2000. This review covers the framework for the math assessment administered in 2006,7 in which fifty-six OECD and non-OECD countries participated. The PISA test is administered to fifteen year-olds since this is the age "in most OECD countries [where] students are approaching the end of compulsory schooling." PISA's objective is to test "not so much in terms of mastery of the school curriculum, but in terms of important knowledge and skills needed in adult life." The PISA framework "defines the contents [sic] that students need to acquire, the processes that need to be performed and the contexts in which knowledge and skills are applied." Though not intended exclusively for school-based education, this framework thus has the same intention and performs essentially the same function as a set of academic standards, and can reasonably be evaluated by the same standards-based criteria. Further, a number of American educators, policymakers, and standard setters look to PISA as a benchmark for what should be required and/or expected of American schools. Since the PISA standards are used for these purposes, it is important to appraise their content. The framework devotes over forty pages to summarizing what mathematical literacy should mean for fifteen year-olds. PISA breaks content into four categories: "space and shape; change and relationships; quantity; and uncertainty." (In addition to those four categories, PISA discusses four "situation types," eight "competencies," and three "competency clusters," but these are vague and do nothing to clarify the content that students should have mastered.) For each of the four content categories, there is a general discussion about the mathematics in the category, but again, these discussions do not describe content to be covered. Each category features a short list of "key aspects" that are as close to standards as the framework gets, but these, too, are vague and non-specific. They total only twenty-three and do not supply clear guidance for readers or users (e.g., teachers, curriculum developers, test developers, mathematicians). Sample problems are also supplied in each category, and the real content to be covered must often be inferred from these examples (as well as from released items from actual PISA exams, discussed below). The sample problems illustrate the very low level of mathematics content that is required. The exam is primarily a problem solving exam, and seldom requires the highest level of mathematics that a fifteen year-old would be expected to know. The second document reviewed here, the released items, is 106 pages long and contains fifty problems with descriptive names (such as Walking, The Best Car, and Postal Charges). Many of these problems consist of several distinct test items, all related to the same theme. All PISA problems are "in context" (i.e., they are intended as real world problems). Review The PISA framework is evaluated on two dimensions, "content and rigor" and "clarity and specificity," and this review will address these dimensions under four sections. First, the content addressed by the PISA materials is compared with the content that should be covered by a mathematics test for fifteen year-olds (see "Math Content-Specific Criteria," page 11), and the results are presented under "content covered" and "content missing." Specific problems with the content are discussed in the third section, followed by a discussion of the released items in the fourth section. The remainder of the review sums up the content and rigor of the PISA framework and released items and considers the clarity and specificity of the former. The last section provides an overall summary and final grade. Content Covered Any mathematics assessment for fifteen year-olds should thoroughly cover the arithmetic of rational numbers, including decimals, and should also cover rates, ratios, proportions, and percentages. Fractions or rational numbers are not mentioned anywhere, but we do find that understanding the meaning of operations "includes the ability to perform operations involving comparisons, ratios and percentages." The standard covering "number sense" does explicitly include proportional reasoning. Under the content category "quantity," we find some standards that are potentially relevant: "understanding the meaning of operations" and "elegant computations." These are much too vague, however, to assure coverage of the arithmetic of rational numbers. Typically, standards are self-explanatory. However, PISA standards require one to read the lengthy prose surrounding them to gain some insight into their meaning (more on this below). Volume, area, and perimeter are mentioned under the category "quantity," and triangle appears in the preliminary discussion and in an example problem. "Similar" occurs in the discussion, but it is not the technical similarity of geometry. Coordinates are also mentioned in the discussion. The most guidance we get in the "space and shape" content category is from the rather vague standard: "recognizing shapes and patterns." This, in principle, could cover much of the material of geometry, but the standard is far too general to be useful. Students should have covered a full year of algebra by age fifteen. Under the content category "change and relationships," we have two potentially relevant standards: "representing changes in a comprehensible form" and "understanding the fundamental types of change." These do not give clear guidance to readers, though. In the accompanying discussion, we find absolute value, linear equations, and inequalities and linear functions. Note that the discussion material included in the PISA framework is not an obvious attempt to clarify individual standards. Rather, it is a very general discussion about the uses of mathematics in the real world. Here is an example: Every natural phenomenon is a manifestation of change and the world around us displays a multitude of temporary and permanent relationships among phenomena. Examples are organisms changing as they grow, the cycle of seasons, the ebb and flow of tides, cycles of unemployment, weather changes and stock exchange indices. Some of these change processes involve and can be described or modeled by straightforward mathematical functions: linear, exponential, periodic or logistic, either discrete or continuous. But many relationships fall into different categories and data analysis is often essential to determine the kind of relationship that is present. Mathematical relationships often take the shape of equations or inequalities, but relations of a more general nature (e.g. equivalence, divisibility, inclusion, to mention but a few) may appear as well. Though PISA is given credit here for covering linear equations and inequalities, this credit might be considered quite generous. For one does not find explicit guidance about content so much as suggestions that certain content should be covered—linear equations for example. Content Missing The core content of the arithmetic of rational numbers is missing, as is any coverage of rates. Most of the expected geometry is missing. There are no similar and congruent triangles; circles; angles associated with triangles and parallel lines; or computation of areas and perimeters of rectangles. The Pythagorean Theorem is missing. Roots, reciprocals, and powers are absent, as is the arithmetic of polynomials and rational expressions. Factoring is also missing. Quadratics are not mentioned. Content problems Relatively recent recommendations concerning school curricula are unanimous in suggesting that statistics and probability should occupy a much more prominent place than has been the case in the past. Although publications from 1982 to 2000 are cited to support this sentiment, "unanimous" is a dramatic overstatement. In the sidebar on data analysis, statistics, and probability, "How much DASP do students need?" on page 12 of this report, the case is made that these content areas should receive only limited attention. Simply put, there is an overemphasis on this content category in the PISA standards. Released items In addition to the sample problems included in the PISA Framework, PISA also makes available a set of released items (test problems used in previous years). Because of the imprecise nature of the standards and the discussion surrounding them, it is important to review the released items to see if they give further guidance about content. The most striking feature of the released items is that thirty-five of the fifty involve the use of a picture, table, or graph. "Data display" falls under the content category of "uncertainty," and this means that the "uncertainty" content category is highly overrepresented, even if the problems go on to test content covered by other categories. There are only a few real formulas and equations, and only about 10 percent of the released items use any algebra. There is only one question that expects students to produce a formula of their own from the information given in a problem. This question requires students simply to multiply a given formula by 0.8 in order to arrive at the new formula. The lack of algebra illustrates the low level of mathematical content knowledge expected by the PISA assessment. Nor are arithmetic skills in much demand in the problems. Even where such skills might be useful, calculators are allowed. The level of geometry used in a few of the released items is significantly higher than that suggested by the standards and the discussion surrounding them (as is the case with similarity and congruence). However, as with algebra, only a few test items call upon this level of geometry. Most of the released items are focused on problem solving and use fairly low-level mathematics content. However, the problems can be quite complex; about 20 percent of them are multi-step problems. So, even though the content is undemanding, the sample problems can be quite difficult due to the number of steps required to solve them. A significant number of problems could be taken to task for errors, misleading statements, or imprecise questions. Enumerating all of these issues is beyond the scope of this review. Content and Rigor Conclusion Arithmetic, most of the geometry, and much of the algebra that is listed in the "Math Content-Specific Criteria," (page 11) are missing from the PISA framework. More than half of the important content is never mentioned, which would normally result in a score of three for content and rigor—that is, if the framework were all that's being evaluated. However, the inclusion of the released items affects PISA's grade for content and rigor. Even though there are very few geometry problems in the released items, those that do appear require more geometry content than the standards suggest. This raises PISA's content and rigor score from three to four as it now appears that at least 50 percent of the content is covered, but certainly more than 35 percent is missing (see the "Common Grading Metric," page 16). Clarity and Specificity The actual standards included in the PISA framework are non-specific and not testable. They give almost no guidance to readers and users—i.e., teachers, students, parents, curriculum designers, test makers, textbook developers, standards writers, policymakers, or others. The discussions around the standards are rambling accounts of mathematics in the real world, and although they mention various bits of content (for which coverage has been generously credited in this review), these discussions, too, give scant guidance to readers at any level. According to the "Common Grading Metric" (page 16), a score of zero for clarity and specificity means: The standards are incoherent and/or disorganized. The standards are not helpful to users. The standards are completely lacking in detail. It is as if the PISA framework were written to illustrate the kind of standards that should merit a score of zero. Accordingly, its score for clarity and specificity score is zero. Summary and Grade The PISA assessment tests mathematical literacy, not knowledge and understanding of grade-level mathematical content. The standards and their explanations do not cover the appropriate grade-level material and the released items indicate that the exam is quite weak in mathematical content. It is a problem-solving test, and although mathematics is used, it is somewhat incidental. Many problems have no apparent mathematical content at all and are, at best, small logic puzzles. Because of this low level of required content knowledge, the claim that PISA tests "preparedness for further study" is in doubt. The test itself is unbalanced, with glaring overemphasis on data display. Most of the content that is expected of a fifteen year-old in PISA is what younger students should have already mastered. As a serious problem solving test using elementary mathematics, the PISA assessment might function nicely, although its unbalanced nature would limit its usefulness here. Certainly schools should teach problem solving in their mathematics curricula, and it should be a major embarrassment for a country to perform poorly on this test. Still, results from PISA ought not to be used to interpret how successful a school system is at imparting grade-level mathematical knowledge and understanding, nor are the PISA framework and released items a suitable model for U.S. standards setters at any level.
To this point we've only looked as solving single differential equations. However, many "real life" situations are governed by a system of differential equations. Consider the population problems that we looked at back in the modeling section of the first order differential equations chapter. In these problems we looked only at a population of one species, yet the problem also contained some information about predators of the species. We assumed that any predation would be constant in these cases. However, in most cases the level of predation would also be dependent upon the population of the predator. So, to be more realistic we should also have a second differential equation that would give the population of the predators. Also note that the population of the predator would be, in some way, dependent upon the population of the prey as well. In other words, we would need to know something about one population to find the other population. So to find the population of either the prey or the predator we would need to solve a system of at least two differential equations. The next topic of discussion is then how to solve systems of differential equations. However, before doing this we will first need to do a quick review of Linear Algebra. Much of what we will be doing in this chapter will be dependent upon topics from linear algebra. This review is not intended to completely teach you the subject of linear algebra, as that is a topic for a complete class. The quick review is intended to get you familiar enough with some of the basic topics that you will be able to do the work required once we get around to solving systems of differential equations. Here is a brief listing of the topics covered in this chapter. Review : Systems of Equations The traditional starting point for a linear algebra class. We will use linear algebra techniques to solve a system of equations. Review : Matrices and Vectors A brief introduction to matrices and vectors. We will look at arithmetic involving matrices and vectors, inverse of a matrix, determinant of a matrix, linearly independent vectors and systems of equations revisited.
Schaum's Outline of Theory and Problems of Matrix Operations 9780070079786 ISBN: 0070079781 Pub Date: 1988 Publisher: McGraw-Hill Summary: Master matrix operationsIf you d...on't have a lot of time but want to excel in class, this book helps you: Brush up before tests Find answers fast Study quickly and more effectively Get the big picture without spending hours pouring over lengthy textbooksInside, you will find: 363 detailed problems with step-by-step solutions Clear, concise explanations of matrix operations Help with Eigenvalues and the QR Algorithm A solved-problem approach that teaches you with hands-on help Exercises for improving your problem-solving skillsIf you want top grades and a thorough understanding of matrix operations, this powerful study tool is the best tutor you can have!Chapters include: Basic Operations Simultaneous Linear Equations Square Matrices Matrix Inversion Determinants Vectors Eigenvalues and Eigenvectors Functions of Matrices Canonical Bases Similarity Inner Products Norms Hermitian Matrices Positive Definite Matrices Unitary Transformations Quadratic Forms and Congruence Nonnegative Matrices Patterned Matrices Power Methods for Locating Real Eigenvalues The QR Algorithm Generalized Inverses Answers to Supplementary Problems[read more
European Partners The European graduate program (Europäisches Graduiertenkolleg) will be interdisciplinary combining aspects from discrete mathematics and computer science. Its scientific program ranges from more theoretical areas like combinatorics and discrete geometry via algorithmics and optimization to application areas like computer graphics or geographic information systems. The program is split into four basic research areas, namely combinatorics, geometry, optimization, and algorithms and computation. The major scientific goal of the program is to intensify the cooperation and interaction between discrete mathematics, algorithmics, and application areas.
Go to Mathematics (minor subject) For more than two millenia, the study of mathematics has developed through a marriage of abstraction and application. Some mathematical theories and techniques are developed directly intended for problem solving in other areas of study, such as: astronomy physics chemistry statistics computer science economy technology Others, however, are derived from the mathematician's interest in abstraction and the inner structure of the field and its aesthetic values. The expression "the unreasonable usefulness of mathematics" exists because it has been shown, time and again, that the sometimes very abstract thought constructions mathematicians create in this manner without utility in mind, later prove to be especially useful in other subjects. One example is numbers theory which forms the basis for cryptology. The programme is primarily for students with majors in other scientific subjects within the natural sciences.
Develop a rich understanding of the rudiments of algebra in a relaxed and supportive learning environment. This course will help you understand some of the most important algebraic concepts, including orders of operation, units of measurement, scientific notation, algebraic equations, inequalities with one variable, and applications of rational numbers. Learning Objectives: Identify types of numbers, practical applications, and history of algebra. You will need to create a login for your online classroom. Go to Find your course by browsing the catalog or using the search bar. Click the "Enroll Now" button. Select your start date and then create a Username and Password. At the end, you will be asked to click on the "Already Paid" button if you have already paid. You must make an 80 or higher on the final exam (online) to successfully complete the course. You may only take the exam once. If you are a certified teacher in Georgia and are interested in taking this course for PLUs, please complete the PLU notification form once you have registered. This course grants 2 PLUs upon successful completion. If you have questions about this course, please contact the online coordinator at 770-499-3355 or online@kennesaw.edu.
An investigation of topics including the history of mathematics, number systems, geometry, logic, probability, and statistics. There is an emphasis throughout on problem solving. Recommended for General Education. For some of you this course might serve to satisfy the math competency requirement, for others this will be just one of the mathematics courses required by your major/minor program. In any case, the main goal of this course is to expose you to a variety of areas of mathematics and thus give you an idea of the importance of mathematics in today's world and a multitude of ways it is being used in practice. We will learn some elements of mathematical logic, set theory, geometry, statistics, probability, consumers mathematics, and some basic algebra. The content and the methods of this course are designed in accordance with general education objectives and the work in this course should help you in developing a number of skills included in the NCTM (National Council of Teachers of Mathematics) `standards'' for mathematics education, and also being among the general education objectives at Viterbo. The main emphasis throughout the course will be on problem solving and developing thinking skills. This includes: writing numbers and performing calculations in various numeration system, solving simple linear equations, exploring the mathematical model of simple and compounded interest rates, and learning how to use those ideas in solving the problems of loan payments, exploring a few major concepts of Euclidean Geometry, focusing especially on the axiomatic-deductive nature of this mathematical system, including a variety of different proofs of the Pythagorean Theorem, develop an ability to use deductive reasoning, in the context of the rules of logic and syllogisms, i.e., learn how to make/recognize a valid argument, some basics of probability and statistics ... Mastering this material requires to learn how to reason mathematically, and also how to communicate mathematics. In learning how to do so (on exams, essays, portfolio, and in oral presentations), you will also develop a confidence in your ability to do mathematics. Other benefits of this course include: cultural skills (appreciation of the history of mathematics and its role in today's world, learning how to handle simple loans, learning how to reason correctly and make a valid argument), appreciate the beauty and intellectual honesty of deductive reasoning, thereby adding to life value and aesthetic skills. I encourage you to read the text at: - the Viterbo critical thinking web page Text Robert Blitzer, Thinking Mathematically, Prentice-Hall, 2000. Format Class sessions will consist of lectures, work in small groups, exams, and individual presentations. I expect students to work out the recommended practice problems and ask for help whenever needed. Resources Please do not hesitate to contact me for any question you might have; do not let a feeling such as ``I am lost ...'' to last. Internet and the blackboard software. There is a lot of material on my web page. I will use the Blackboard to communicate with you, so please check your e-mail regularly. I would also like to encourage you to explore, and use numerous capabilities of that (Blackboard) software. Learning center. Library. Note that a video set that covers your textbook exists. Grading The following grading scale applies to individual exams, and to the overall grade as well: A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%. The following exceptions to that scale are possible: An A on the final exam (more than points) will raise your grade up, one letter, i.e., a B will turn into an A, a BC will become AB, .... An outstanding presentation, or an outstanding portfolio can raise your grade up a half letter, i.e., a C will turn into a BC, .... If one is failing the course by the end of the semester, but has over average on exams, and earns at least points on the final, he/she can get a D for the final grade. If one is passing the course by the time of the final exam, but earns less than points (a score less than), that will result in an F for the final grade. Assignments Recommended practice: First, middle and the last problems from each Practice Exercises set in each section that we cover; at least one or two of the Application Exercises, at least one of the Writing in Mathematics Exercise, and at least two of the Critical Thinking Exercises. These practice problems will not be graded. However, fell free to ask me for help with any difficulty you might have with those problems. Two essays, points each: Autobiography: Introduce yourself to me in a 2-3 pages essay. State your name, and where (city/state) you are coming from. The reason you are taking this course, and what mathematics courses you have had before. What was your experience from those courses and what are your expectations, if any, from this course? This assignment is due Friday, January 18. World without mathematics: another 2-3 pages 20 points essay. Try to imagine, and describe, a world without mathematics. Due: Friday, January 25. Homework At the end of each chapter, there is a Chapter Test. Each one of those tests will be due second class period after the corresponding chapter is covered, and each problem on the ``test''is worth 1 point. Exams There will be three in-class exams, worth points each. An exam will typically cover three chapters worth of material. The exams will be closed notes, closed book. However, a calculator and a formula sheet (but not any worked out problem) is allowed. Before each exam, I will give you a take-home practice exam, which will be very much like the actual exam coming. I will grade (25 points) the first one of those, i.e., the ``Exam 1 - Practice'', but not the others. I will also allow a makeup (up to 50%) of the lost credit for the exam 2. This makeup will be oral, and will apply to those under points on the test, and is to be done within two weeks after the exam. Final Exam Final exam is a 2-hour, cumulative exam, and is worth points. Portfolio It should consist of 5 problems, but no two problems should be of the same type (from the same section). Format: You state a problem, write a complete/correct solution to it, and then write a paragraph (or more) explaining why did you choose that particular problem, what did you learn from it, etc.. The portfolio will be worth points. The problems you choose for the portfolio should illustrate the progress in learning mathematics, the change of the perception (if any) of what mathematics is about, the change (if any) in your perception about your abilities to do mathematics. In-class Presentation: The presentation of a proof of the Pythagorean Theorem found on the Internet. Typically, the explanations you will find on the Internet are a bit sketchy. So part of your job will be to make sure you really understand the proof you are going to present (including filling in the gaps, i.e., the reasons not entirely spelled out in the Internet write-up), and then to clearly explain that proof to your class mates. Sometimes, some people, may find this part quite difficult. Of course, I am here to help you understand and overcome those difficulties, and so please do not hesitate to ask me for help. You should also be prepared for the questions from the audience (myself and/or other students), and it is expected that you listen closely to other presentations and ask any question you might have. The presentation will be worth points. In addition to that, one certain problem for one of the exams, and for the final exam is going to be: State and prove the Pythagorean Theorem. Group Labs: At a number of points during the course you will be working on a ``lab'' in small groups. Even though you will be working in a group of three or four people, each person should turn in a paper. It is important that each person contributes their input into these labs. However, I expect you to write the turn-in paper all by yourself. Americans with Disability Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me and Wayne Wojciechowski in Murphy Center Room 320 (796-3085) within ten days to discuss your accommodation needs.
Upcoming Events Courses Courses offered in the past four years. ▲indicates offered in the current term ▹indicates offered in the upcoming term[s] MATH 0100 - A World of Mathematics ▲ A World of Mathematics How long will oil last? What is the fairest voting system? How can we harvest food and other resources sustainably? To explore such real-world questions we will study a variety of mathematical ideas and methods, including modeling, logical analysis, discrete dynamical systems, and elementary statistics. This is an alternative first mathematics course for students not pursuing the calculus sequence in their first semester. The only prerequisite is an interest in exploring contemporary issues using the mathematics that lies within those issues. (This course is not open to students who have had a prior course in calculus or statistics.) 3 hrs lect./disc. MATH 0109 - Mathematics for Teachers ▹ Mathematics for Teachers What mathematical knowledge should elementary and secondary teachers have in the 21st century? Participants in this course will strengthen and deepen their own mathematical understanding in a student-centered workshop setting. We will investigate the number system, operations, algebraic thinking, measurement, data, and functions, and consider the attributes of quantitative literacy. We will also study recent research that describes specialized mathematical content knowledge for teaching. (Not open to students who have taken MATH/EDST 1005. Students looking for a course in elementary school teaching methods should consider EDST 0315 instead.)▲▹ 0122 - Calculus II ▲▹ Calculus II A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations. (MATH 0121 or by waiver) 4 hrs. lect./disc. MATH 0190 - Math Proof: Art and Argument ▹ Mathematical Proof: Art and Argument Mathematical proof is the language of mathematics. As preparation for upper-level coursework, this course will give students an opportunity to build a strong foundation in reading, writing, and analyzing mathematical argument. Course topics will include an introduction to mathematical logic, standard proof structures and methods, set theory, and elementary number theory. Additional topics will preview ideas and methods from more advanced courses. We will also explore important historical examples of proofs, both ancient and modern. The driving force behind this course will be mathematical expression with a primary focus on argumentation and the creative process. (MATH 0122 or MATH 0200) 3 hrs. lect. MATH 0200 - Linear Algebra ▲▹ 0217 - Elements of Math Bio & Ecol ▲ Elements of Mathematical Biology and Ecology Mathematical modeling has become an essential tool in biology and ecology. In this course we will investigate several fundamental biological and ecological models. We will learn how to analyze existing models and how to construct new models. We will develop ecological and evolutionary models that describe how biological systems change over time. Models for population growth, predator-prey interactions, competing species, the spread of infectious disease, and molecular evolution will be studied. Students will be introduced to differential and difference equations, multivariable calculus, and linear and non-linear dynamical systems. (MATH 0121 or by waiver) MATH 0223 - Multivariable Calculus ▲▹ Multivariable Calculus The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications. (MATH 0122 or MATH 0200 or by waiver) 3 hrs. lect./disc. MATH 0247 - Graph Theory ▹ Graph Theory A graph (or network) is a useful mathematical model when studying a set of discrete objects and the relationships among them. We often represent an object with a vertex (node) and a relation between a pair with an edge (line). With the graph in hand, we then ask questions, such as: Is it connected? Can one traverse each edge precisely once and return to a starting vertex? For a fixed k/, is it possible to "color" the vertices using /k colors so that no two vertices that share an edge receive the same color? More formally, we study the following topics: trees, distance, degree sequences, matchings, connectivity, coloring, and planarity. Proof writing is emphasized. (MATH 0122 or by waiver) 3 hrs. lect./disc. MATH 0250 - Ethnomathematics Ethnomathematics: A Multicultural View of Mathematical Ideas and Methods* What are the cultural roots of the mathematics we study and use today? Even though it has been developed by individuals from widely varying cultural contexts, we take the verity, consistency, and universality of mathematics for granted. How does the western tradition stand in comparison to the mathematics developed by indigenous societies, labor communities, religious traditions, and other groups that can be studied ethnographically? By examining the cultural influences on people and the mathematics they practice, we shall deepen our understanding of mathematics and its relationship to society. 3 hrs. lect/disc. MATH 0261 - History of Mathematics ▹ History of Mathematics This course studies the history of mathematics chronologically beginning with its ancient origins in Babylonian arithmetic and Egyptian geometry. The works of Euclid, Apollonius, and Archimedes and the development of ancient Greek deductive mathematics is covered. The mathematics from China, India, and the Arab world is analyzed and compared. Special emphasis is given to the role of mathematics in the growth and development of science, especially astronomy. European mathematics from the Renaissance through the 19th Century is studied in detail including the development of analytic geometry, calculus, probability, number theory, and modern algebra and analysis. (MATH 0122 or waiver) MATH 0311 - Statistics ▲ Statistics An introduction to the mathematical methods and applications of statistical inference. Topics will include: survey sampling, parametric and nonparametric problems, estimation, efficiency and the Neyman-Pearsons lemma. Classical tests within the normal theory such as F-test, t-test, and chi-square test will also be considered. Methods of linear least squares are used for the study of analysis of variance and regression. There will be some emphasis on applications to other disciplines. (MATH 0310) 3 hrs. lect./disc. MATH 0315 - Mathematical Models Mathematical Models in the Social and Life Sciences An introduction to the role of mathematics as a modeling tool and an examination of some mathematical models of proven usefulness in problems arising in the social and life sciences. Topics will be selected from the following: axiom systems as used in model building, optimization techniques, linear and integer programming, theory of games, systems of differential equations, computer simulation, stochastic process. Specific models in political science, ecology, sociology, anthropology, psychology, and economics will be explored. (MATH 0200 or waiver) 3 hrs. lect./disc. MATH 0318 - Operations Research ▲ 0335 - Differential Geometry Differential Geometry This course will be an introduction to the concepts of differential geometry. For curves in space, we will discuss arclength parameterizations, Frenet formulas, curvature, and torsion. On surfaces, we will explore the Gauss map, the shape operator, and various types of curvature. We will apply our knowledge to understand geodesics, metrics, and isometries of general geometric spaces. If time permits, we will consider topics such as minimal surfaces, constant curvature spaces, and the Gauss-Bonnet theorem. (MATH 0200 and MATH 0223) 3 hr. lect./disc. MATH 0345 - Combinatorics Combinatorics Combinatorics is the "art of counting." Given a finite set of objects and a set of rules placed upon these objects, we will ask two questions. Does there exist an arrangement of the objects satisfying the rules? If so, how many are there? These are the questions of existence and enumeration. As such, we will study the following combinatorial objects and counting techniques: permutations, combinations, the generalized pigeonhole principle, binomial coefficients, the principle of inclusion-exclusion, recurrence relations, and some basic combinatorial designs. (MATH 0200 or by waiver) 3 hrs. lect./disc. MATH 0402 - Topics In Algebra ▹ Topics in Algebra A further study of topics from MATH 0302. These may include field theory, algebraic extension fields, Galois theory, solvability of polynomial equations by radicals, finite fields, elementary algebraic number theory, solution of the classic geometric construction problems, or the classical groups. (MATH 0302 or by waiver) 3 hrs. lect./disc. 0423 - Topics in Analysis Topics in Analysis In this course we will study advanced topics in real analysis, starting from the fundamentals established in MA401. Topics may include: basic measure theory; Lebesgue measure on Euclidean space; the Lebesgue integral; total variation and absolute continuity; basic functional analysis; fractal measures. (MATH 0323 or by waiver) 3 hrs. lect./disc. MATH 0432 - Elementary Topology ▹ Elementary Topology An introduction to the concepts of topology. Theory of sets, general topological spaces, topology of the real line, continuous functions and homomorphisms, compactness, connectedness, metric spaces, selected topics from the topology of Euclidean spaces including the Jordan curve theorem. (MATH 0122 or by waiver▲▹ MATH 1001 - The Game of Go The Game of Go Go is the most ancient of all East Asian board games and the most challenging in play. It combines the logic of mathematics with the aesthetic appeal of music and art. Despite this, it is a simple game to learn and is enjoyed by approximately 35 million enthusiasts the world over including over 1000 dedicated professionals. The course will involve playing, recording, analyzing, and critiquing our games and learning about its history and the cultures in which it flourishes. We will also read and write about various related Japanese arts and traditions. (Not open to students who have taken FYSE 1175). MATH 1004 - The Shape of Space The Shape of Space We know that the earth we live on is a sphere, but consider the three-dimensional shape of the universe. Does it go on forever, or could it wrap back on itself in some way? In this course we will consider the shape of space. We will learn how topologists and geometers visualize three-dimensional spaces, with a goal of learning about the eight three-dimensional shapes that form the building blocks of all three-dimensional spaces. In the process, we will learn about the celebrated Poincare Conjecture. The ideas we encounter will be deep, but we will study them in a hands-on way. MATH 1005 - Mathematics for Teachers Mathematics for Teachers What mathematical knowledge should elementary and secondary teachers have? We will investigate recent research that describes specialized mathematical content knowledge for teaching. Participants in this course will also strengthen and deepen their own mathematical understanding in a student-centered workshop setting. Readings include Knowing and Teaching Elementary Mathematics by Liping Ma as well as readings from the Journal for Research in Mathematics Education. Anyone interested in mathematics education at any level is welcome. (Not open to students who have taken MATH 1003) MATH 1006 - Heart of Mathematics The Heart of Mathematics Wrestling with the infinite, tiling a floor, predicting the shape of space, imagining the fourth dimension, untangling knots, making pictures of chaos, conducting an election, cutting a cake fairly; all of these topics are part of the landscape of mathematics, although they are largely excluded from the calculus-centric way that the subject is traditionally presented. Following the acclaimed text, The Heart of Mathematics, by Ed Burger and Michael Starbird, we will dive headfirst into ideas that reveal the beauty and diverse character of pure mathematics, employing effective modes of reasoning that are useful far beyond the boundaries of the discipline. MATH 1007 - Combinatorial Gardner The Combinatorial Gardner It has been said that the Mathematical Games column written by Martin Gardner for Scientific American turned a generation of children into mathematicians and mathematicians into children. In this course we will read selections from three decades of this column, focusing on those that deal with combinatorics, the "science of counting," and strive to solve the problems and puzzles given. An example problem that illustrates the science of counting is: what is the maximum number of pieces of pancake (or donut or cheesecake) one can obtain via n linear (or planar) cuts? (MATH 0116 or higher; Not open to students who have taken FYSE 1314). MATH 1038 - Combinatorial Games & Puzzles Combinatorial Games and Puzzles Games and puzzles with a combinatorial flair (based on counting and arrangement) have entertained and frustrated people for millennia. Mathematicians have developed new areas of research and discovered non-trivial mathematics upon examining these amusements. Students will play games (including nim, hex, dots-and-boxes, clobber, and Mastermind®) and be presented with puzzles (including instant insanity and mazes) in an attempt to develop strategy and mathematics during play. Basic notions in graph theory, design theory, combinatorics, and combinatorial game theory will be introduced. Despite the jargon, this course will be accessible to all regardless of background. MATH 1095 - Statistical Computing with R Statistical Computing with R This course offers an intensive introduction to the R statistical programming environment. Students will learn to use a modern programming language that incorporates object-oriented programming. Topics will include data frames, the R environment, the graphics system, probability distributions, descriptive statistics, statistical tests and confidence intervals, regression, ANOVA, tables of counts, simulation, and selected topics in statistical programming. (One course in statistics or one course in computer programming).
To prepare students for the New York Eighth-Grade Test beginning in March 2006. This book provides a complete review of the four strands tested on the New York Eighth-Grade exam: Number Sense & Operations, Algebra, Geometry, and Measure
Book Description: This book can be used individually or as a set with Chenier's Practical Math Dictionary. This book is designed to parallel and enhance any practical math class from general education through college level programs. Many of these math concepts are left out of traditional math books and are relevant to many different trades, occupations, do-it-yourselfers, home owners, home schools, etc. This book includes testing material, economical hands-on projects that simulate industry (use with sticks of wood, chalk lines, flip chart paper, etc.), the answers, and many different unique modules for projects, classroom situations, self-study, industry, etc. All have been proven in the classroom and on-the-job. It's size is 8 1/2" x 10 1/4", has perforated pages and 3 hole drilled. Buyback (Sell directly to one of these merchants and get cash immediately) Currently there are no buyers interested in purchasing this book. While the book has no cash or trade value, you may consider donating it
Product Description The final stop for Saxon's middle school math, Math 8/7 continues teaching students the way they learn best...through incremental development of new material and continual review of the old. Following Math 7/6, concepts such as arithmetic calculation, measurements, geometry and other skills are reviewed, while new concepts such as pre-algebra, ratios, probability and statistics are introduced as preparation for upper level mathematics. Lessons contain a warm-up, introduction to new concepts, lesson practice where the new skill is practiced, and mixed practice, which is comprised of old and new problems. Product Reviews Math 87, Third Edition 4.7 5 34 34 Great Math Series! I purchased Saxon Math 8/7 for my 4th grader after she finished Saxon Math 7/6. We use it as part of our homeschool curriculum. I love that Saxon Math has such great explanations of new concepts, and constant repetition of those concepts throughout the book. The result is that when a child finishes a Saxon book, they thoroughly know all the concepts taught in that book. Great series! March 28, 2013 Best Math text for middle school!! Saxon Math 8/7 is the best I have seen for teaching middle school math students. The lessons flow at a comfortable pace; while the Mixed Practice keeps past lessons fresh in students minds during the introduction of new facts and procedures. September 18, 2012 Solid math program with plenty of explanation and review. Student can use textbook to "self teach". Lots of drill practice which has been very helpful. Very little teacher prep required! September 7, 2012 I LOVE the saxon program I am very happy with that. However it took over 2 weeks for it to get to my house. I should have bought it from amazon. And the book was not manufactured right when it was binded the pages were doubled over. And since I wanted this program before my son started school this year I carefully fixed it myself with some scissor's and tape. I am more unhappy with the time it took to get to me however that was riduclous. July 27, 2012
Math I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education. Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin's Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). The list is updated almost on a daily basis, so, if you want to bookmark this page, use the button in the upper right corner. Click here for a pdf copy of the entire book, or get the chapters individually below. I: Fundamentals II: How to Prove Conditional Statements III: More on Proof IV: Relations, Functions and Cardinality This web site is hosted in part by the Software and Systems Division , Information Technology Laboratory . This is a dictionary of algorithms, algorithmic techniques, data structures, archetypal problems, and related definitions. Algorithms include common functions, such as Ackermann's function . Problems include traveling salesman and Byzantine generals . Some entries have links to implementations and more information. Index pages list entries by area and by type . Algebra Help Math Sheet This algebra reference sheet contains the following algebraic operations addition, subtraction, multiplication, and division. It also contains associative, commutative, and distributive properties. There are example of arithmetic operations as well as properties of exponents, radicals, inequalities, absolute values, complex numbers, logarithms, and polynomials. This sheet also contains many common factoring examples.
Beginning Algebra - 2nd edition ISBN13:978-0073312675 ISBN10: 0073312673 This edition has also been released as: ISBN13: 978-0073028712 ISBN10: 0073028711 Summary: New Features NEW! Problem Recognition Exercises Developmental math students are sometimes conditioned into algorithmic thinking to the point where they want to automatically apply various algorithms to solve problems, whether it is meaningful or not. These exercises were built to decondition students from falling into that trap. Carefully crafted by the authors, the exercises focus on the situations where...show more students most often get "mixed-up." Working the Problem Recognition Exercises, students become conditioned to Stop, Think, and Recall what method is most appropriate to solve each problem in the set. NEW! Skill Practice exercises follow immediately after the examples in the text. Answers are provided so students can check their work. By utilizing these exercises, students can test their understanding of the various problem-solving techniques given in the examples. NEW! The section-ending Practice Exercises are newly revised, with even more core exercises appearing per exercise set. Many of the exercises are grouped by section objective, so students can refer back to content within the section if they need some assistance in completing homework. Review Problems appear at the beginning of most Practice Exercise Sets to help students improve their study habits and to improve their long-term retention of concepts previously introduced. NEW! Mixed Exercises are found in many of the Practice Exercise sets. The Mixed Exercises contain no references to objectives. In this way, students are expected to work independently without prompting --which is representative of how they would work through a test or exam. NEW! Study Skills Exercises appear at the beginning of the Practice Exercises, where appropriate. They are designed to help students learn techniques to improve their study habits including exam preparation, note taking, and time management. NEW! The Chapter Openers now include a variety of puzzles that may be used to motivate lecture. Each puzzle is based on key vocabulary terms or concepts that are introduced in the chapter. 20067.75 +$3.99 s/h Acceptable BookSleuth Danville, CA Fast Shipping ! Used books may not include access codes, CDs or other supplements. $16.99 +$3.99 s/h Acceptable Penntext Downingtown, PA Ships within 24 hrs of your order. Open Mon - Fri. May have some notes/highlighting, slightly worn covers, general wear/tear
Book Description: The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.
MathWare Ltd. Software and books for algebra, geometry and calculus: Derive, MathPert, Scientific Notebook, Cyclone; books for use with the TI Graphing Calculators, and a book and CD for Mathematica. Also an Interactive Math Dictionary on CD-ROM with biographical entries, ...more>> The MathWorks, Inc. From the makers of MATLAB, a technical computing environment for high-performance numeric computation and visualization, and Simulink, an interactive environment for modeling, analyzing, and simulating a wide variety of dynamic systems, including discrete,MathZapper - Russel Timmins Libraries of smartboard video tutorials for the student of mathematics, organized into GCSE, IGCSE, A Level, and International Baccalaureate (IB). ...more>> Multidimensional Analysis - George W. Hart A brief introduction to Multidimensional Analysis, a generalization of linear algebra that incorporates ideas from dimensional analysis. The central idea is that vectors and matrices as used in science and engineering can be thought of as having elements ...more>> My World of Linear Algebra - Thomas S. Shores Linear algebra resources, including Applied Linear Algebra and Matrix Analysis, a textbook for an introductory linear algebra course; and tutorial notebooks in Maple and Mathematica, some of which are the basis for linear algebra projects. ...more>> New Mathwright Library - James White; Bluejay Lispware An Internet-based library of interactive workbooks on topics commonly encountered in undergraduate mathematics, from college algebra and precalculus through multivariable calculus, differential equations, and mathematical modelling. Workbooks, together ...more>> OpsResearch - DRA Systems A collection of Java classes for developing operations research programs and other mathematical applications. The site includes documentation and tutorials, and software download is free. Also features a bookstore and related links. ...more>> Paul Nevai Paul Nevai researches orthogonal polynomials and approximation theory. Many of his articles are available here in .dvi format or as html documents. ...more>> PC Calc Calculator program for Windows 2000/XP. Offers calculations, graphing, matrices, steps in solving equations, and more. Features listed and a free 30-day demo is available on the site. ...more>> Pearson Math Stats - Pearson Learning Pearson Learning brings you free activities for students to use with Wolfram\|Alpha. The questions teach input and command syntax or reinforce answer interpretation or the concepts of intermediate algebra. ...more>> Penn State University Mathematics Department Course home pages and instructional material; seminars, colloquia, and conferences; and subject area pages for Penn State research centers, with preprints and links to other resources on the Web: Algebra and Number Theory; Dynamical Systems; Mathematical ...more>> Personal Algebra Tutor (PAT) - CyberEd, Inc. An algebra problem solving program for high school and college algebra. Students can enter a wide variety of algebra problems and PAT will solve them step-by-step, with detailed explanations. PAT solves, simplifies, graphs, etc. Solutions with side-by-side ...more>>
Vector Calculus, CourseSmart eTextbook, 4th Edition Description For undergraduate courses in Multivariable Calculus. Vector Calculus, Fourth Edition, uses the language and notation of vectors and matrices to teach multivariable calculus. It is ideal for students with a solid background in single-variable calculus who are capable of thinking in more general terms about the topics in the course. This text is distinguished from others by its readable narrative, numerous figures, thoughtfully selected examples, and carefully crafted exercise sets. Colley includes not only basic and advanced exercises, but also mid-level exercises that form a necessary bridge between the two. Instructors will appreciate the mathematical precision, level of rigor, and full selection of topics. Table of Contents 1. Vectors 1.1 Vectors in Two and Three Dimensions 1.2 More About Vectors 1.3 The Dot Product 1.4 The Cross Product 1.5 Equations for Planes; Distance Problems 1.6 Some n-dimensional Geometry 1.7 New Coordinate Systems True/False Exercises for Chapter 1 Miscellaneous Exercises for Chapter 1 2. Differentiation in Several Variables 2.1 Functions of Several Variables;Graphing Surfaces 2.2 Limits 2.3 The Derivative 2.4 Properties; Higher-order Partial Derivatives 2.5 The Chain Rule 2.6 Directional Derivatives and the Gradient 2.7 Newton's Method (optional) True/False Exercises for Chapter 2 Miscellaneous Exercises for Chapter 2 3. Vector-Valued Functions 3.1 Parametrized Curves and Kepler's Laws 3.2 Arclength and Differential Geometry 3.3 Vector Fields: An Introduction 3.4 Gradient, Divergence, Curl, and the Del Operator True/False Exercises for Chapter 3 Miscellaneous Exercises for Chapter 3 4. Maxima and Minima in Several Variables 4.1 Differentials and Taylor's Theorem 4.2 Extrema of Functions 4.3 Lagrange Multipliers 4.4 Some Applications of Extrema True/False Exercises for Chapter 4 Miscellaneous Exercises for Chapter 4 5. Multiple Integration 5.1 Introduction: Areas and Volumes 5.2 Double Integrals 5.3 Changing the Order of Integration 5.4 Triple Integrals 5.5 Change of Variables 5.6 Applications of Integration 5.7 Numerical Approximations of Multiple Integrals (optional) True/False Exercises for Chapter 5 Miscellaneous Exercises for Chapter 5 6. Line Integrals 6.1 Scalar and Vector Line Integrals 6.2 Green's Theorem 6.3 Conservative Vector Fields True/False Exercises for Chapter 6 Miscellaneous Exercises for Chapter 6 7. Surface Integrals and Vector Analysis 7.1 Parametrized Surfaces 7.2 Surface Integrals 7.3 Stokes's and Gauss's Theorems 7.4 Further Vector Analysis; Maxwell's Equations True/False Exercises for Chapter 7 Miscellaneous Exercises for Chapter 7 8. Vector Analysis in Higher Dimensions 8.1 An Introduction to Differential Forms 8.2 Manifolds and Integrals of k-forms 8.3 The Generalized Stokes's Theorem True/False Exercises for Chapter 8 Miscellaneous Exercises for Chapter 8 Suggestions for Further Reading Answers to Selected Exercises
Synopsis Need some serious help solving equations? Totally frustrated by polynomials, parabolas and that dreaded little x? THE MATH DUDE IS HERE TO HELP!Jason Marshall, popular podcast host known to his fans as The Math Dude, understands that algebra can cause agony. But he's determined to show you that you can solve those confusing, scream-inducing math problems--and it won't be as hard as you think! Jason kicks things off with a basic-training boot camp to help you review the essential math you'll need to truly "get" algebra. The basics covered, you'll be ready to tackle the concepts that make up the core of algebra. You'll get step-by-step instructions and tutorials to help you finally understand the problems that stump you the most, including loads of tips on: Working with fractions, decimals, exponents, radicals, functions, polynomials and more Solving all kinds of equations, from basic linear problems to the quadratic formula and beyond Using graphs and understanding why they make solving complex algebra problems easier Learning algebra doesn't have to be a form of torture, and with The Math Dude's Quick and Dirty Guide to Algebra, it won't be. Packed with tons of fun features including "secret agent math-libs," and "math brain games," and full of quick and dirty tips that get right to the point, this book will have even the biggest math-o-phobes basking in a-ha moments and truly understanding algebra in a way that will stick for years (and tests) to come. Whether you're a student who needs help passing algebra class, a parent who wants to help their child meet that goal, or somebody who wants to brush up on their algebra skills for a new job or maybe even just for fun, look no further. Sit back, relax, and let this guide take you on a trip through the world of algebra
Courses in Mathematics and Statistics Honours timetable MT2001 MATHEMATICS This is a core second level mathematics module which must be taken by anyone planning to pursue Honours level material in Mathematics, Statistics or Physics. Aims To extend the knowledge and skills gained by students in the module MT1002 in preparation for Honours study. In particular - to enhance their skills and understanding of calculus to include functions of several variables; - to develop an appreciation of basic ideas in Linear Algebra. Objectives By the end of the course a student should be able to demonstrate: - how to determine the Taylor series of a function and estimate the error associated with a Taylor polynomial; - an understanding of the concepts of continuity and differentiability of functions of several variables; partial differentiation and an appreciation of the difference between partial and ordinary differentiation; - how to determine the local behaviour of surfaces at stationary points; - how to determine integrals over areas and volumes (and when necessary use different co-ordinate systems including cylindrical and spherical polars); - an ability to work with the fundamental concepts of vector space theory; - an ability to manipulate matrices in order to find inverses and determinants; - how to determine the eigenvalues and eigenvectors of a square matrix (and diagonalisation when appropriate); - how to obtain a Fourier series representation of simple function and interpret its properties.
Precalculus 9780321531988 ISBN: 0321531981 Edition: 4 Pub Date: 2008 Publisher: Pearson Summary: - By Judith A. Penna - Contains keysroke level instruction for the Texas Instruments TI-83 Plus, TI-84 Plus, and TI-89 - Teaches students how to use a graphing calculator using actual examples and exercises from the main text - Mirrors the topic order to the main text to provide a just-in-time mode of instruction - Automatically ships with each new copy of the text
Differential Geometry, Gauge Theories, and Gravity Using a self-contained and concise treatment of modern differential geometry, this book will be of great interest to graduate students and researchers in applied mathematics or theoretical physics working in field theory, particle physics, or general relativity. The authors begin with an elementary presentation of differential forms. This formalism is then used to discuss physical examples, followed by a generalization of the mathematics and physics presented to manifolds. The book emphasizes the applications of differential geometry concerned with gauge theories in particle physics and general relativity. Topics discussed include Yang-Mills theories, gravity, fiber bundles, monopoles, instantons, spinors, and anomalies
MATH 111 – Mathematics for Educators I Description: This course teaches students to communicate and represent mathematical ideas, how to solve problems, and how to reason mathematically. Material covered includes operations and their properties, sets, counting, patterns, and algebra.
Learning Matlab - Essentials Skills (2012) - FREE SHIPMENT The advertisement posted in this page is already inactive and it is possible that the details here are already invalid. Content posted in this page is only provided for reference and does not constitute fact. Please be guided accordingly. In this video series 7 Hour, Jason Gibson Teaches you How to use this Software package with Step-by Step video tutorials. The lessons begin with Becoming familiar with the user interface and Understanding How to interact with Matlab. Then you'll learn about variables, Functions, and How to Perform Basic Calculations. Next, Jason Will guide you in Learning How to do Algebra, trigonometry, and Calculus computations Both numerically and symbolically. The course wraps up with basic plotting in Matlab. Take the mystery out of Matlab and improve your productivity with the software immediately! 1. Introduction Sect 1: Overview of the User Interface - Part 1 Sect 2: Overview of the User Interface - Part 2 Sect 3: Overview of the User Interface - Part 3 Sect 4: Using the Help Menus 2. Basic Calculations Sect 5: Basic Arithmetic and Order of Operations Sect 6: Exponents and Scientific Notation Sect 7: Working with Fractions and the Symbolic Math Toolbox - Part 1 Sect 8: Working with Fractions and the Symbolic Math Toolbox - Part 2
This course introduces the student to the study of linear algebra. Practically every modern technology relies on linear algebra to simplify the computations required for internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. This free course may be completed online at any time. See course site for detailed overview and learning outcomes. (Computer Science 105, Mathematics 211
Price Specifications StudyWorks! Middle School Deluxe Math delivers colorful, interactive lessons, animations and activities that reinforce all the fundamentals needed before entering high school. This package delivers over 120 complete lessons, covering all the major topics in sixth, seventh and eighth grade math - in greater depth than any other package! Each unit is comprehensive, interactive and motivational so that middle school students can absorb key concepts step - by - step, master each and move on. Middle School Deluxe Math covers what middle school students need for success on the math portion of almost every state's standardized exams, with correlations to the majority of state curriculum standards. ??Designed to be always up - to - date and useful every day - because it includes unlimited access to deluxe Web services at StudyWorksOnline! This includes online testing, live online homework help, and integrated access to rich Web resources and learning activities.
An entire online text for Precalc from the Univeristy of Houston. An entire online text for Precalc from the Univeristy of Houston. It's the best free source I've seen. It's the project of grad students and is a comprehensive online text that has everything you'd expect from a textbook. And on top of that, there are streaming lectures.
Summary PREALGEBRA, 5/e, is a consumable worktext that helps users 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 book. This book's strength lies in the Aufmann Interactive Method, which enables users 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 users. 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 books ideal for users: 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 exercises. Table of Contents AIM for Success: Getting Started Whole Numbers Introduction to Whole Numbers Addition and Subtraction of Whole Numbers Multiplication and Division of Whole Numbers Solving Equations with Whole Numbers The Order of Operations Agreement Focus on Problem Solving: Questions to Ask Projects and Group Activities: Surveys Applications of Patterns in Mathematics Salary Calculator Subtraction Squares Integers Introduction to Integers Addition and Subtraction of Integers Multiplication and Division of Integers Solving Equations with Integers The Order of Operations Agreement Focus on Problem Solving: Drawing Diagrams Projects and Group Activities: Multiplication of Integers Closure Fractions Least Common Multiple and Greatest Common Factor Introduction to Fractions Multiplication and Division of Fractions Addition and Subtraction of Fractions Solving Equations with Fractions Exponents, Complex Fractions, and The Order of Operations Focus on Problem Solving: Common Knowledge Projects and Group Activities: Music Using Patterns in Experimentation Decimals And Real Numbers Introduction to Decimals Addition and Subtraction of Decimals Multiplication and Division of Decimals Solving Equations with Decimals Radical Expressions Real Numbers Focus on Problem Solving: From Concrete to Abstract Projects and Group Activities: Customer Billing Variable Expressions Properties of Real Numbers Variable Expressions in Simplest Form Addition and Subtraction of Polynomials Multiplication of Monomials Multiplication of Polynomials Division of Monomials Verbal Expressions and Variable Expressions Focus on Problem Solving: Look for a Pattern Projects and Group Activities: Multiplication of Polynomials First-Degree Equations Equations of the Form x + a = b and ax = b Equations of the Form ax + b = c. General First-Degree Equations Translating Sentences into Equations The Rectangular Coordinate System Graphs of Straight Lines Focus on Problem Solving: Making a Table Projects in Mathematics: Collecting, Organizing, and Analyzing Data Measurement And Proportion The Metric System of Measurement Ratios and Rates The U.S. Customary System of Measurement Proportion Direct and Inverse Variation Focus on Problem Solving: Relevant Information Projects in Mathematics: Earned Run Average Percent Percent The Basic Percent Equation Percent Increase and Percent Decrease Markup and Discount Simple IntereSt. Focus on Problem Solving: Using a Calculator as a Problem-Solving Tool
Humble Trigonometry almost guarantee that you will have ?Aha! So that?s how it works!? moments as algebra becomes more familiar and understandable. Algebra 2 builds on the foundation of algebra 1, especially in the ongoing application of the basic concepts of variables, solving equations, and manipulations such as factoring. ...In the course of tutoring and counseling, study skills are just part of the package. This includes 'how to' read for success. Many people (young and old) do not know how to 'critically' read information to precisely answer questions about a particular passage.A small amount of elementary word problems are also included. Precalculus reveiws algebra 1 and algebra 2. It also covers a bit of trigonometry because calculus uses trigonometric functions as part of the basic curriculum.
Graphs are mathematical entities whose theory facilitates the discussion of the relationships between the elements of a set. Introduced about 1870 by the pioneers of combinatorics who were then still known as geometers, graphs have recently been recognized as the most adaptable tools for certain organizational problems. After first being used in operations research, graphs have now been introduced into information theory where they have proved to ba of use in many different areas. ... This book stresses the connection between, and the applications of, design theory to graphs and codes. Beginning with a brief introduction to design theory and the necessary background, the book also provides relevant topics for discussion from the theory of graphs and codes. ...Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails. As graph theory continues its explosive growth, conjectures are proved and new theorems formed. The techniques involved, which have applications in a broad spectrum of mathematics, ranging from analysis to operations research, have become more sophisticated if not more manageable. This new edition, therefore, includes new theorems (e.g. the Perfect Graph Theorem, due to Lovasz) as well as new proofs of classical results. A number of sections have been significantly revised. ... Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book's primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted.The Graphs and Models series by Bittinger, Beecher, Ellenbogen, and Penna is known for helping students "see the math" through its focus on visualization and technology. These books continue to maintain the features that have helped students succeed for years: focus on functions, visual emphasis, side-by-side algebraic and graphical solutions, and real-data applications. This book is concerned with the relations between graphs, error-correcting codes and designs, in particular how techniques of graph theory and coding theory can give information about designs. A major revision and expansion of a previous volume in this series, this *** includes many examples and new results as well as improved treatments of older material. So that non-specialists will find the treatment accessible the authors have included short introductions to the three main topics. This book will be welcomed by graduate students and research mathematicians and be valuable for advanced courses in finite combinatorics. ...
Placement in Mathematics A First-year student without Advanced Placement calculus credit or previous college mathematics credits will take one or more of the following mathematics courses as determined by the placement criteria explained below: UCOR 1200 Mathematics and Quantitative Reasoning MATH 110 Functions and Algebraic Methods (fall, winter, spring) MATH 118 College Algebra for Business (fall, winter, spring) MATH 120 Precalculus: Algebra (fall, winter, spring) MATH 121 Precalculus: Trigonometry (fall, winter, spring) MATH 130 Calculus for Business (fall, winter, spring) MATH 131 Calculus for Life Sciences (spring, winter) MATH 134 Calculus I (fall, winter, spring) MATH 141 Statistics for Life Sciences (fall, spring) The course in which you will enroll depends upon your intended major as well as your SAT or ACT mathematics score, or Mathematics Placement Exam score. To determine math course selection: use the SAT or ACT column to find the range that includes student score; check first column for eligible courses; then, based on major and personal preference, determine which course to take. [Note: if the SAT or ACT test has been taken more than once, use the highest math score.] Math Courses SAT score ACT score S.U. Math Placement Scores UCOR 1200, 110 450 to 530 18 to 22 Algebra score 4 to 14 118, 120 540 to 630 23 to 27 Algebra score 15 to 24 121, 130 620 or higher 27 or higher Algebra score 23 or higher 131 620 or higher and Trig score 4 or higher* 27 or higher and Trig score 4 or higher* Algebra score 23 or higher and Trig score 4 or higher* 134 640 or higher and Trig score 6 or higher* 28 or higher and Trig score 6 or higher* Algebra score 25 or higher and Trig score 6 or higher* (*Corequisite for MATH 134 and 131 is MATH 121 or indicated score on trig. placement exam) Mathematics Department Placement Exam You are placed in mathematics based on your SAT or ACT score. If you do not like where you are placed based on this score, you can take the SU placement exam. Students in majors in the College of Science and Engineering, Albers School of Business, Sports and Exercise Science and BS Criminal Justice with a specialization in Forensic Science are best served by placing into MATH 118/120 or above. Math Placement Exams will be offered at Summer in Seattle sessions and welcome week. Note: There is no charge to take the Mathematics Department Placement Exam.
Graph paper Graph paper is paper that is printed with fine lines making up a grid. It is typically used for drawing diagrams, with the lines being used as guides for the drawing. It is commonly found in engineering settings for quick drawings and sketches. It can also be used for plotting mathematical functions manually. Some specialized forms of graph paper have logarithmic scales or allow plotting in polar coordinates. Quad paper is a common form of graph paper in imperial unit countries where the grid is a quarter inch apart printed in light blue and right to the edge of the paper. 3D graph paper is also available, but fairly rare. It uses a series of three guidelines forming a 60-degree grid, so that the paper is covered with small triangles. It is used for drawing isometric views. Graph paper is typically sold in pads of 100 sheets with a cardboard backing to draw on. It is also sold in hardcover books for use as engineering notebooks. Fact-index.com financially supports the Wikimedia Foundation. Displaying this page does not burden Wikipedia hardware resources. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.
Site Navigation EDMA502 Mathematics Curriculum Studies (Pakistan) 10 cp This unit will focus on developing a practical understanding of content, methods and strategies appropriate to the teaching of Mathematics in secondary schools in Pakistan. It includes topics related to the syllabi for Years 8,9,10 (Matric) and Cambridge O levels. This unit will develop a critical and reflective approach towards the teaching of Mathematics.
Displays lines and surfaces defined algebraically in 3D space in many forms, including z=f(x,y), cylindrical polar coordinates, and parametric definitions with one (giving a line) and two (surface) parameters. View controls move the viewpoint through 3D space, using keyboard and mouse. There are options to display a surface, a mesh or a combination. The number of 'steps' on each edge (level of detail) can be controlled. Displays graphs of algebraic functions in a variety of forms. These include polar and cartesian co-ordinates, parametric and implicit functions. A wide range of functions are built-in, from simple trig and hyperbolic functions to things such as the ceil and gamma functions. On-screen HTML help is bulit-inGraphSight is a feature-rich comprehensive 2D math graphing utility with easy navigation, perfectly suited for use by high-school an college math students. The program is capable of plotting Cartesian, polar, table defined, as well as specialty graphs. Importantly, it features a simple data and formula input format, making it very practical for solving in-class and homework problems. The program comes with customizable Axis options, too
Maths To support students enrolled in maths and statistics courses, The Learning Centre provides a range of activities associated with mathematics and learning skills. Semester 2, 2013 workshops Success in Maths for Statistics (SIMS) Topics include formulas, arithmetic, calculator, basic statistics, graphing. See the workshop program (PDF68kb). Complete the online readiness testing (UConnect username and password required) or complete the first CMA on your STA2300 course webpage, to self assess your knowledge and determine whether you need to attend this workshop. Completing the first CMA is part of your first assignment in STA2300. You will need to bring a copy of the SIMS workbook (PDF 1.34mb) with you to the workshop. If you are unable to attend a workshop, the workbook will still be useful for your data analysis studies.
At Fresh we love to share helpful tools and here's a website we know you're going to love playing around with. Wolfram Alpha is basically a website (they call it an engine) for computing answers to a wide range of questions and it displays or provides the knowledge and/or answer in a variety of ways. It essentially works by using its vast store of expert-level knowledge and algorithms to automatically answer questions, do analysis, and generate reports. How cool is that?! Whether you're at school and have a tricky equation you want to solve AND see the answer displayed in different graph forms or whether you're just trying to calculate the amount of interest you're going to pay on a mortgage or loan, this resource will soon have you addicted to putting in all kinds of information!
Selected topics of current interest in mathematics are researched and presented. Students and faculty share in the presentations. Prerequisite: consent of instructor and junior or senior status. Text:Dunham, William. Journey through Genius. (first published by John Wiley & Sons, 1990). Course Goals: 1.This course is something of a "capstone" course for mathematics majors and as such gives students the opportunity to read and present a variety of mathematical topics. The major goals here are to absorb by reading mathematical content and to prepare presentations for the group, in short, to learn to do "mathematical research" and to make mathematical presentations. 2.This course also is the one place in the curriculum where students encounter in a formal way the history of mathematics. Even though this is good for any mathematics major, it is required by the DPI for any math-ed majors, and this course serves the purpose of satisfying that requirement. Course Objectives: The text includes chapters on the following topics. They cover a wide range of history and mathematical content: 1.Hippocrates and the Quadrature of the Lune 2.Euclid's Proof of the Pythagorean Theorem. 3.Euclid and the Infinitude of Primes 4.Archimedes' Determination of Circular Area 5.Heron's Formula for Triangular Area 6.Cardano and the Solution of the Cubic 7.Isaac Newton and the Binomial Theorem 8.The Bernoullis and the Harmonic Series 9.Leonard Euler and His Infinite Sums 10.Euler's Number Theory 11.Cantor and the Non-denumerability of the Continuum 12.Cantor and the Transfinite Realm In addition to going through these topics we will try to consider a few other topics. Course Procedures: At the first meeting we will assign chapters to each member of the class and then the weekly activity will be a presentation by a student of the material in the assigned chapter. You should present the material as if you were giving a "lecture", using appropriate blackboard/overhead skills and being prepared to answer questions anyone might ask. Grades will be determined by your level of participation and the quality of your presentations; you want to show that you have read the material with understanding and that you can explain it in detail.
Nature tries to minimize the surface area of a soap film through the action of surface tension. The process can be understood mathematically by using differential geometry, complex analysis, and the calculus of variations. This book employs ingredients from each of these subjects to tell the mathematical story of soap films. The text is fully self-contained, bringing together a mixture of types of mathematics along with a bit of the physics that underlies the subject. The development is primarily from first principles, requiring no advanced background material from either mathematics or physics. Through the Maple® applications, the reader is given tools for creating the shapes that are being studied. Thus, you can "see" a fluid rising up an inclined plane, create minimal surfaces from complex variables data, and investigate the "true" shape of a balloon. Oprea also includes descriptions of experiments and photographs that let you see real soap films on wire frames. The theory of minimal surfaces is a beautiful subject, which naturally introduces the reader to fascinating, yet accessible, topics in mathematics. Oprea's presentation is rich with examples, explanations, and applications.
30 Mathematics Lessons Using the TI-15 Helps younger learners grasp mathematical concepts and skills with lessons that integrate calculator use. This book provides step-by-step mathematics lessons that incorporate the use of the TI-15 calculator throughout the learning process. The lessons present mathematics in a real-world context and cover each of the five strands, including numbers and operations, geometry, algebra, measurement, and data analysis and probability. Teacher Resource CD Included
Synopses & Reviews Publisher Comments: This edition of Swokowski's text is truly as its name implies: a classic. Groundbreaking in every way when first published, this book is a simple, straightforward, direct calculus text. It's popularity is directly due to its broad use of applications, the easy-to-understand writing style, and the wealth of examples and exercises which reinforce conceptualization of the subject matter. The author wrote this text with three objectives in mind. The first was to make the book more student-oriented by expanding discussions and providing more examples and figures to help clarify concepts. To further aid students, guidelines for solving problems were added in many sections of the text. The second objective was to stress the usefulness of calculus by means of modern applications of derivatives and integrals. The third objective, to make the text as accurate and error-free as possible, was accomplished by a careful examination of the exposition, combined with a thorough checking of each example and exercise
Chapter 1: Numbers and Points The word number typically refers to a quantity, which may be definite as in ``the number of people in line'', indefinite as in ``a number of students took the class''; the dictionary definition typically lists over a dozen common meanings. In this course, a variety of types of numbers will be studied, including the numbers in each of the following sets: The positive integers or natural numbers : 1, 2, 3, .... The integers which includes the positive integers, zero, and the negative integers: -1, -2, -3, .... The rational numbers is the set of numbers which can be written as m/n where m and n are integers with n non-zero. Rational numbers can be expressed as finite or repeating decimals. The real numbers include the rational numbers as well as irrational numbers such as and . Every real number has a possibly infinite decimal expansion. The complex numbers which include the real numbers as well as the . Every complex number can be expressed in the form where and are real numbers. On each of these sets, is defined the binary operations addition and multiplication. For each ordered pair of numbers from one of these sets, there are well defined numbers and called the sum and the product of and in the same set. One expresses this property by saying that the sets are closed under addition and multiplication. These operations have simple geometric interpretations. For example, if two line segments are of length x and y respectively, then connecting the two together end-to-end yields a line segment of length x + y. Similarly, the area of a rectangle with length x and width y is precisely xy. By repeatedly combining numbers via addition and multiplication, one can make complicated expressions such as . By substituting various values of x and y into this expression, one gets numerical values for the expression. One finds that regardless of which values of x and y you use, the numerical value is the same as that obtained by the expression . The process of verifying this is one of the skills that you have already mastered in earlier algebra courses. The idea is that you can use a number of properties of numbers to successively simplify the first expression until you get to the second one. Amongst these properties are: Addition and Multiplication are commutative: a + b = b + a and ab = ba. These properties are referred to as the commutative, associative, and distributive laws. Before going on, we should clear up some ambiguity. We said that was obtained by a succession of additions and multiplications. This is true, but there are many different ways of doing this, e.g. after one gets the value of , , and , one could add the third to the sum of the first two or add the first to the sum of the last two. Of course, the result would be the same, and it is the associative law for addition that guarantees this. For this reason, one typically does not even bother to specify the order by adding in parentheses. Similarly, one didn't put parentheses to indicate the order of evaluation of the product of the three factors in . Another ambiguity occurs in the expression . Does this mean add the product of 4 and x to y or does it mean multiply 4 times the sum of x and y? This is more serious because the two ways of evaluating the expression give different answers. This is resolved by the rules for order of evaluation of expressions. In this case, the rules say that you evaluate multiplications before additions. So, the meaning of the expression is and not . The rules for order of evaluation will be stated in detail at the end of the next section. Exercise 1.1: Using the geometric interpretation of addition and multiplication given above for positive real numbers, give geometric interpretations of the commutative, associative, and distributive laws. For example, the commutative law for addition might be illustrated by saying that if you join a line segment of length x end-to-end with a line segment of length y, then the length of the result is the same whether you measure it from one end or from the other. Here are two more important properties of the rational numbers , the real numbers , and the complex numbers : There is a number 0 such that a + 0 = a for all numbers a and there is a number 1 not equal to 0 such that for all numbers a. Any such numbers are called additive or multiplicative identities. For any number , there is a number such that . Such a number is called an additive inverse of . Similarly, if is number other than 0, then there is a number called a multiplicative inverse or reciprocal of such that . The numbers 0 and 1 are unique: Proposition 1.1: There is at most one additive identity and at most one multiplicative identity. Proof: Suppose and are two additive identities. Letting play the role of in the definition of an additive identity , we have . Similarly, one has by letting play the role of in the definition of being an additive identity. Because of the commutative law, one has The same argument shows that multiplicative identities are also unique. We can define subtraction by and division by if b is non-zero. Alternative ways of denoting division are: . Exercise 1.2: Which of the five sets of numbers , , , , and are closed under subtraction? Which are closed under division? Are subtraction and/or division commutative and/or associative? Does addition distribute over multiplication? Does division distribute over addition? Do the natural numbers and/or the integers have additive and/or multiplicative identities? What about inverses? Any set closed under addition and multiplication which satisfies the commutative, associative, and distributive laws as well as has identities and inverses is referred to as a field. Here is the formal definition: Definition 1.1: A field is a set F together with with two binary operations '+' and '·' called addition and multiplication which satisfy the following conditions: F is closed under addition and multiplication. Addition and multiplication are commutative. This means that and for all a and b in F. Addition and multiplication are associative. This means that and for all a, b, and c in F. Multiplication distributes over addition. This means that for all a, b, and c in F. (Identities) There is an element 0 in F such that a + 0 = a for all . There is an element 1 in F different from 0 such that for all . (Inverses) For every in F, there is an element denoted in F such that . For every in F other than 0, there is an element denoted in F such that . Remark: If all the conditions except for the requirement that multiplicative inverse exist are true, then F is called a commutative ring with identity. Many of the common rules of algebra follow from the fact that one is working in a field. One very important example is: Proposition 1.2: Let F be a field. If is in F, then . If and in F satisfy , then either or are zero. Proof: Let be an arbitrary element of F. Then . Let b be the additive inverse of . Then applying it to the last equation, we get Suppose ab = 0. If a is zero, there is nothing more to prove. On the other hand, if , then a has a multiplicative inverse c and so . Corollary 1.1: If a is an element of a field F, then -a = (-1)a. Proof: . Corollary 1.2: 0 has no multiplicative inverse. Proof: If this were false, then one would have But, Combining these, we conclude that . But this contradicts the property that 0 and 1 are not equal. Because the field properties imply most of the rules of algebra, they help us understand why these rules are true. For example, you know that the product of two negative numbers is always positive. But why should this be true? Why not make a new rule, e.g. that the product of two negative numbers should be negative. The answer is that you could do this, but then one of the field properties would no longer be true. Exercise 1.3: Show that in a field, one has (-a)(-b) = ab. Hint: Fill in reasons for each of the following steps: Here are some more common rules of algebra: Proposition 1.3: Let F be a field containing a, b, c, and d where b and d are non-zero. Then if and only if If c is also non-zero, then Proof: (i) By the definition of multiplicative inverse, one must show that Starting from the left hand side, one can simplify the expression as shown here. Each of the above steps involves one of the fundamental properties of fields; make sure that you can justify each step with the appropriate property. (ii) This like any "if and only if" statement is really two assertions: If then If then To prove assertion (a), assume that To see that , multiply both sides of by to get . Now simplify each side: and Combining results, one gets (As before, make sure that you can justify each step.) Now, let's show assertion (b). Assuming that and that both and are non-zero, we know that both and have multiplicative inverses. So, we need only multiply both sides of by and simplify. Here are the details: and (Label each line with the property that justifies the step.) (iii) Simplify starting from the left hand side: Did you justify each step? (iv) One has: (v) This is the usual rule for adding fractions. Notice that property (iv) allows one to convert the fractions so that they have a common denominator . Here are the detailed steps: (vi) This is the rule for simplifying fractions of fractions. It is the same as Using property (ii), this is the same as showing But, we can see this by simplifying each side: and This completes the proof of Proposition 1.3. Exercise 1.3: (i) Redefine addition on the real numbers: With the new definition of addition and the usual multiplication, do the real numbers still form a field? Which of the field properties are true and which are not. We defined multiplication by making be the area of a rectangle with sides of length m and n respectively. Suppose that we based things on the area of a triangle with given base and height instead. With the new definition of multiplication (and the usual definition of addition), do the real numbers form a field? What is the multiplicative identity? 1.2.1 Mathematical Induction The positive integers are the numbers 1, 2, 3, .... Every positive integer n has a successor n' = n + 1. Every positive integer n except 1 is the successor of a unique positive integer, viz. n - 1. The most important property of the set of positive integers is Principle of Mathematical Induction: If is any property of the positive number such that is true. If is true for a positive integer , then is true for the successor of Then is true for all positive integers This principle can be used for both definitions as well as for theorems. For example, one can use it to define the usual operations on the positive integers using only the successor operation: Let be the property of the positive integer n that m + n is defined. Then (i) assures us that is true and (ii) assures us that is true of n' if it is true of n. So, is true for all positive integers n. Similarly, one sees that we can define multiplication by: Definition 2.2: The product of two positive integers and is defined by: for all positive integers If is defined, then One can also use the principle of mathematical induction to prove theorems. For example, we can verify that our addition operation is associative: Proposition 2.1: Addition is associative. Proof: Let be the property of the positive integer n that (a+ b) + n = a + (b + n) for all positive integers a and b. In the case where n = 1, is true because (Why is each step true?) Assuming that is true for n, let's show it for n'. One has So, must be true for all positive integers Proposition 2.2: Addition is commutative. Proof: Let be the property of the positive integer n that a + n = n + a for all positive integers a. We will show that is true by using the principle of mathematical induction. Let be the property of the positive integer a that a + 1 = 1 + a. We know that is true because 1 + 1 = 1 + 1. If is true for some positive integer a, then and so is true for all positive integers and so is true. Now suppose that is true for n. Then and so is true for all positive integers n. (Did you justify each step?) Proposition 2.3Multiplication distributes over addition. Proof: Let be the property of the positive integer n that a(b + n) = ab + an for all positive integers a and b. Then is true. because Suppose that is true for n. Then it is also true for n' because and so is true for all positive integers n. Exercise 2.1:Use induction to prove the following results: Multiplication is associative. (a + b)c = ac + bc for all positive integers a, b, and c. Multiplication is commutative. Exercise 2.2: (Laws of Exponents) Use induction to define where a is number (real or complex) and n is a positive integer. Use induction to prove that where a is a number and m and n are positive integers. Use induction to prove that where a is a number and m and n are positive integers. Assuming that you want these properties to hold for all integers m and n, what are the only possible definitions for ? What about where n is a negative integer? Exercise 2.3: (Sums of Powers) Show by induction that the sum of the first n positive integers is precisely n(n+1)/2. Show by induction that the sum of the squares of the first n positive integers is precisely n(n + 1)(2n + 1)/6. 1.2.2 Binomial Theorem It will often be useful to expand powers of a binomial a + b. For small positive integer value of n, you can do this using the distributive law. For example, Exercise 2.4: Show that It should be clear that for any given positive integer n, we can obtain a formula for ; but also that it is becoming more and more tedious as n increases. The goal is to find a general formula. Here is a first approximation: Proposition 2.4: (Binomial Theorem) For each positive integer n, one can write where the coefficients are all positive integers. In fact, one has for k = 1, ..., n-1 and Proof: Define by induction using the formulas in the statement of the proposition. Let be the property of the positive integer n which is true provided that the can be expressed as in the statement of the proposition. Then is clearly true because Assume that is true for n so that So is true for all positive integers n. The equations defining the coefficients appear quite complicated, however, if we write them down in the form of a triangle with the row containing for k = 0, 1, ..., n from left to right: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 From the diagram, we see that the formulas for the simply says that each row begins and ends with a 1 and every other entry is the sum of the two numbers in the row above it immediately to its left and to its right. This triangle is called Pascal's triangle and gives a simple way of computing the coefficients. Exercise 2.5: Define n! (pronounced n factorial) by 0! = 1! = 1 and (n+1)! = (n+1)n!. So, n! is the product of all positive integers less than or equal to n. Another interpretation of the number is the number of combinations of objects taken at a time which is a formal way of saying the number of subsets of elements which can be formed from a given set of elements. Exercise 2.6: In Pascal's triangle, note that each row reads the same from left to right as from right to left, i.e. is palindromic. Express this relation as an equation involving the . Explain why the relation is true. 1.2.3 Order of Operation The rules for order of operations are used to determine the precise order in which operations are to be carried out when evaluating an expression. Using them allows one to write expressions with far fewer parentheses making them both more readable and less error prone. On the other hand, the rather large number of operators makes for a rather complicated set of rules. The goal is to make your expressions readable and correct; sometimes it is better and clearer to add a set of parentheses even though the rules indicate that they are not really necessary. Here are the rules. You should read through the rules now and start using them. Because they are complicated, you will need to refer back to them a few times before they become clear. I am purposely not specifying exactly which numbers we are talking about because we want the material to apply to any of several types of algebraic quantities. Variables like x or y are letters that stand for numbers. By parentheses we mean any of the usual types of parentheses seen in algebraic expressions; they include rounded parentheses (), square brackets [], and curly brackets {}; in the rules below, we will use the rounded parentheses, but any other kind can be used as well. Definition 2.4: An expression is a particular kind of string of numbers, variables, operators, and parentheses. The following rules can be used to determine if such a string is an expression: Every number and every variable is an expression. If E is an expression, then so is ( E ). If E is an expression, then so is -E. If E1 and E2 are expressions and op is any of the binary relations '+', '-', '·', '÷', and '^'then E1 op E2 is also an expression. The only strings which are expressions are those which can be shown to be expressions by applying the above rules a certain number of times. Some fine points: Sometimes we will omit the multiplication operator ·. You can handle these expressions either by adding in the operator or by defining a new operator which is simply a space character. One can add the alternative slash '/' character as an alternate for the division symbol '÷'. Division is also often indicated with a horizontal bar. To handle this, simply replace the expression with the numerator between parentheses followed by ÷ and the denominator between parentheses. Sometimes, one does not assume that the exponentation operator is evaluated from left to right. In such cases, you need to be sure to put in additional parentheses so that the order of evaluation is explicit. Any expression has a well defined value. To guarantee this property, one needs to agree on the order in which an expression is evaluated. For example, if one evaluates the addition in 4 + 5 · 3 before the multiplication, the value is 27; but evaluating the multiplication first gives a value of 19. In order to avoid this kind of ambiguity, expressions are always evaluated using the so-called order of operations. The rule is: Evaluate parenthesized subexpressions starting from the first innermost parenthesized expression. If E has value v, then the value of (E) is also v. Use the remaining rules in order to evaluate a subexpression with no parentheses. The value of a number is the number itself; the value of a variable is the number which it represents. Powers should be evaluated first, from left to right. Unary minus operations should be evaluated next starting from the innermost. Multiplications and divisions should be done next; do this left to right. Additions and subtractions should be done next; again do this left to right. Lemma 3.1: Let r be a positive integer. There is no positive integer m such that Proof: We prove this by induction using the property which is true of m provided that We know that is true because 1 is not the successor of any positive integer. Now suppose that is true for If were not true for its successor then But then and so because each positive integer is the successor of at most one positive integer. This contradiction shows that must be true for and so the lemma is true. We can use the lemma to show that no two of the conditions , , and can be simultaneously true. There are three cases: If and , then If and , then If and , then In all three cases, we get a contradiction with the assertion of the lemma and so no two of the conditions can be simultaneously true. It remains to show that at least one of the three conditions must always be true. Let m be a fixed positive integer and be the property of a positive integer n that is true provided that at least one of and is true. is true because either or else for some positive integer p; in the second case, and so If is true for the positive integer n, then there are three cases: If then there is an r with But then and so If then and so . If there is a positive integer with If then On the other hand, if is not 1, then it is the successor of some positive integer, say and so One has and so In all three cases, we see that the condition is true and so holds for all positive integers and so trichotomy is true. Exercise 3.1: Prove that if then Definition 3.2: If then the difference is defined to be 1.3.2 Descent Descent is a variant on the principle of mathematical induction: Descent Principle: Let be a property which may or may not be true of the positive integer n. If is false for at least one positive integer then there is a smallest integer for which is false, i.e. is false for and true for all with . Proof: Suppose the descent principle is false for some property . Let be the property of the positive integer which is true provided that is true for all positive integers less than or equal to must be true; otherwise, would be false and clearly n = 1 would be the smallest number for which would be false because 1 is smaller than every positive integer except 1. Now suppose that is true for If were not true, then would be false but would be true for all numbers k less than . But then would be the smallest positive integer for which was false which contradicts the assumption that the descent principle is false for . 1.3.3 Unique Factorization Definition 3.2:Let and be positive integers. is a factor of (or is a multiple of ) if there is a positive integer such that A factor of is a proper factor if is prime if it is not equal to 1 and has no proper factors. Examples of primes are 2, 3, 5, 7, 11,.... Exercise 3.2: Replace the order relation in the statement of Proposition 3.1 with the property that be a factor of Which of the conclusions of the Proposition are still true? Which are false? Exercise 3.3: Show that if is a factor of then Proposition 3.2 Every positive integer can be factored into a product of primes. In other words, if is a positive integer, then there are prime positive integers ..., (where is a non-negative integer) such that Proof: This is a proof by descent. If the Proposition is false, then there is a smallest positive integer for which the assertion is false. cannot be equal to 1 or prime and so it can be written as product of proper factors: Since each of the factors is smaller than , one has and where the and are primes. But then Proposition 3.3 (Euclid) There are infinitely many primes. Proof: If not, then there are only finitely many. Let ..., be the complete list of them. Consider the positive integer Let be any prime factor of we know that for some = 1, 2, ..., Then But then which means that is a prime factor of 1. By Exercise 3.3, it follows that is less than or equal to 1 and so must be equal to 1 contrary to the assumption that is prime. Clearly, positive integers might be factored in more than one way as product of primes, e.g. As it turns out, the factorization is unique except for the order of the factors (and the way it is parenthesized): Proposition 3.4 (Division Theorem) If and are positive integers, then there are non-negative integers and such that and Furthermore, the numbers and are unique. Proof: (i) Existence: Let and be positive integers for which there are no such numbers and For the fixed value choose to be the smallest such that and do not exist. We cannot have because that would allow for Also, one cannot have as this would allow for By trichotomy, it follows that and so for some positive integer But then and so there must be non-negative numbers and such that and and But then contrary to the assumption that no appropriate and exist. (ii) Uniqueness: Suppose where and are non-negative integers with and If then and so Otherwise, assume that (swapping the roles of q, r and u, v if necessary. Re-arranging, we get which is a contradiction because the right hand side is smaller than n and the left hand side is greater than equal to n. Theorem 3.1 (Fundamental Theorem of Arithmetic) Every positive integer can be factored as a product of primes and this factorization is unique up to order of the factors. Proof: By Proposition 3.2, we need only show the uniqueness assertion. Let us begin with Lemma 3.2 If is a prime factor of the product then is a factor of either or (or both). Proof: If this is not true in general, then let be the smallest prime for which there is a counter-example. Of all such counter-examples choose one with the smallest possible value for , and of all these choose one with the smallest possible value for n. Using the division theorem, we know that one can write and where and are smaller than Since , we see that is a factor of and cannot be a factor of either or of without being a factor of or respectively. So, we can assume by the choice of and that they are both smaller than Since is a factor of there is a with . Further, and so But then every prime factor of must be smaller than . We have a factor of and so by the minimality of it must be that is either a factor of or of In either case, we could divide both sides of by and get a contradiction with the minimality of the choice of either or So, it must be that has no prime factors and so But then contradicting the fact that is a prime. This proves the lemma. Corollary 3.1: If is a prime factor of then is a prime factor of at least one of the factors ..., Proof: Suppose not. Choose a counter-example with the smallest possible value of . Then divides and so Lemma 3.2 tells us that is either a factor of or of . But, it can't be a factor of the second quantity because this would give us a counter-example with a smaller . So, must be a factor of contrary to assumption. If factorization can be non-unique, then let be the smallest positive integer which has at least two factorizations into products of primes differing other than in the order of the factors. No prime can occur in both factorizations; otherwise, would be a smaller positive integer with at least two factorizations. If is any prime factor of the first factorization, then is a factor of some prime of the second factorization by Corollary 3.1. So for some positive integer Since is a prime, it must be that and so contrary to assumption. A numeral is a symbol used to represent a number. The standard numeration system is the Arabic numeral system. A base 10 Arabic numeral is a sequence of digits 0, 1, 2, 3, ..., 9. The numeral where each is a digit represents the number For example, 3147 means . Decimal numerals are similar, e.g. . Sometimes it is more convenient to use bases other than 10. A base b Arabic numeral (where b is a positive integer bigger than 1) uses digits 0, 1, ..., b-1 and the numeral where each is a digit represents the number The commonly used bases other than 10 are 2 and 16. Base 2 numerals are called binary numbers and base 16 numerals are called hexadecimal numbers. Hexadecimal numbers use the letters A through F (or a through f) to denote the digits 10 through 15 respectively. For example, the decimal number 3147 is equal to the Number C4B in hexadecimal because 11 + 16(4) + 162(12) = 3147. You can check that in binary, the same number is 110001001011. One can also work with decimals in other bases as well; for lack of a better term, base 2 decimals will be referred to as binary decimals. The usual algorithms for doing operations with base 10 numbers work fine in other bases. However, most people find it easier to convert the numbers to base 10, do the computation in base 10, and convert the answer back to the desired base. Doing base conversions is easy. We have already seen how to convert to base 10. If you wanted to convert to say base 16, then take the number and divide by 16. The remainder is the lowest digit. Dividing the quotient by 16 gives a remainder which is the next digit, etc. Example 4.1: To convert the base 10 number 3145 to hexadecimal, divide 3145 by 16 to get So the lowest digit is eleven which is denoted B. Next divide 196 by 16 to get and so the next digit is 4. Finally, and so the high order digit is twelve or C. The number 3147 in base 10 is equal to the number C4B in base 16. Example 4.2:Calculate First, let's calculate it using the same algorithm as one uses in school. Here is the calculation: C4B C4B --- 8739 312C 9384 ------ 971DF9 To understand this, recall that and so by the distributive law: The three terms on the right are written above in the three lines between the horizontal bars. Instead of writing multiplying the lines by 10 (hex) and (hex), the numbers are simply written one and two columns to the left of the where they would normally be written. To see how one calculates , write this as Verify that this is precisely what you do when you do multiplication in base 10; then check the other two subproducts and the final addition. Now check your work: We know that C4B is 3145 in base 10. So its square in base 10 is in base 10 which converted to base 16 is 971DF9. Check the computation! Of course, if the two answers do not match, there must be an error in the computation. Repeating decimals are possibly infinite decimals whose digits eventually start to repeat themselves, numbers like 3.14141414... Such numbers are actually rational numbers. To see this, we will need: Proposition 4.1: (Geometric Series) One has In particular, if x is a number less than 1 in absolute value, then Proof: Use the distributive law to expand out the left side of the first equation. All but two of the terms subtract out leaving the two terms on the right hand side. To see the second assertion, divide both sides of the first equation by . As grows larger and larger, the term approaches 0 because . We can use the Proposition to evaluate our infinite decimal: A similar argument shows that any repeating decimal represents a rational number. Conversely, if you have a rational number p/q, then you can expand it out into a decimal by using the standard long division algorithm. At each stage in the computation, the remainder is a number between 0 and q - 1. As soon as the remainder repeats, the sequence of digits also repeat; so, the decimal expansion of a rational number is repeating. The formal result is: Proposition 4.2: Every repeating decimal represents a rational number and every rational number has a repeating decimal expansion. Although we argued in base 10 numerals, everything works anlaogously regardless of which base we work in. Corollary 4.1: There exist real numbers which are not rational. Proof: Any decimal expansion which is not repeating represents a real number which is not rational. Remark 4.1: The square root of 2 is irrational. For otherwise, we could write where and are integers with non-zero. Squaring both sides and multiplying through by , one gets . Now express and as products of primes and substitute these into this last equation. Because integers factor uniquely, we have a contradiction, because the left side of the equation would have an odd number of factors of 2 and the right side would have an even number of factors of 2. Definition 5.1: An ordered field F is a field (i.e. a set with addition and multiplication satisfying the conditions of Definition 2) with a binary relation < which satisfies: (Trichotomy) For every pair of elements a and b in F, exactly one of the following is true: a < b, a = b, and b < a. (Transitivity) Let a, b, c be arbitrary elements of F. If a < b and b < c, then a < c. If a, b, and c in F satisfy a < b, then a + c < b + c. If a, b, and c in F satisfy a < b and 0 < c, then ac < bc. Fact: If F is an ordered field, then 0 < 1. Proof: By Definition 2, and so by trichotomy, if the the fact were wrong, then we would have a field F with 1 < 0. By property iii, we would have 1 + (-1) < 0 + (-1) and so 0 < -1. But then using property iv, we would have . By Proposition 2, the left side is 0 and so . This contradicts trichotomy and so the assertion must be true. If F is an ordered field, an element a in F is called positive if 0 < a. Proposition 5.1: The set P of positive elements in an ordered field F satisfy: (Trichotomy) For every a in F, exactly one of the following conditions holds: a is in P, a = 0, and -a is in P. (Closure) If a and b are in P, then so are a + b and ab. Proof: (i) By property i of the Definition 3, exactly one of a < 0, a = 0, and 0 < a must be true. If a < 0, then by property iii of Definition 3, we have a + (-a) < 0 + (-a) and so 0 < -a. Conversely, if 0 < -a, adding a to both sides gives a < 0. So the three conditions are the same as -a is in P, a = 0, and a is in P. (ii) Suppose a and b are in P. Then 0 < a and by property iii of Definition 3, we have 0 + b < a + b and . Since 0 < b and b < a + b, transitivity implies that 0 < a + b. Since by Proposition 2, we have 0 < ab. Remarks: i. In one of the exercises, you will show that, if a field has a set P of elements which satisfy the conditions of Proposition 8, then the field is an ordered field assuming that one defines a < b if and only if b - a is in P. ii. An element a of an ordered field F is said to be negative if and only if a < 0. iii. It is convenient to use the other standard order relations. They can all be defined in terms of <. For example, we define a > b to mean b < a. Also, we define to mean either a < b or a = b and similarly for . iv. The absolute value function is defined in the usual way: Proposition 5.2: Let a and b be elements of an ordered field F. \|-a\| = \|a\| (i.e. and ) (Triangle Inequality) Proof: i. By Trichotomy, we can treat three cases: a > 0, a = 0, and a < 0. If a > 0, then -a < 0 and so \|a\| = a and so \|-a\| = -(-a) = a. If a = 0, then -a = 0 and so \|a\| = 0 = \|-a\|. If a < 0, then -a > 0 and so \|a\| = -a and \|-a\| = -a. In all three cases, we have \|a\| = \|-a\|. ii. Again, we can treat three cases: If a > 0 or a = 0, then \|a\| = a and so . If a < 0, then adding -a to both sides gives 0 < -a and so a < -a by transitivity. In this case we have \|a\| = -a and so a < \|a\|. We could argue the other inequality the same way, but notice that we could also use our result replacing a with -a. (Since it holds for all a in F, it holds for -a.) The result says , where we have used assertion i. Adding a - \|a\| to both sides of the inequality gives the desired inequality. iii. Once again, do this by considering cases: If , then \|a + b\| = a + b. Since and , we can add b to both sides of the first inequality and \|a\| to both sides of the second one to get and . Using transitivity, we get as desired. Now suppose that . Then adding -a - b to both sides of the inequality gives -a + (-b) > 0. Applying the result of the last paragraph, we get . But a + b < 0 means that \|a + b\| = -(a + b) and so where we have used assertion i for the last step. This completes the proof. As noted earlier, one can write numerals in any integer base b > 1. The choice b = 10 has the advantage of being most familiar; but choosing b = 2 often makes the proofs a bit simpler - basically, each additional digit cuts an interval in two equal pieces which is easier to handle than 10 equal pieces. For this section, we will use base 10; but after reading it, you should go back and verify that everything works regardless of the choice of the base. In some later sections, we will use base 2 in order to have simpler proofs. We start by formalizing our notion of infinite decimal. Definition 6.1: i. An infinite decimal is an expression of the type , where is an integer, and is an infinite sequence of decimal digits (i.e. integers between 0 and 9). ii. Every such infinite decimal defines a second sequence of finite decimals where . iii. One says that the infinite decimal represents the number r (or has limit r) if can be made arbitrarily close to zero simply by taking k sufficiently large. Definition 6.2: i. An ordered field F is said to be Archimedean if, for every positive a in F, there is a natural number N with a < N. ii. An Archimedean ordered field F is called the field of real numbers if every infinite decimal has a limit in F. Given any element a in F, we can form an infinite decimal for a. First, we can assume that a is positive, since the case where a = 0 is trivial, and if a < 0, then we can replace a with -a. Next, we see why we needed to add the Archimedean property to the above definition. Without it, we would not know how to get the integer part of a: Since F is Archimedean, the set of natural numbers N with a < N is non-empty and so there it has a smallest element b. Let . Then if . Choose to be the decimal digit such that and let , so that again . Assuming that we have already defined for some natural number k, the quantities and with , define by induction the digit so that and let , so that . The infinite decimal was defined so that with . So this infinite decimal has limit a. We say that this is the infinite decimal expansion of the element a in F. Proposition 6.1:i. Every element a in F is the limit of the infinite decimal expansion of a. ii. The decimal expansion of every rational number is a repeating decimal, i.e. except for an initial segment of the decimal, the decimal consists of repetitions of a single string of digits. iii. Every repeating decimal has limit a rational number. Proof: The first assertion has already been proved. For the second assertion, note that the definition of the sequence of digits is completely determined by the value of . If a = r/s is rational with r and s integers, then is a rational number with denominator (a factor of ) s. Furthermore, since , if is rational with denominator s, then so is . By induction, it follows is rational with denominator s for every k. Since lies between 0 and 1 and is rational with denominator s, it follows that there are at most s possible values for . The following principle is called the pigeonhole principle: If s + 1 objects are assigned values from a set of at most s possible values, then at least two of the objects must be assigned the same value. By the pigeonhole principle, there are subscripts i and j with such that . As indicated at the beginning of the proof, it follows that the sequence of digits starting from must be the same as the sequence of digits starting from and so the decimal repeats over and over again the cycle of values . The third assertion is easy to prove -- it is essentially the same as our calculation of the limit of the infinite decimal expansions of 1/3 and 1/7. The formalities are left as an exercise. Example 2: The field of real numbers contains many numbers which are not rational. All we need to do is choose a non-repeating decimal and it will have as its limit an irrational number. For example, you might take where at each step one adds another zero. Proposition 6.2: Every a > 0 in the field of real numbers has a positive -root for every natural number n, i.e. there is a real number b with . Proof:It is easy to show by induction that, if , then for every natural number n. So the function is an increasing function. By the Archimedean property, we know that there is a natural number M > a. Again by induction, it is easy to see that . Now consider the set S of all numbers m with . Clearly 0 is in this set. If every successor of an element of S lies in S, then the principle of mathematical induction would imply that S would be the set of all non-negative numbers contrary to the fact that we have already identified a number M not in S. So, let be an element of S such that is not in S. We know that the -root of a must lie between and . Next evaluate for integers j from 0 to 10. The values start from a number no smaller than a and increase to a number larger than a. Let be the largest value of j for which the quantity is at most a. Repeating the process, one can define by induction an infinite decimal such that the -power of the finite decimal differs from a by no more than . Let b be the limit of the infinite decimal, and be the values of the corresponding finite decimals. Then we have and and so it is reasonable to expect that . This is in fact true. Using the identity for geometric series, we see that: . But then the triangle inequality gives where C is a positive constant which does not depend on k. Since this holds for all positive integers k, it follows that . If a is a positive element of any ordered field, we know that because the set of positive numbers is closed under multiplication. Since we also have , it follows by trichotomy that the square of any element in an ordered field is always non-negative. In particular, such a field cannot contain a solution of . We would like to have a field where all polynomial equations have a root. We will define a field called the field of complex numbers which contains the field of rational numbers and which also has a root, denoted i, of the equation . In a later chapter, it will be shown that, in fact, contains a root of any polynomial with coefficients in . This result is called the Fundamental Theorem of Algebra. Let us first define the field of complex numbers. Since it is a field which contains both the field of real numbers and the element i, it must also contain expressions of the form z = a + bi where a and b are real numbers. Furthermore, there is no choice about how we would add and multiply such quantities if we wanted the field axioms to be satisfied. The operations can only be: and where we have used the assumption that . It is straightforward, but a bit tedious to show that these operations satisfy all the field axioms. Most of the verification is left to the exercises. But let us at least indicate how we would show that there are multiplicative inverses. Let us proceed heuristically -- we would expect the inverse of a + bi to be expressed as but this does not appear to be of the desired form because there is an i in the denominator. But our formula from geometric series shows how to rewrite it: We have . This is just what we need: Of course, we have proven nothing. But we now have a good guess that the multiplicative inverse might be . It is now an easy matter to check that this does indeed work as a multiplicative inverse. Proposition 7.1: The set of all expressions a + bi, where a and b are real and i behaves like , is a field if we define operations as shown above. We have already seen that the field cannot be ordered. Nevertheless, we can define an absolute value function by . The conjugate of a complex number is a + bi is defined to be a - bi and is denoted . Proposition 7.2: Let w and z be complex numbers. Then \|w\| = \|-w\| \|wz\| = \|w\|\|z\| \|z\| = 0 if and only if z = 0. (Triangle Inequality) . . If r is a real number, its absolute value is the same as a real number as it is if it is considered to be the complex number . Proof: These are all left as exercises except for the triangle inequality. To prove the triangle inequality, let and where and are real numbers. To show the triangle inequality, it is enough to show that Substituting in the values for and one sees that this will hold provided that This simplifies to the equivalent inequality This in turn would be true if the square of the left side were less than or equal to the square of the right side, i.e. which is equivalent to But subtracting from both sides and factoring gives which is obviously true. So, the triangle inequality is also true. Exercise 7.1 (i) Prove the rest of Proposition 7.2 Prove that in the triangle inequality, one has equality if and only if one of and is a non-negative real multiple of the other. Example 8.1: To solve the general linear equation ax + b = c for x where a, b, and c are constants with a non-zero, one can assume that x is a solution so that ax + b = c. Add -b to both sides of the equation to get (ax + b) + (-b) = c + (-b). This simplifies to ax = c - b. Multiplying both sides by a-1 and simplifying gives x = a-1(c - b). We have shown that this is the only possible solution. Substituting it into the original equation and simplifying verifies that this value is indeed a solution. So, the equation has the unique solution x = a-1(c - b). Example 8.2: To solve the general quadratic equation ax2 + bx + c = 0, we can restrict ourselves to the case where a is non-zero. (Otherwise, the equation is linear and Example 3 applies.) If x is a solution of the equation, then one can divide both sides by a and simplify to get x2 + (b/a)x + c/a = 0. If 2 is not zero, then one can complete the square to get (x + b/(2a))2 = (b/(2a))2 - c/a. If the right side was square, then we could solve for x to get Furthermore, one can check that this actually is a solution of the original equation. This last equation is known as the quadratic formula. In particular, if we are working in the field of real numbers, then we have completely solved the quadratic; there are two, one, or no solutions when b2 - 4ac is positive, zero, or negative respectively. In the case of the field of complex numbers, things are even simpler as we will show that all complex numbers have square roots. Another example is the general system of 2 linear equations in two unknowns x and y: ax + by = e, cx + dy = f. What makes the general system of 2 linear equations appear difficult is that both equations involve both variables. There are two approaches: One could solve the first equation for one of the variables in terms of the other. Then substitute this into the second equation giving an equation in only the second variable. Then proceed as in the easy case. One could subtract an appropriately chosen multiple of one equation from the other in order to obtain an equation involving only one variable. As before, one needs to check that the possible solutions one obtains do indeed satisfy the original equations. Example 8.3: Consider the system: 2x + 3y = 5, 4x - 7y = -3. Assume that x and y are satisfy both equations. One can proceed using either method: Solving for x using the first equation, gives x = (5/2) - (3/2)y. Substituting this into the second equation gives 4((5/2) - (3/2)y) - 7y = -3, which can be used to find y = 1. Substituting this back into our expression for x yields x = 1. One then checks that the values x = 1, y = 1 do indeed satisfy the original equations. If one subtracts twice the first equation from the second equation, one gets (4x - 7y) - 2(2x + 3y) = -3 - 2(5) or -13y = -13. This gives y = 1 and substituting this value back into the first equation gives 2x + 3 = 5 or x = 1. As before, we need to check that x = 1, y = 1 does indeed satisfy the original two equations. Example 8.4: Find all the solutions of the system of equations: , . Assume that one has a solution x, y. Solving the second equation for y, one gets a value y = 1 -x which when substituted into the firs equation gives: or . Collecting terms, we get a quadratic . Factoring and using the second part of Proposition 1, gives x = 0 or x = 1. Substituting these values into our expression for y, gives two possible solutions (x,y) = (0, 1) and (x,y) = (1, 0). Substituting each of these into the original equations, verifies that both of these pairs are solutions of the original system of equations. If we have more than two variables and more equations, we can apply the same basic strategies. For example, if you have three linear equations in three unknowns, you can use one of them to solve for one variable in terms of the other two. Substituting this expression into the two remaining equations gives two equations in two unknowns. This system can be solved by the method we just described. Then the solutions can be substituted back into the expression for the first variable to find all possible solutions. When you have these, substitute each triple of numbers into the original equations to see which of the possibilities are really solutions. One can also use the second approach as is illustrated by the next example. Example 8.5: Solve the system of equations: x + y + z = 0, x + 2y + 2z = 2, x - 2y + 2z = 4. Assume that (x, y, z) is a solution. Subtracting the first equation from each of the other two equations gives y + z = 2, -3y + z = 4. Now subtracting the first of these from the second gives -4y = 2 or y = -1/2. Substituting this into the y + z = 2 gives z = 5/2. Finally, substituting these into the first of the original equations gives x = -2. So the only possible solution is (x, y, z) = (-2, -1/2, 5/2). Substituting these values into the original equations shows that this possible solution is, in fact, a solution of the original system. This section is an informal review of some elementary notions of analytic geometry. Let's start with the number line. Choose a line and mark off two points 0 and 1 on the line. By marking off segments of length equal to that between 0 and 1, one can define points 2, 3, etc. Moving in the other direction, one gets -1, -2, etc. To each rational number, one can associate a point on the line; e.g. 1/2 is the point midway between 0 and 1; 7/4 is the point a quarter of the way between 0 and 7, etc. Real numbers can be represented as infinite decimals; we can associate them with the points obtained as limits of the finite decimals obtained by throwing away the tail end of the decimal. For example, 1.2121212... is the point which is the limit of the points 1, 1.2, 1.21, 1.212, etc. At least intuitively, it appears that we have defined a one-to-one correspondence between the real numbers and the set of points on the line. To name the points in the plane, simply use the cartesian product of the real numbers with itself. Geometrically, this corresponds to taking two perpendicular number lines intersecting at 0. These are called the x-axis and y-axis respectively. Each point on the plane can be projected perpendicularly onto each of the two axes. The name the point is the ordered pair (x, y) where x is the number associated with the projection of the point on the x-axis and y is the number associated with the projection of the point on the y-axis. The diagram below shows the point (3, 2): Recall that the complex numbers can be written in the form a + b i where and a and b are real numbers. So, a complex number is essentially the same thing as an ordered pair (a, b) of real numbers. This allows one to think of the plane as being the set of real numbers. The number a is called the real part and the number b is called the imaginary part of the complex number a + b i. In the above diagram, the point (3, 2) corresponds to the complex number 3 + 2i. Proposition 9.1: (Distance Formula) If and are two points in the plane, then the distance between them is . Proof: Let . Then is a right triangle with right angle at . The legs have length and respectively. The distance formula now follows by the Pythagorean Theorem. (This theorem will be proved in a next chapter.) If (a, b) and (c, d) are two points in the plane, then one can define their sum to be (a + c, b + d). Note that this is precisely the point corresponding to the sum z + w of the complex numbers z = a + bi and w = c + di corresponding to the two points. The addition is the so-called parallelogram law Let L be the line segment from the origin (0, 0) to (a, b) and M be the line segment from the origin to (c, d). If we move M parallel to itself so that it starts at (a, b), then its other end-point will be at (a + c, b + d). So the line segments R from (a, b) to (a + c, b + d) and S from (c, d) to (a + c, b + d) combine with L and M to make a parallelogram whose diagonal starts from the origin and ends at the sum of z = (a, b) and w = (c, d). You can also think of this as a triangle rule: To add z and w, start with a line segment from the origin to (a, b), then move the line segment from the origin to (c, d) parallel to itself until it is starting from (a, b); the final end-point is the sum z + w. Let and be any two complex numbers with We can think of as defining a line segment M from the origin to If is a positive real number, then corresponds to a line segment obtained by stretching M by a factor of t. If is a negative real number, then the segment is streched by the a factor of but goes in the oppositive direction as does M. When you add to it, the complex numbers in the set make up a line through and parallel to M. Definition 9.1 A line is any set of points of the form where and are complex numbers with The corresponding set of points in the plane are also referred to as a line. Proposition 9.2: (Midpoint Formula) If and are two points in the plane, then the point . is on a line through and and the distance between and either end-point is equal to half the distance between the end-points. Proof: The line is the one defined by and Letting t = 1/2 gives the point M. The assertions about the distance are easy to verify using the distance formula. Intuitively speaking, a function is a rule which associates with each element of a set an element of a set . The set is called the domain of the function and the set is called the codomain of the function. A function with domain and codomain is often denoted and the number in associated by the rule to the element in is denoted The range of is the set of all elements of that are associated to at least one element of the domain For example, function which squares each real number is a function with domain and codomain both equal to the field of real numbers. For each real number one has and the range of the function is the set of non-negative real numbers. On the other hand, the square root function has domain the non-negative real numbers and The codomain might be the set of non-negative real numbers or any set containing this set. For any function one can form the set of all ordered pairs where is in the domain of This set is called the graph of the function. In the special case where the function has domain and codomain contained in the set of real numbers, the graph of f can be thought of as a subset of the number plane. Remark 1.10.1 One should think of the graph of a function as a visual representation of the function. In the special case in which the domain and codomain are subsets of the real numbers, one has: The domain is the set of x-coordinates of points in the graph. The range is the set of y-coordinates of points in the graph. Every vertical line intersects the graph in at most one point. The last property is often referred to as the vertical line test Example 1.10.1 Not every subset of the number plane is the graph of a function. For example, the circle with center at the origin and radius 1 is not the graph of a function. This is because there are vertical lines which intersect the graph in more than one point. For example, the y-axis intersects the circle at which means that there cannot be a rule which associates to 0 a single number and still have both of these points in the graph. On the other hand, the part of the circle which has y-coordinate non-negative is the graph of the function which assigns to every in the closed interval [-1, 1] the value Many times the function will be ambiguously specified by simply giving the rule for associating elements of the domain with elements of the codomain, without specifying a domain and codomain. In this case, one normally assumes one is dealing with the largest domain for which the rule makes sense, and this domain is referred to as the natural domain. For example, the natural domain of the squaring function is the set of all real numbers, even though there are other functions with domains any specific subset of the real numbers. In some cases, one needs to intuit the meaning of what one means by the largest domain for which the rule makes sense; for example, the squaring function is also defined on the field of complex numbers. Remark 1.10.2 Because it is difficult to formalize what one means by a rule, a formal definition of function usually is a definition of the graph of the function. For example, one could say that a function with domain and codomain is any subset of the set of all ordered pairs where and such that for every there is exactly one ordered pair in first coordinate equal to One then writes to indicate that 1.10.1 Operations on Functions Let and be functions with domains some set codomains some field Then one can combine and to form new functions: The sum of and is defined by The difference of and is defined by The product of and is defined by The quotient of and is defined by Note that the domain of the first three functions is but the domain of the fourth function is the set of for which Finally, if and are two functions, then the composition of the two functions is defined by Example 1.10.2 If is the squaring function and is the cubing function, then is the function Also, is the function with The function is the reciprocal function, which is only defined for non-zero real numbers. Finally, the composition is the function which raises numbers to their sixth power. Exercise 1.10.1 Show that the operations sum and product are commutative but that the other three operations are not. Show that the operations sum, product, and composition are associative, but the other two are not.
1. Use basic matrix operations and the algebra of matrices in practical problems. Possible applications may be drawn from areas such as Kirchoff's laws, Leontieff model of an interacting economy, Markov chains, method of least squares, singular value decomposition and fourier coefficients of a function. 2. Understand the concepts of vector spaces, subspaces, basis, independence and dependence, dimension, coordinates, rank of a matrix, inner product. 3. Use the dependency relationship algorithm and the Gram-Schmidt orthogonizational process. 4. Understand linear transformations, range and null space of a linear transformation, the correspondence principle and similarity. 5. Understand properties of the determinant function and the cofactor expansion of determinants. 6. Understand the concepts of eigenvalues and eigenvectors. 7. Understand the concepts of quadratic formsMethods of presentation can include lectures, discussion, experimentation, audio-visual aids, small-group work and regularly assigned homework. Calculators/computers will be used when appropriate. Mathematica, Derive and TI-92 calculators are available for use at the College at no charge. Course may be taught as face-to-face, media-based, hybrid or online course. VIII. Course Practices Required (To be completed by instructor) IX. Instructional Materials Textbook information for each course and section is available on Oakton's Schedule of Classes. Within the Schedule of Classes, textbooks can be found by clicking on an individual course section and looking for the words "View Book Information". Textbooks can also be found at our Mathematics Textbooks page. A computer algebra system is required. X. Methods of Evaluating Student Progress (To be determined and announced by the instructor) Evaluation methods can include grading homework, chapter or major tests, quizzes, individual or group projects, calculator/computer projects and a final examination
Description This workbook includes over 25 activities and is written to fully take advantage of the functionality of the TI-83 Plus/TI-84 Plus Families of handhelds and the Cabri Jr handheld software application. Topics in this workbook encompass five major areas of Geometry: 1) Points-Lines-Angles, 2) Transformations, 3) Triangles, 4) Quadrilaterals-Circles and 5) Algebraic Geometry
Science Books MATLAB: An Introduction with Applications 2nd Edition Assuming no prior MATLAB experience, this clear, easy-to-read book walks readers through the ins and outs of this powerful software for technical computing, including: Generously illustrated computer screen shots and step-by-step tutorials applied in the areas of mathematics, science, and engineering Clearly shows how MATLAB is used in science and engineering Includes a completely new chapter on Symbolic Math Thoroughly updated to match Matlab's newest release, Matlab 7 .Simulation Of Mechanical Systems(September 2, 2005) — Assistant lecturer at the Public University of Navarra, José Javier Gil Soto, is the author of the thesis "Preprocesser for dynamic simulation of multibody systems based on symbolic algebra". ... > read more Contributing to the Nuclear Fusion Project(April 27, 2012) — Many regard nuclear fusion as the main energy source of the future. Among others, the ITER project is seeking to turn this venture into reality and is making use of the Tokamak reactor for this ... > read more A Picture Is Worth A Thousand Locksmiths(October 31, 2008) — Computer scientists have built a software program that can perform key duplication without having the key. Instead, the computer scientists only need a photograph of the ... > read more
Three components contribute to a theme sustained throughout the Coburn-Herdlick Series: that of laying a firm foundation, building a solid framework, and providing strong connections. In the Graphs and Models texts, the authors combine their depth of experience with the conversational style and the wealth of applications that the Coburn-Herdlick texts have become known for. By combining a graphical approach to problem solving with algebraic methods, students learn how to relate their mathematical knowledge to the outside world. The authors use technology to solve the more true-to life equations, to engage more applications, and to explore the more substantial questions involving graphical behavior. Benefiting from the feedback of hundreds of instructors and students across the country, College Algebra: Graphs & Models emphasizes connections in order to improve the level of student engagement in mathematics and increase their chances of success in college algebra
Mathematics Courses MTH 013. Basic Math Skills with Algebra.3 hours credit, fall, spring A review of basic concepts to prepare students who are deficient in basic mathematics skills. Includes basic arithmetic skills and beginning topics of algebra such as signed numbers, linear equations, exponents, polynomials and word problems to form an introduction to intermediate algebra. This course does not fulfill any general education mathematics requirement. MTH 123. Mathematics for Educators.3 hours credit, spring The study of set theory, logic, numeration systems, whole numbers, integers, rational numbers, number theory, and geometry, presented as a foundation for elementary school mathematics. Required of ALL elementary education majors. MTH 133. Intermediate Algebra.3 hours credit, fall, spring A study of fundamental algebraic operations, polynomials, graphing, pairs of linear equations, roots and radicals, ratios and proportions, and their applications. Designed to prepare the student for college algebra and to satisfy the math requirements for medical technology, other allied health-related disciplines, home economics and similar programs. A student may not earn credit for this course after passing MTH 163, MTH 173, or MTH 184. MTH 145. Math in the Real World.3 hours credit, fall, spring A practical course introducing basic concepts of logic, set theory, finance, functions, statistics and probability as they relate to events commonly encountered. This course will meet the General Education requirements for all students who are not required to take College Algebra or Calculus for their chosen degree. MTH 163. Functions and Modeling.3 hours credit, fall, spring A study of linear, exponential, logarithmic and polynomial functions and their graphs with emphasis on modeling, rates of change, and data analysis as applied to the natural sciences, business, the behavioral sciences, and social issues. Prerequisites: MTH 133 or two years of high school algebra with a grade of B or better; minimum ACT Math score of 22 or SAT Math score of 520. MTH 173. College Algebra.3 hours credit, fall, spring A study of sets, relations and functions, exponential and logarithmic functions, systems of equations and inequalities, matrices and determinants, theory of equations, sequences, permutations, and combinations, the binomial theorem and introduction to the theory of probability. Prerequisite: at least two units of high school algebra with B's or better, or MTH 133 with a C or better. MTH 181. Trigonometry.1 hour credit, fall Covers the standard trigonometric functions, their inverses, identities, relationship to the unit circle, along with basic applications such as the laws of sine and cosine. Prerequisite: MTH 133 with a C or better, or 2 years of high school algebra with B's or better. MTH 184. Pre-Calculus.4 hours credit, fall Study of linear, polynomial, power, exponential and logarithmic functions and their graphs; systems of equations and inequalitites, matrices, and determinants. Covers the standard trigonometric functions, their inverses, identities, relationship to the unit circle, along with basic applications such as the laws of sine and cosine. Prerequisites: MTH 133 with a C or better, or 2 years of high school algebra with B's or better. MTH 203. Discrete Mathematics. 3 hours credit, fall Covers fundamentals of discrete mathematics including sets, proofs, induction, logic, relations and functions, algorithms, graph theory, and combinatorial counting principles. Prerequisites: MTH 163, MTH 173, or MTH 184. MTH 213. College Geometry.3 hours credit, odd years, spring An extension of high school geometry. Includes construction, foundations, and methods of proof in Euclidean geometry and solid geometry. Prerequisite: High School Geometry. MTH 214. Analytic Geometry and Calculus I.4 hours credit, fall Plane analytic geometry, differentiation of algebraic functions, applications of derivatives, integration and its applications. Prerequisite: MTH 173 or 184, or two years of high school algebra with B's or better and one unit of Trigonometry. Co-requisite: for those weak in trigonometry take MTH 181 Trigonometry. MTH 243. Introduction to Mathematical Thought.3 hours credit, even years, fall A course to prepare the serious mathematics student for the more advanced courses in abstract algebra, and analysis. This course is designed to bridge the gap between applied mathematics courses and proof oriented abstract mathematics courses. The course will emphasize the logical skills required for mathematical proof. Prerequisite: MTH 224. MTH 313. Probability and Statistics.3 hours credit, odd years, spring Probability as a mathematical system, random variables and their distributions, limit theorems, and topics in statistical inference. This course is designed primarily for mathematics majors. Prerequisite: MTH 224.
user-friendly, Web-based software package makes it easy to help you develop the essential math skills you need to succeed in construction trades. An innovative, interactive teaching and learning resource, BUILDING MATH SKILLS ONLINE FOR CONSTRUCTION allows you to quickly assess baseline skill levels and assign personalized courses of study, with lessons and problems targeted to your individual needs and career interests. In addition to a strong focus on practical applications for real-world construction tasks and hands-on projects, instructional content is correlated to the Common Core Standards. BUILDING MATH SKILLS ONLINE FOR CONSTRUCTION, is a powerful new resource to support your success—in the classroom and your future career
Provide students with practice in skills areas required to understand pre-algebra conceptsBasic skills review in the first part of the book and more specific algebra topics introduced throughout the bookEachInstructional Fair Pre-Algebra Resource BookGrades 5 - 8Provide students with practice in skills areas required to understand pre-algebra conceptsBasic skills review in the first part of the book and more specific algebra topics introduced throughout the bookInstructional Fair Pre-Algebra Resource Book Fair Pre-Algebra Resource Book0.00
Red Rock, AZ Algebra at explaining math in different ways if one approach is not working. I provide real world examples for things that are sometimes abstract.What will a student learn in physics: - laws that deal with the motion of an object in one and two dimensions. They are also known as ... ...Probability has found its way into all the sciences including the daily weather reports, biology and even theoretical physics, and that is why you need to know about it. Linear algebra is the study of simultaneous linear equations. They can be solved with one of two methods: substitution or elimination of a variable.
First three weeks of class: Goldstein, et al., ``Brief Calculus and its Applications'' (chapter 7, which is bundled with the main text) The remainder of the term: Goldstein, et al., ``Finite mathematics and its applications'' Calculator: A graphing calculator is required. Recommended: TI-83. Actually, most graphing calculators will work. Sharing of calculators during exams is not permitted. Prerequisites: A grade of C-- or better in MATH 214. Web page: Email list: math215@yahoogroups.com (when you fill out your information you may elect to be on this list) Objectives: The student should be able: To maximize or minimize a function of many variables, even including constraints; To compute probabilities for some discrete random processes; To use Bayes' Theorem to compute conditional probabilities; To compute and interpret mean and standard deviation; To solve a system of linear equations to find all solutions, or to show when no solution exists. Goals: The student should develop: An ability to translate a problem which is well-suited to mathematical solutions into mathematical language; A familiarity with functions of several variables; A practical knowledge of probability that will be used in BA 216; An ability to manipulate matrices and use them when it is well-suited to the problem; A confidence about doing mathematics to overcome ``math anxiety''; An ability to think and reason in a structured logical manner. Homework: Homework will be assigned daily, and due two class times later, at the beginning of class. Some homework may be due the next class period, and this will be specified on the homework. Each assignment will include both problems not to be submitted, and problems to be submitted. The problems not to be submitted are those which have answers in the book. You are to do these problems to best prepare you for the rest of the assignment, and if you find that you understand how to do these problems before you finish them all, you may skip to the problems to be submitted. Not all homework problems will be graded--only a select few, representative of the different kinds of problems. Those seeking solutions to past problem sets may request them after the homework is due. Homework will be graded primarily on your ability to complete the problem, and secondarily on its correctness. The five lowest homework scores will be disregarded. Remember that the primary purpose of the homework is to prepare for the exams, so treat it primarily as a training program for yourself, and only secondarily as something you need to score highly on. Late assignments: No late homework is accepted. Exceptions can be granted, if you must give me notice that you are going to turn in an assignment late at least the class before the assignment is due. You must also have a good reason. These reasons will be treated on a case-by-case basis. When you obtain permission to turn in an assignment late, we will discuss a new due date for that homework. Extra credit: If you find a way to apply any of the material to something you are interested in, you can discuss with me the possibility of getting extra credit. It should involve some research outside the material in the book and be written in a professional manner. Collaboration: You are encouraged to collaborate on all homework assignments, unless otherwise specified. This means you work on it independently before discussing it with each other, and it means you must thoroughly understand how to do the problem before writing it up. You must write up your answers separately; you cannot turn in one homework for more than one person, nor can you simply include photocopies of other students' work. There is no limit to the size of a group for collaboration, although 3-5 people tends to be an efficient size. You should also use these groups to ask questions of each other to better understand the material. If you do not see each other frequently, you should set up a regular time and place to meet to work on assignments. If you do not have a group, talk to me and I can place you in a group. If you do not wish to work in a group, that is your prerogative but this will be a disadvantage to you. Comments: You should include comments about the class at the top of your homework assignments. These comments can be ``You go too fast'', ``You say `um' too often'', ``I like this chapter'', ``This is too easy/hard'', ``Can we have more applications to Marketing'', ``Everything's okay'', and so on. You will not be graded on these comments, but they will affect how I teach the class, and may make the class more enjoyable for you. Class participation: You are expected to actively participate in class. Many students view learning as a passive act, where the teacher takes the only active role, and the student simply listens, or at most takes notes. This view is not advisable in this class. Here, you will need to take an active role in learning the material. {\em You} are in charge of your education, and {\em you} should take responsibility to learn the material as thoroughly as you can. Part of this involves asking questions in class, even questions that may sound ``stupid''. A question clearing up a point you do not understand is, by definition, not stupid. Similarly, when I ask the class questions, you should try to answer them, even if you're not sure of the answer. Your best guess is, by definition, not stupid. The effect of class participation on your grade is noted under ``quizzes'' below. Pre-class preparation: You are expected to read through the section of the book we are covering before you come to class. If you don't understand something, write down specific questions you have to ask in class. Quizzes: There will be no regular quizzes, but to ensure you have read through the section beforehand, I will, from time to time, give out pop quizzes at the beginning of class. These will be short and only test a superficial knowledge of the material. In this way, they are not useful for indicating what an exam will be like. They will be used to decide borderline cases in the final grade, as will class participation. Remember that since there are 12 grades (counting +'s and --'s), almost everyone in the class will be a borderline case. There is no make-up for quizzes. Attendance: Attendance is important simply due to the difficulty of the course. Missing one class may have the effect of your not being able to follow any of the classes for the rest of the term. Furthermore, those who do not attend classes will have poor scores on class participation and cannot take quizzes, and these will also affect grades. In short, skip class at your peril. Exams: There will be three midterms, and one final. Each midterm counts for 20% of your grade, and the final counts for 30%. The remaining 10% is your homework grade. The final exam grade will substitute for your lowest midterm grade if this is to your advantage. Note that borderline cases will be resolved by quiz grades and class participation, as noted above. Midterms will be in-class, and the final will be at a separate time as listed below. All final exams follow the schedule listed at There are no make up quizzes or exams. If you must miss an exam due to a major emergency, you must make arrangements with me beforehand, and exceptions may be granted on a case-by-case basis. If granted, your final exam score will be used to calculate the score for the missed exam. Midterm 1 February 3 during class Midterm 2 February 24 during class Midterm 3 March 31 during class Final (sec. 3) April 25 1:30 p.m. -- 4:00 p.m. Final (sec. 4) April 26 1:30 p.m. -- 4:00 p.m. I will hold review sessions before each, at a time that is popular with the class. Holidays: Conference Jan. 12--13 Martin Luther King, Jr. Day Jan. 16 Spring break Feb. 27--Mar. 3 Grading: A grade of C indicates an ability to do homework-like problems, and memorization of all techniques and definitions. In order to receive a B, a student must demonstrate a deeper knowledge of the material, being able to apply the course material to new circumstances where applicable. An A student must demonstrate this kind of deep understanding in all of the covered topics, as well as be able to draw new conclusions from known facts in a logical manner, and must also demonstrate persistence and dilligence. In the other direction, a grade of D shows only superficial understanding of the material, and shows inconsistency to do straightforward problems. An F grade indicates that the student has severe gaps in even superficial understanding of the material in the course. Although this is the philosophy, grading will be done by counting points received on each problem, as usual. But the difficulty level of the problems will be arranged in order to achieve the above grading scale. Christian attitude: Although not part of the grading for this course, you are expected to approach this class with a Christian attitude, being willing to help your fellow classmates to understand the material outside of class, being willing to be corrected by your fellow classmates when you see they are right, but firm in your conviction otherwise, being bold to ask questions without feeling ashamed of looking foolish, encouraging one another in love, being patient with those who are asking questions, and preferring a grasp of the material, which is enduring and becomes part of you, over a grade, which is transient, external, and shallow. You should diligently devote the time you spend on this class as to the Lord. As cheating harms both the cheater and the rest of the class (though in different ways), you should not cheat, nor should you provide temptations for others to cheat. For my part, I commit to approaching this class with a Christian attitude, viewing my role as that of a servant, being concerned first for your personal, especially intellectual, development. I will also seek to produce an environment of encouragement and love, that fosters a sense of community and understanding. I commit to reporting grades that accurately and honestly reflect the level of work done in the class, as described in the paragraphs above. I also commit the time I spend preparing for this class as to the Lord, and I will pray for all individuals in the class on a regular basis, understanding that even as I may seek to educate, God provides the true transformation.
Line GraphsPlotting and analysis of line graphs from various areas of Science. A range of topics and difficulties included. 21 worksheets. Sorry, but the solutions are not included at this point. From ORB Education. This version includes the fixed PDF worksheet files as well as the editable Word copies. PLEASE NOTE: ORB Education resources have been edited professionally. See for a full set of previews. Compressed Zip File Be sure that you have an application to open this file type before downloading and/or purchasing. 987.47 KB \| 21 Pages Was $13

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.