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Mathematics Report 2002
The mathematics department is one of the largest in the school with all learners taking mathematics as a subject from Grade 8 to Grade 12. The grade 8,9 and 10 have eight 55-minute periods in a two-week cycle and Grades 11 and 12 have nine 55-minute periods. Additional mathematics is offered as a seventh subject. Functional mathematics for Grade 12 was re-introduced at the beginning of 2001 on an attempt to aid those who have difficulty with mathematics. Judging from results thus far this has been a success, with all learners passing.
In grade 8 and 9 an OBE approach to teaching mathematics has been introduced with some cross-curricular work but in the light of uncertainty with what follows an emphasis has been kept on content. In Grade 9 (and in Grade 8) the learners are taught the basics in Algebra and Geometry which they will require in their senior years at school. There are a few variations to this. For example, the Grade 9 learners do an introductory section of statistics in which they do an assignment where they collect and analyse data. The external CTA (common task for assessment) was introduced in 2002 for the first time to the Grade 9
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Nuffield Advanced Mathematics
This Nuffield Foundation Project was a collaboration which drew on the expertise of the Supported Self Study Unit in Northumberland, the Mechanics in Action project in Manchester, teachers in several Local Authorities as well as the insights of team members at the King's College London and the West Sussex Institute of Higher Education.
The course focused on the differing needs of individuals by offering students the opportunity to follow their own interests, to apply mathematics in solving real problems and to study a range of options.
Key features
Nuffield Advanced Mathematics course had the following features built into it.
• Encouragement for students to co-operate, to take responsibility for their own learning, and to use resources in an independent way
• An emphasis on building good intuitive understanding, through experimenting with graphical and numerical approaches before or in parallel with more formal analytical approaches.
• The use of graphics calculators and spreadsheets
• Mathematical modeling
• Exploratory data analysis based on large datasets
• The use of algorithms to improve students' understanding of mathematics.
• Investigative and exploratory work with students being expected to write about mathematics, give presentations, and use technology appropriately.
• Wider reading about mathematics with the help of a Mathematics Reader
• Aspects of the history of the development of mathematical concepts to add extra insight and interest.
Course structure
The AS and A-level courses were designed in units of learning which were planned to take about 10 to 20 hours each. This included time spent both in and out of the classroom.
The AS course assumed a total of 300 hours of student learning time, and was based on 15 units of work plus some work of the student's own choice. The A-level course involved an additional nine units plus one option. An option was intended to take about 75 hours of student learning timeThe teacher's notes for Nuffield Advanced Mathematics were designed to help teachers organise and to support their work with students.
The student materials were designed to support a flexible approach to teaching and learning mathematics. The teacher's notes made suggestions for varying teaching approaches. They encouraged…
The five books covering the core course of Nuffield Advanced Mathematics each consisted mainly of activities through which students could develop their understanding of mathematical ideas and results, or could apply their knowledge and understanding to problems of various kinds. The activities were designed to make full use of graphics…
As part of the Nuffield Advanced Mathematics A-level course, each student had to choose an option to study from a list of eight titles. Students could take Further Mathematics at AS or A level by taking a further three or six options.
The option books were intended to be as suitable for individual work and independent study as…
In addition to the core text books and the options, the Nuffield Advanced Mathematics project published two other resources for students to support their learning and broaden their appreciation of mathematics.
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What is the difference between matrix theory and linear algebra? - MathOverflow most recent 30 from is the difference between matrix theory and linear algebra?kolistivra2010-01-13T17:17:41Z2010-04-04T18:10:56Z
<p>Hi,</p>
<p>Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the difference,if any, between matrix theory and linear algebra?</p>
<p>Thanks!</p>
by Steve Huntsman for What is the difference between matrix theory and linear algebra?Steve Huntsman2010-01-13T17:23:10Z2010-01-13T17:23:10Z<p>The difference is that in matrix theory you have chosen a particular <a href=" rel="nofollow">basis</a>.</p>
by Qiaochu Yuan for What is the difference between matrix theory and linear algebra?Qiaochu Yuan2010-01-13T18:29:28Z2010-01-13T18:29:28Z<p>Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences.</p>
<p>In linear algebra, however, you instead talk about <strong>linear transformations,</strong> which are <strong>not</strong> (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a <strong>choice of basis.</strong> Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.</p>
by Anweshi for What is the difference between matrix theory and linear algebra?Anweshi2010-01-24T14:00:45Z2010-03-30T21:57:38Z<p>Matrix theory is the specialization of linear algebra to the case of finite dimensional vector spaces and doing explicit manipulations after fixing a basis. More precisely: The algebra of $n \times n$ matrices with coefficients in a field $F$ is isomorphic to the algebra of $F$-linear homomorphisms from an $n$-dimensional vector space $V$ over $F$, to itself. And the choice of such an isomorphism is precisely the choice of a basis for $V$. </p>
<p>Sometimes you need concrete computations for which you use the matrix viewpoint. But for conceptual understanding, application to wider contexts and for overall mathematical elegance, the abstract approach of vector spaces and linear transformations is better.</p>
<p>In this second approach you can take over linear algebra to more general settings such as modules over rings(PIDs for instance), functional analysis, homological algebra, representation theory, etc.. All these topics have linear algebra at their heart, or, rather, "is" indeed linear algebra..</p>
by Konrad Waldorf for What is the difference between matrix theory and linear algebra?Konrad Waldorf2010-03-30T22:06:44Z2010-03-30T22:06:44Z<p>Let me quote without further comment from Dieudonné's "Foundations of Modern Analysis, Vol. 1".</p>
<blockquote>
<p>There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices. </p>
</blockquote>
by John Stillwell for What is the difference between matrix theory and linear algebra?John Stillwell2010-03-31T08:06:31Z2010-03-31T08:06:31Z<p>A counter-quotation to the one from Dieudonné:</p>
<blockquote>
<p>We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.</p>
</blockquote>
<p>(Irving Kaplansky, writing of himself and Paul Halmos)</p>
by zhaoliang for What is the difference between matrix theory and linear algebra?zhaoliang2010-04-01T14:45:47Z2010-04-01T14:45:47Z<p>My opinion: matrix theory mostly deals with matrix of a paticular kind , or a few relevant ones. But linear algebra cares about the general, underlying structrue. </p>
by Jon for What is the difference between matrix theory and linear algebra?Jon2010-04-01T21:08:12Z2010-04-04T18:10:56Z<p>Although some years ago I would have agreed with the above comments about the relationship between Linear Algebra and Matrix Theory, I DO NOT agree any more! </p>
<p>See, for example Bhatia's "Matrix Analysis" GTM book. For example, doubly-(sub)stochastic matrices arise naturally in the classification of unitarily-invariant norms. They also naturally appear in the study of quantum entanglement, which really has nothing to do with a basis. (In both instances, all sorts of NONarbitrary bases come into play, mainly after the spectral theorem gets applied.)</p>
<p>Doubly-stochastic matrices turn out to be useful to give concise proofs of basis-independent inequalities, such as the non-commutative Holder inequality:</p>
<p>tr |AB| $\le$ $||A||_p$ $||B||_q$</p>
<p>with 1/p+1/q=1, $|A|=(A^*A)^{1/2}$, and $||A||_p = (tr |A|^p)^{1/p}$</p>
by XX for What is the difference between matrix theory and linear algebra?XX2010-04-02T02:24:30Z2010-04-02T02:24:30Z<p>I'm with Jon. Matrices don't always appear as linear transformations. Yes, you can look at them as linear transformations, but there are times when it's better not to and study them for their own right. Jon already gave one example. Another example is the theory of positive (semi)definite matrices. They appear naturally as covariance matrices of random vectors. The notions like schur complements appear naturally in a course in matrix theory, but probably not in linear algebra.</p>
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Complex Numbers from A to ...Z2994
FREE
About the Book
It is impossible to imagine modern mathematics without complex numbers. Complex Numbers from A to . . . Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics.
The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.
The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.
The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics.
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This book introduces students with diverse backgrounds to various types of mathematical analysis that are commonly needed in scientific computing. The subject of numerical analysis is treated from a mathematical point of view, offering a complete analysis of methods for scientific computing with appropriate motivations and careful proofs.
In an engaging and informal style, the authors demonstrate that many computational procedures and intriguing questions of computer science arise from theorems and proofs. Algorithms are presented in pseudocode, so that students can immediately write computer programs in standard languages or use interactive mathematical software packages.
This book occasionally touches upon more advanced topics that are not usually contained in standard textbooks at this level.
An instructor's manual for this title is available electronically. Please send email to textbooks@ams.org for more information.
Readership
Undergraduate and graduate students interested in applied mathematics and scientific computing.
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QAX - The Complete Mathematics Solution
What is QAX?
QAX Mathematics can be used for your school's entire mathematics program. It covers the secondary curriculum for each Australian state, negating the need for other teaching resources. Each topic within the curriculum has a supply of questions and an introduction including examples of questions with full working out.
Teachers can create work programs or question sets (called Q Sets) for each topic from the 50,000 available questions.
Students complete questions online or as printed worksheets.
The program assists students by providing example questions, hints and solutions.
A fully worked solution is shown to students once they have answered the question correctly or made two incorrect attempts.
Students who have answered incorrectly are shown the correct solution and then given similar questions to answer.
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Prerequisites: MATH20132 Calculus of Several Variables; MATH20222 Introduction to Geometry (optional); MATH31051 Introduction to Topology (optional)
Future topics requiring this course unit: differentiable manifolds are used in almost all areas of mathematics and its applications, including physics and engineering.
Details of prerequisites: standard calculus and linear algebra; familiarity with the statements of the implicit function theorem the existence and uniqueness theorem for ODEs (students may consult any good textbooks in multivariate calculus and differential equations); some familiarity with algebraic concepts such as groups and rings is desirable, but no knowledge of group theory or ring theory is assumed (students may refresh basic definitions using, e.g., J. Fraleigh, A First Course in Abstract Algebra); some familiarity with basic topological notions (topological spaces, continuous maps, Hausdorff property, connectedness and compactness) may be beneficial, but is not assumed (a good source is M. A. Armstrong, Basic Topology).
Last modified: Monday 8 (21) May 2012. (Refresh the browser to get the updated page.)
Differentiable manifolds are among the most fundamental notions of modern mathematics. Roughly, they are geometrical objects that can be endowed with coordinates; using these coordinates one can apply differential and integral calculus, but the results are coordinate-independent.
Examples of manifolds start with open domains in Euclidean space Rn, and include "multi-dimensional surfaces" such as the n-sphere Sn and n-torus Tn, the projective spaces RPn and CPn, and their generalizations, matrix groups such as the rotation group SO(n), etc. Differentiable manifolds naturally appear in various applications, e.g., as configuration spaces in mechanics. They are arguably the most general objects on which calculus can be developed. On the other hand, differentiable manifolds provide for calculus a powerful invariant geometric language, which is used in almost all areas of mathematics and its applications.
In this course we give an introduction to the theory of manifolds, including their definition and examples; vector fields and differential forms; integration on manifolds and de Rham cohomology.
Textbooks:
No particular textbook is followed. Students are advised to keep their own lecture notes and use my notes posted on the web. There are many good sources available treating various aspects of differentiable manifolds on various levels and from different viewpoints. Below is a list of texts that may be useful. More can be found by searching library shelves.
Exam structure
The exam paper will consist of 4 questions (for 10 credit version) or 5 questions (for 15 credit version). You will have to answer any 3 questions out of Questions 1 to 4 (everybody). Those taking 15 credit version will also have to answer Question 5 (compulsory). Therefore, those taking 10 credit version will altogether answer 3 questions and those taking 15 credit version, 4 questions.
Questions 1 to 4
Each of them will consist of parts (a), (b), and (c):
(a) A definition or a group of related definitions and a question concerning a simple statement or example directly related with the definition.
(b) An important statement from the course (a theorem, or a lemma, or a part of a theorem). Possibly requiring some definition(s). You will be asked either to give the full statement and prove it; or quote a general statement and then prove some part of it; or you will be given a statement and you will have to give a proof; and/or you may be asked to deduce something from a general statement.
(c) A problem where you will have to calculate something or to show something about a concrete example.
Compulsory question 5 (15 credit version only)
(a) A definition and an important statement. You will have to quote some statements and give a proof.
(b) and (c) An advanced problem, subdivided into two parts.
Main exam topics
Questions 1 and 2. Charts, atlases, smoothness. Smooth manifolds and smooth maps. Diffeomorphism. Algebra of smooth functions. Specifying manifolds by equations. Submanifolds. Products. Tangent vectors and tangent spaces. Velocity of a parameterized curve. The natural basis of tangent vectors associated with a coordinate system. Tangent bundle. Partitions of unity.
Questions 3 and 4. Derivations of an algebra and derivations over an algebra homomorphism. Vectors and vector fields as derivations. Commutator of vector fields. Exterior differential: axiomatic definition and properties. Integration of forms and Stokes theorem. Closed and exact forms. De Rham cohomology: definition and examples. Pull-back and homotopy invariance of de Rham cohomology. Application to distinguishing manifolds.
Question 5. Embedding manifolds into RN. Existence of an embedding. Corollary from Sard's Lemma. Reducing the dimension of the ambient space (Whitney's Theorem). See § 4.3 of the online notes. Nota bene: This question will also include a more advanced problem concerning differential forms, Stokes theorem and de Rham cohomology.
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...Algebraic math is a major stepping stone to multiple sciences and must be mastered to facilitate future academic progress in the sciences. As algebra skills consolidate, we move toward calculus (differential and integration calculus) which employs extremely powerful math skills. Pre-calculus is the bed rock of algebraic math upon which calculus stands
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This book is an excellent introduction to the Fourier analysis. It is the first volume of the four planned volumes based on a series of four one-semester courses taught at Princeton University whose purpose was to present, in an integrated manner, the core areas of analysis. As the authors write: ``The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.''\par The reviewer is in favour of the concept of the authors given in the Preface to Volume I: ``Not to overburden the beginning student with some of the difficulties that are inherent in the subject: a proper appreciation of the subtleties and technical complications that arise can come only after one has mastered some of the initial ideas involved. This point of view has led us to the following choice of material in the present volume:\par {\it Fourier series}. At this early stage it is not appropriate to introduce measure theory and Lebesgue integration. For this reason our treatment of Fourier series in the first four chapters is carried out in the context of Riemann integrable functions.\dots\par {\it Fourier transform}. For the same reasons, instead of undertaking the theory in a general setting, we confine ourselves in Chapter 5 and 6 largely to the framework of test functions.\dots\par {\it Finite Fourier analysis}. This is an introductory subject {\it par excellence}, because limits and integrals are not explicitly present. Nevertheless, the subject has several striking applications, including the proof of the infinitude of primes in an arithmetic progression.''\par The prerequisites are kept to a minimum, only some acquaintance with the notion of the Riemann integral is supposed, but an appendix contains most of the results needed in the text.\par For further orientation here is the list of chapter headings: The Genesis of Fourier Analysis, Basic Properties of Fourier Series, Convergence of Fourier Series, Some Applications of Fourier Series, The Fourier Transform on $\bbfR$, The Fourier Transform on $\bbfR^d$, Finite Fourier Analysis, Dirichlet's Theorem. Furthermore there are Appendix, Notes and References, Bibliography, Symbol Glossary.\par Each chapter ends with Exercises and Problems. If the reader solves these problems, some of them are very hard in spite of the useful hints, he gets a very good practice in analysis.\par The reviewer knows, what the authors hope, that their approach will facilitate their goal: ``To inspire the interested reader to learn more about this fascinating subject, and to discover how Fourier analysis affects decisively other parts of mathematics and science.''\par We warmly recommend this valuable work to everybody, from students of mathematics, physics, engineering and other sciences, to teachers and mathematicians, who is interested in Fourier series. [Laszlo Leindler (Szeged)]
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Algebraic Videogame Programming
Bootstrap is a FREE curriculum for students ages 12-16, which teaches them to program their own videogames using purely algebraic and geometric concepts.
Our mission is to use students' excitement and confidence around gaming to directly apply algebra to create something cool.
We work with schools, districts and tech-educational programs across the country, reaching hundreds of students each semester. Bootstrap has been integrated into math and technology classrooms across the country, reaching thousands of students since 2006.
Programming. Not just writing code.
Knowing how to write code is good, but it doesn't make you a programmer.
Sure, Bootstrap teaches students a programming language. But most
importantly, it teaches solid program design skills, such as
stating input and types, writing test cases, and explaining code to others.
Bootstrap builds these elements into the curriculum in a gentle way
that helps students move from a word problem to finished code.
After Bootstrap, these skills can be put to use in other programming environments, letting students take what they've learned into other programming classes.
Watch the video to hear students, engineers, teachers, and the Bootstrap team describe what excites them about Bootstrap!
Real, Standards-Based Math
Unlike most programming classes, Bootstrap uses algebra as the vehicle for creating images and animations. That means that concepts students encounter in Bootstrap behave the exact same way that they do in math class. This lets students experiment with algebraic concepts by writing functions that make a rocket fly (linear equations), respond to keypresses (piecewise functions) or make it explode when it hits a meteor (distance formula). In fact, many word problems from standard math textbooks can be used as as programming assignments!
The entire curriculum is designed from the ground up to be aligned with Common Core standards for algebra. Bootstrap lessons cover mathematical topics that range from simple arithmetic expressions to the Pythagorean Theorem, Discrete Logic, Function Composition and the Distance Formula. The program is based on cognitive science research and best practices for improving critical thinking and problem solving.
""
—
Our team
Bootstrap is the creation of Emmanuel Schanzer, M.Ed. (in the hat). After earning a bachelors of Computer Science (Cornell University), he worked in the private sector for a number of years as a programmer (Microsoft, Vermonster, and others) until he switched careers and became a math teacher, starting out in Boston Public Schools. He is now a doctoral student at the Harvard Graduate School of Education.
Our Supporters
We would like to thank the following, for their volunteer and financial support over the years: Apple, Cisco, the Entertainment Software Association (ESA), Facebook, Google, as well as the Google Inc. Charitable Giving Fund of Tides Foundation, IBM, Jane Street Capital, LinkedIn, Microsoft, The National Science Foundation, NVIDIA, Thomson/Reuters, and the generous individuals who have given us private donations.
If you would like to support Bootstrap with a donation, send a check made out to Brown University to our PI, Shriram Krishnamurthi,
at his mailing address. Be sure to include this letter, indicating that you wish for the funds to be put towards Bootstrap. Once your check is received, we'll send you a reciept for your tax records.
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
Definite Integral Word ProblemsExample 1 An empty bucket is placed under a tap and filled with water. t minutes after the bucket has been placed under the tap. The rate of flow of water into the bucket is equal to 2.3 - 0.1t gallons per minute. How
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INTRODUCTION TO THE TI-83 AND TI-83 PLUS BasicsKeyboardEach key on the TI-83 and TI-83 Plus accesses up to three objects, operations, or menus. The primary object, operation, or menu is written on the key. Above each key are other objects, operatio
TI-83 (+) Keystrokes for Chapter 4 of Understanding Basic StatisticsItems in boxes are actual keys; other items are menu choices (selected with arrow keys, or the key). Some keys have text above them; this is given [in brackets]. A vertical line li
Just the Basics: Regression on the TI-83 Before You Do Your First Regression on the TI-83: Press CATALOG (2nd 0) and press the D key to jump down to the commands that start with the letter D. Use the down arrow to move the triangle down until it is t
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Journal of Online Mathematics and its Applications
Lite Applets
Introduction
The following image is "live" -- click on it to see what happens. Then think about how you might use this tool in material for mathematics instruction.
This "image" has two parts: a static image (the map), for which you can substitute any image you like, and a "Lite Applet" called Image_and_Cursor, which we will describe in more detail in the following pages. Our goals are
to illustrate the flexibility, adaptability, and ease of use of this and other lite applets, and
to convince you that you can make use of these tools to design your own interactive instructional materials.
Lite applets are flexible and powerful tools that can be used as part of highly interactive curriculum modules that are scientifically and pedagogically sound. The power and flexibility of lite applets are based on three ideas:
Lite applets use parameters to enable a curriculum developer to control their appearance and functionality.
Lite applets can produce output that is easily cut-and-pasted into a spreadsheet or a computer algebra system, in which a student can analyze the data. This provides students with more power and creative control, and it helps them build general purpose skills.
By using Javascript and HTML forms (which are much easier to use than Java), together with applet parameters, curriculum developers can create very interactive materials without knowing Java.
Image_and_Cursor is the first entry in the Lite Applet Collection, an open source collection housed in MathDL. All the files used in this article, including the Java source code files, are freely available and may be downloaded for your own use. Links are provided throughout the article and on our Resources page. We invite you to submit your own lite applets and modules to be considered for inclusion in this collection. Contact any of the authors for further information. We expect to add new lite applets and new modules using those applets to the collection regularly.
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Description
Mathematical proofs, functions, formulas—oh, and word problems too: welcome to Algebra, the branch of math that provides tools to solve for the unknown. This video course will take you through the kinds of topics and problems covered in Algebra I and II courses. It's a great introduction for those who have never taken Algebra and a nice refresher for those who remember the general rules of solving for X, but who want to rediscover their polynomials and parabolas.
Note that given the length of these lessons, you may want to adjust your settings to receive one or two lessons a week.
Opening Lines (Experimental)
Today's Algebra lesson (in video) from the Khan Academy is: Simple Equations: To view other Khan Academy videos, you can find them at their website here: Enjoy! P.S. Note that given the length of these lessons, you may want to adjust your settings to receive one or two lessons a ...
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AcademicsNumber Theory
Course Outline: Number theory is primarily
concerned with the properties of and relationships between
whole numbers. Topics we will study:
1. Prime numbers
2. Modular arithmetic
3. Sums of squares
4. Pythagorean triples
5. Fermat's Last Theorem
6. Magic squares
7. Continued fractions
8. Approximation of reals by rationals
We will also spend a couple of weeks studying cryptography.
In particular, we will look at how the RSA system works.
This relies heavily on some of the number theory we will have
learnt and is behind almost all modern cryptographic systems.
You will need two books for the course: "A Pathway
Into Number Theory" by R.P. Burn and "An Introduction
To Number Theory" by H. Stark. Burn's book will
lead us to discover and prove for ourselves some of the main
results of number theory. Stark's book is more traditional.
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Miles Reid's Undergraduate Algebraic Geometry is an excellent topical (meaning it does not intend to cover any substantial part of the whole subject) introduction. In particular, it's the only undergraduate textbook that isn't commutative algebra with a few pictures thrown in.
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Classics 2123: The Roman Way Lecture 1 ~ Eras of Roman History Reading: BHR 2-4.Spring 2008Roman history is traditionally divided into three phases: monarchy, republic and empire. The dates are as follows (keep in mind that the very earliest date
Classics 2123: The Roman Way Notes to Virgil's Aeneid, Books I-IVSpring 2008Virgil's Aeneid, strongly modeled on Homer's Iliad and Odyssey, was considered even in antiquity the great epic of Rome. It tells the story of the Trojan Aeneas who, afte
Classics 2123: The Roman Way Notes to Virgil's Aeneid, Books V-VIIISpring 2008Book V provides a transition between the high emotion of Book IV and the sombre majesty of the descent to the underworld in Book VI. Most of the book is taken up with g
Classics 2123: The Roman Way Notes to Virgil's Aeneid, Books IX-XIISpring 2008Book IX. War finally breaks out, the full-scale battles spoken of in Book VII. The book divides into three sections: (1) Turnus and the Rutulians attack the Trojan ship
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Binomial Theorem
In the last lecture of Algebra 2, Dr. Eaton ends with Binomial Theorem. She first begins with Pascal's Triangle and how it represents the expansion of a binomial. Next, she moves into the properties as well as a refreshment course on factorials before the binomial theorem and how to find a specific term. Four examples at the end make sure you can apply all that you have learned.
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Binomial Theorem
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
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GCS - Algebra I Curriculum Guide
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gaston county schools algebra i curriculum guide course description ·the purpose of the algebra i course is to formalize and extend the mathematics that students learned in the middle grades this course deepens and extends understanding of linear relationships by contrasting them with exponential and quadratic phenomena and by applying linear models to data that exhibit a linear trend in addition to studying bivariate data students also summarize represent and interpret data on a single count or measurement variable progressing from the geometric experiences in the middle grades students explore more complex geometric situations and deepen their understanding of geometric relationships moving toward formal mathematical arguments ·the standards for mathematical practice apply throughout each course and together with the content standards require that students experience mathematics as a coherent useful and logical subject that makes use of their ability to make sense of problem situations ·successful completion of this course requires a passing score in the class and a level iii or iv on the state mandated math i eoc course i can statements ·i can fluently write interpret and translate between various forms of linear equations and inequalities and use them solve problems ·i can analyze and explain the process of solving an equation inequality and/or system of equations for linear relationships exponentials and quadratics ·i can create analyze interpret and compare linear exponential and quadratic functions from multiple representations ·i can describe geometric relationships using precise vocabulary and appropriate symbolic notation ·i can use the rectangular coordinate system to find distance midpoint and verify geometric relationships including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines ·i can use regression techniques graphical representations and knowledge of context to describe approximate linear relationships between quantities make judgements concerning the appropriateness of a linear model and use residuals to analyze goodness of fit course standard links ·ncdpi http maccss.ncdpi.wikispaces.net/sixth+grade ·gcs middle school math website https sites.google.com/a/gaston.k12.nc.us/gcs-secondary-math/msmath ·gcs common core live binder https incorporated standards ·common core literacy standards http ·information and technology standards http ·english language development standards http ·extended content standards http created by the gcs high school teacher leader team draft revised 5/2012 to be implemented 2012-2013
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gaston county schools timeline 2 weeks unit unit 1 data analysis major concepts dot plots histograms box plots key features of graphs avg rate of change from graph shape center and spread outliers two-way frequency tables joint marginal and conditional relative frequency compute joint conditional and marginal probabilities correlation vs causation ccss-m s.id.1 s.id.2 s.id.3 s.id.5 s.id.7 s.id.8 s.id.9 ccss-m implementation 2012-2013 i can statements i can take a set of data and plot it on the number line i can compare data sets using statistics that are appropriate to the shapes of the distributions i can draw conclusions from the shape and individual points of a data set i can summarize two categorical variables using two-way frequency tables i can interpret the slope and intercept for a linear model i can find and explain the correlation coefficient for a linear model i can describe the difference between correlation and causation essential vocab first six weeks association categorical causation conditional distribution conditional probability distribution correlation explanatory variable five number summary interquartile range joint probability marginal distribution marginal probability maximum/minimum measures of center outliers quantitative relative frequency residual response variable shape center spread skew standard deviation trend two-way table 3 weeks unit 2 solving graphing linear equations simplifying linear expressions write and solve linear equations in one variable algebraic proof graphing in the coordinate plane geometric terms definitions and symbols define and name and lines using appropriate symbols prove and lines using slope writing equations of lines midpoint a.sse.1.a a.sse.1.b a.rei.1 a.rei.3 a.ced.1 a.ced.2 a.ced.3 a.ced.4 a.rei.10 g.co.1 g.gpe.5 g.gpe.6 i can identify the parts of linear expression i can identify and utilize the important parts of a linear expression or formula i can solve a simple linear equation in one variable i can solve linear equations in one variable i can write and solve a linear equation using one variable i can write a linear equation using two or more variables i can graph a linear equation with appropriate labels and scales i can determine whether a solution is viable to the given problem i can solve a literal equation i.e formula for a specified variable i can determine the solution set for a linear equation from a graph i can identify locate and name points in the coordinate plane i can identify and name lines in the coordinate plane i can define and name parallel and perpendicular lines in the coordinate plane i can define and name line segments in the coordinate plane i can determine if two lines are parallel or perpendicular i can create linear equations using slope and any point on the line i can define the midpoint of a line segment i can determine the midpoint of a line segment argument assumption bisect coefficient congruent constant dependent variable equidistant independent variable line line segment literal equation midpoint parallel line perpendicular line properties of equality proof replacement set slope solution set substitution symbolic notation t-chart term variable glencoe algebra i 1.1-1.2 1.4-1.6 2.1-2.4 2.7 8.1-8.8 9.1-9.6 11.1 12.2-12.8 1.3 1.8 3.1-3.5 3.6 3.8 4.1-4.3 4.5-4.8 5.15.6 6.1-6.6 12.9 created by the gcs high school teacher leader team draft revised 5/2012 to be implemented 2012-2013
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gaston county schools timeline 3 weeks unit unit 3 linear functions best fit lines major concepts domain/range linear function arithmetic sequences recursive/explicit formulas rate of change comparing properties of functions linear regression scatter plots ccss-m9 f.bf.3 f.le.5 f.le.2 s.id.6 s.id.7 s.id.8 s.id.9 ccss-m implementation 2012-2013 i can statements i can distinguish between domain and range i can use and interpret equations written in function notation i can relate sequences and functions i can create functions that describe a relationship between two quantities i can model recursive and explicit formulas to create arithmetic sequences i can use recursive and explicit formulas to model and solve real-world and mathematical problems i can identify and interpret features of linear functions i can draw a sketch of linear function in the coordinate plane i can identify domains of a function in the context of a real-world or mathematical problem i can determine the average rate of change i can graph linear functions expressed symbolically i can identify and interpret key features of a linear function i can compare properties of two linear function each represented in a different way i can create functions that describe a relationship between two quantities i can model recursive and explicit formulas to create arithmetic sequences i can use recursive and explicit formulas to model and solve real-world and mathematical problems i can construct a linear regression model i can interpret and explain the parameters of a linear model in context i can create and use scatter plots to describe how two quantitative variables are related i can interpret the slope and intercept for a linear model i can find and explain the correlation coefficient for a linear model i can describe the difference between correlation and causation i can use scatter plots and linear models to draw conclusions and make predictions to solve real-world and mathematical problems essential vocab arithmetic sequences association causation correlation domain explanatory variable explicit fitted function function function notation horizontal shift impractical intercepts joint probability marginal distribution marginal probability parent graph practical range rate of change recursive regression model relative frequency response variable residual scatter plot transformation trend two-way table horizontal shift parent graph transformation translation vertical shift bounded dependent variable independent variable literal equation solution unbounded second six weeks 1 weeks unit 4 linear inequalities simplifying linear expressions write and solve linear equations in one variable algebraic proof graphing in the coordinate plane geometric terms definitions and symbols define and name and lines using appropriate symbols prove and lines using slope writing equations of lines midpoint a.rei.3 a.rei.12 a.ced.1 a.ced.3 i can solve linear inequalities in one variable i can can determine the solution set for a linear inequality from a graph i can graph linear inequalities in the coordinate plane with appropriate labels and scales i can write and solve a linear inequalities using one variable i can write a linear inequalities using two or more variables i can determine whether a solution is viable to the given problem glencoe algebra i 6.1-6.6 created by the gcs high school teacher leader team draft revised 5/2012 to be implemented 2012-2013
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gaston county schools 3 weeks unit 5 non-linear prerequisites geometry ccss-m implementation 2012-2013 laws of exponents rational exponents simplifying radicals operations with polynomials factoring expressions geometric terms definitions and symbols coordinate geometry proofs geometric formulas n.rn.1 n.rn.2 a.apr.1 f.if.8.a g.co.1 g.gpe.4 g.gpe.7 g.gmd.1 g.gmd.3 i can explain the definition of rational exponents and how it relates to algebra i pacing guide radicals i can rewrite expressions with rational exponents as radical expressions and vice versa i can simplify radical expressions involving square and cube roots i can use the properties of integers operations to simplify polynomials of a higher degree i can solve real-world and mathematical problems involving polynomial expressions and equations i can create equivalent expressions by factoring i can solve a quadratic equation i can define and name an angle i can define and name a circle i can define and name line segments i can identify polygons based on defining characteristics i can use coordinates to prove or disprove statements about plane figures i can compute perimeters and areas of polygons in the coordinate plane i can explain area circumference and volume formulas i can use volume formulas for cylinders pyramids cones and spheres to solve problems angle area of a circle circle circumference cone cylinder distance line segment monomial plane figures point polynomial properties of polygons pyramid radical expression rational exponent ray sphere volume created by the gcs high school teacher leader team second six weeks draft revised 5/2012 to be implemented 2012-2013
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gaston county schools timeline 2 weeks unit unit 6 non-linear models major concepts extend concept of function to quadratics and exponentials create solve graph interpret exponentials/quadratics build functions using geometric sequences interpret and find key features of graphs choose appropriate domains and parameters for graphs compare properties of functions represented various ways add and subtract standard function types interpret vertical/horizontal shifts graphically/algebraically recognize and compare rates of linear/exponential growth ccss-m a.sse.1.a a.sse.1.b a.sse.2 a.sse.3 a.ced.1 a.ced.2 a.rei.107.e f.if.8.a f.if.8.b f.le.5 f.if.9 f.bf.3 f.le.1 f.le.2 f.le.3 ccss-m implementation 2012-2013 i can statements i can write and solve exponential functions i can write and graph linear quadratic and exponential functions i can understand that the ordered pairs on a line are the solutions to the equation being graphed i can identify quadratics and exponentials as functions i can describe the domain and range of a function i can evaluate fx for non-linear functions i can interpret key features of graphs and tables for example intercepts increasing and decreasing intervals relative maximums and minimums and symmetries i can choose appropriate domains and parameters for non-linear functions i can find the average rate of change of a non-linear model i can graph exponential and quadratic functions show intercepts maxima minima and end behavior using technology for the more complicated cases and by hand in simple cases i can use the properties of exponents to interpret expressions for exponential functions for example classify exponential growth and decay i can compare properties of two functions linear exponential and quadratic each represented in a different way for example algebraically and graphically i can write a function that describes a relationship between two quantities i can write arithmetic and geometric sequences and use them to model situations i can translate exponential functions i can identify the differences between exponential quadratic and linear models including observing that exponential functions grow at a faster rate than linear or quadratic functions i can construct exponential functions given different representations of data i can interpret the parameters in an exponential function in terms of a context i can solve systems of linear equations by elimination substitution and graphing i can estimate the solution in the terms of the system when graphing i can solve systems of exponential equations using technology by graphing or a table of values i can solve systems of inequalities by graphing and explain the meaning of the shaded and non-shaded regions in terms of the system i can create constraints linear inequalitites and apply those constraints to solve real world applications linear programming i can interpret the maximum and minimum values yielded by my objective function essential vocab function function notation domain range exponential quadratic geometric sequence growth/decay algebra i pacing guide textbook correlations glencoe algebra i 10.1-10.6 third six weeks 2 weeks unit 7 systems represent constraints by systems of equations and inequalities and interpret viability of solutions in context prove the process for solving a system of equations by substitution algebraically solve systems of linear equations exactly and approximately find approximate solutions to intersecting curves using technology and/or tables of values explain the meaning of two curves in the coordinate plane graph systems of linear inequalities in two variables a.ced.3 a.rei.5 a.rei.6 a.rei.11 a.rei.12 elimination substitution additive inverse consistent independent dependent inconsistent constraint system of equations system of inequalities intersection infinitely many solutions glencoe algebra i 7.1-7.5 5.6 created by the gcs high school teacher leader team draft revised 5/2012 to be implemented 2012-2013
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Maths from Scratch for Biologists
Numerical ability is an essential skill for everyone studying the biological sciences but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Having taught introductory maths and statistics for many years, Alan Cann understands these challenges and just how invaluable an accessible, confidence building textbook could be to the fearful student. Unable to find a book pitched at the right level, that concentrated on why numerical skills are useful to biologists, he wrote his own. The result is Maths from Scratch for Biologists , a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological sciences.
Features:
* An accessible, jargon-busting approach to help readers master basic mathematical, statistical and data handling techniques in biology
* Numerous end of chapter problems to reinforce key concepts and encourage students to test their newly acquired skills through practise
* A handy, time-saving glossary
* A supplementary website with numerous problems and self-test exercises
Alan Cann has worked in both the UK and USA, and in addition to teaching undergraduate and postgraduate biologists and medical students, he runs an active research laboratory at the University of Leicester, UK, studying the molecular biology and pathogenesis of viruses. He has been awarded numerous grants for educational research and was the inaugural winner of the Society for General Microbiology UK Wildy prize for Education in 2001.
Biology students will gain an appreciation of the basic mathematical, statisical and data handling techniques that they will need throughout their undergraduate career.
These skills are introduced using a problem-solving approach that emphasises the biological background of the book rather than the mathematical theory.
This book is written for biologists by a biologist with a proven track record as the author of a successful virology textbook.
Students purchasing the book are able to access a password-protected web site for the book that includes numerous problems and self-help exercises, to help them develop confidence in their mathematical abilities through practise.
Available Versions
Maths from Scratch for Biologists
by Alan J. Cann
ISBN 978-0-471-49835-3
December 2002
Paperback, 240 pages
US $55.95Add to CartThis is a Print-on-Demand title. It will be printed specifically to fill your order. Please allow an additional 5-6 days delivery time. The book is not returnable.
Maths from Scratch for Biologists
by Alan J. Cann
ISBN 978-0-471-49834-6
December 2002
Hardcover, 240 pages
US $239.95Add to CartThis is a Print-on-Demand title. It will be printed specifically to fill your order. Please allow an additional 5-6 days delivery time. The book is not returnable.
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Course Description:This course is built on the basic concepts and principles in Algebra I. Students will work extensively with nonlinear functions while continuing to develop their power to think and work systematically. The content outline for Algebra II and Algebra II Honors is the same, but the Algebra II Honors class is structured to meet the needs of the more advanced mathematics students. The objectives in the content outline are covered in greater depth and in a more rigorous manner. The Algebra II Honors course is for those students who wish to advance to Calculus during their senior year
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Monday, June 25, 2012
Real math! Romance! Karate! That's what the back of this unusual math book promises to its readers. Odd as it may seem to teach math with a comic book, The Manga Guide to Linear Algebra does a pretty good job of teaching the basics through comics and some light storytelling.
The book starts off with an introduction to the story's main characters. Reiji wants to learn martial arts so he can stop being a wimp. The head of the karate club, Tetsuo, agrees to let him join on one condition: he has to tutor Tetsuo's sister, Misa, in linear algebra. Most the book's instruction is given from the perspective of Reiji teaching Misa, and the story of Reiji's efforts to get stronger in karate and woo Misa are sprinkled in between the lessons.
I read this book from the perspective of someone who enjoyed and did well in linear algebra. I still use many of the basics today, but have not studied the minute details for a while. (For example, I haven't had to do Gaussian elimination in ages.) So I read this book as a bit of a refresher for myself, and tried to guess how easily someone new to the subject would grasp the concepts.
For my purposes, it was great. I definitely understood everything explained in the book, and even learned a few new tricks for solving problems. For new learners, I had some mixed feelings, but felt positively about it overall.
The comic book form worked really well for setting up an informal conversation for each lesson as Reiji explained the math to Misa. This allowed for more colloquial language and some back-and-forth questioning and discussion that you'd never include in a formal textbook. The illustrations were also helpful, allowing for extra imagery, even if just to set up a useful metaphor (such as a character using a broom for the sweeping concept in Gaussian elimination). The images were especially helpful when explaining vectors, which I thought was done particularly well.
On the other hand, the content was not as dense as a regular textbook. Most of the time I didn't think this mattered, especially since the book is openly intended to be supplementary. Even still, there were times I felt like a little extra explanation might help. This was especially true in the non-comic sections between chapters. For example, one discussed combinations and permutations through example, but I felt like I'd be kind of lost after just those few pages on the topic.
As for the story, it successfully got me interested in the characters and their well-being. I wanted Misa to do well in math, and I wanted Reiji to excel in karate. This did give some motivation for reading the math parts in order to get to the next piece of the story.
However, the story didn't really integrate much into the mathematical content beyond one character teaching another. Some examples did include the story's characters and one even related to karate, but really, the story could have been anything and it would have worked just as well. This isn't necessarily a bad thing, since engagement is a good reason to use story. But it would be neat to see an example where the story itself could actually help teach the content better than could be done without any story at all.
Overall, I really enjoyed this book and got through it surprisingly quickly. Despite any small weaknesses discussed above, I would wholeheartedly recommend it to students just learning the subject as well as anyone needing to brush up
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Description
Goldstein's BriefCalculus and Its Applications, Twelfth Edition is a comprehensive print and online program for students majoring in business, economics, life science, or social sciences. Without sacrificing mathematical integrity, the book clearly presents the concepts with a large quantity of exceptional, in-depth exercises. The authors' proven formula–pairing substantial amounts of graphical analysis and informal geometric proofs with an abundance of exercises–has proven to be tremendously successful with both students and instructors. The textbook is supported by a wide array of supplements as well as MyMathLab® and MathXL®, the most widely adopted and acclaimed online homework and assessment system on the market.
This text is designed for a one-semester course in applied calculus.
CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
Table of Contents
0. Functions
0.1 Functions and Their Graphs
0.2 Some Important Functions
0.3 The Algebra of Functions
0.4 Zeros of Functions–The Quadratic Formula and Factoring
0.5 Exponents and Power Functions
0.6 Functions and Graphs in Applications
1. The Derivative
1.1 The Slope of a Straight Line
1.2 The Slope of a Curve at a Point
1.3 The Derivative
1.4 Limits and the Derivative
1.5 Differentiability and Continuity
1.6 Some Rules for Differentiation
1.7 More About Derivatives
1.8 The Derivative as a Rate of Change
2. Applications of the Derivative
2.1 Describing Graphs of Functions
2.2 The First and Second Derivative Rules
2.3 The First and Second Derivative Tests and Curve Sketching
2.4 Curve Sketching (Conclusion)
2.5 Optimization Problems
2.6 Further Optimization Problems
2.7 Applications of Derivatives to Business and Economics
3. Techniques of Differentiation
3.1 The Product and Quotient Rules
3.2 The Chain Rule and the General Power Rule
3.3 Implicit Differentiation and Related Rates
4. Logarithm Functions
4.1 Exponential Functions
4.2 The Exponential Function ex
4.3 Differentiation of Exponential Functions
4.4 The Natural Logarithm Function
4.5 The Derivative of ln x
4.6 Properties of the Natural Logarithm Function
5. Applications of the Exponential and Natural Logarithm Functions
5.1 Exponential Growth and Decay
5.2 Compound Interest
5.3 Applications of the Natural Logarithm Function to Economics
5.4 Further Exponential Models
6. The Definite Integral
6.1 Antidifferentiation
6.2 Areas and Riemann Sums
6.3 Definite Integrals and the Fundamental Theorem
6.4 Areas in the xy-Plane
6.5 Applications of the Definite Integral
7. Functions of Several Variables
7.1 Examples of Functions of Several Variables
7.2 Partial Derivatives
7.3 Maxima and Minima of Functions of Several Variables
7.4 Lagrange Multipliers and Constrained Optimization
7.5 The Method of Least Squares
7.6 Double Integrals
8. The Trigonometric Functions
8.1 Radian Measure of Angles
8.2 The Sine and the Cosine
8.3 Differentiation and Integration of sin t and cos t
8.4 The Tangent and Other Trigonometric Functions
9. Techniques of Integration
9.1 Integration by Substitution
9.2 Integration by Parts
9.3 Evaluation of Definite Integrals
9.4 Approximation of Definite Integrals
9.5 Some Applications of the Integral
9.6 Improper
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For the faculty members whose main focus is teaching
undergraduate students, PCMI offers the opportunity to renew excitement about mathematics,
talk with peers about new teaching approaches, address some challenging research
questions, and interact with the broader mathematical community.
Each year the theme of the UFP bridges the research and education
themes of the Summer Institute. In 2002, Daniel Schaal, South Dakota State University,
will lead the UFP program Directing Undergraduate Research Programs in Mathematics with
an emphasis on combinatorics.
In this program we will address some common problems faced by anyone
directing an undergraduate research program in mathematics. We will consider how to select
and recruit students and how to find appropriate problems for students to investigate. We
will discuss how to quickly teach students the background material they will need and how
to get students started on research. We will also discuss possible sources of funding and
the application procedures of some funding agencies. We will consider large and small
programs, summer and academic year programs, and programs with or without students from
outside the director's home institution.
To illustrate the ideas of problem selection, student preparation
and problem investigation, we will conclude the program with a short investigation in
Ramsey theory. Ramsey theory is an area of graph theory and combinatorics that deals with
colorings of the edges of a graph and colorings of the natural numbers. A coloring is
simply a function that partitions the domain of the function into equivalence classes.
Among other things, Ramsey theory addresses the question of how big the domain of the
coloring must be to guarantee that at least one equivalence class contains a
"special" subset. Ramsey theory is an especially good area for undergraduate
research due to the small amount of background material needed and the abundance of open
problems. We will briefly cover the basics of Ramsey theory and some proof techniques that
are often used in this area. We will then investigate some open problems and may even
solve some.
It is not the goal of this program to prepare the participants to
direct undergraduate research in Ramsey theory. The brief investigation of Ramsey theory
is intended as an example of how an undergraduate research program can be structured. The
techniques of problem selection, student preparation and problem investigation that will
be illustrated in this investigation are applicable to undergraduate research in all areas
of mathematics. One of the programs for undergraduate students at the PCMI will
investigate expander graphs, another area of graph theory and combinatorics. Participants
in this program may also choose to attend some or all of the sessions on expander graphs.
The program on expander graphs can serve as a second example of how an undergraduate
research program can be structured.
All mathematics faculty members interested in undergraduate research
are invited to apply to this program. No knowledge of combinatorics or prior experience
directing undergraduate research is expected or required. However, faculty members with
experience directing undergraduate research are encouraged to apply and will be able to
enrich the program by sharing their experiences with the other participants.
The UFP explores one central course of topic in the
undergraduate curriculum from the dual perspectives of the mathematics itself and its
teaching. Also, the UFP is one of the fundamental sources of meaningful
interaction between PCMI's constituent groups and programs. Some UFP participants
attend courses of the Graduate Summer School each year. A large number are attracted to
the Undergraduate Program, both for the interesting mathematics in the courses and for the
kind of research experiences for undergraduates that the mathematics of the courses
typically generates. Finally some engage teachers of the High School Teachers program in
the examination of transitional issues between high school and early undergraduate
mathematics instruction.
College faculty with a strong interest in undergraduate
education are encouraged to apply to PCMI's Undergraduate Faculty Program.
Prerequisites: two years of undergraduate mathematical
teaching experience. This program is generally not for graduate students or new
PhD's.
The Coordinator of PCMI's Undergraduate Faculty
Program, Daniel Goroff, is Professor of the Practice of Mathematics at Harvard
University and Associate Director of the Derek Bok Center for Teaching and Learning.
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Combinatorics is an area of mathematics that is frequently looked on as one that is reserved for a small minority of mathematicians: die-hard individualists who shun the limelight and take on problems that most would find boring. In addition, it has been viewed as a part of mathematics that has not followed the trend toward axiomatization that has dominated mathematics in the last 150 years. It is however also a field that has taken on enormous importance in recent years do its applicability in network engineering, combinatorial optimization, coding theory, cryptography, integer programming, constraint satisfaction, and computational biology. In the study of toric varieties in algebraic geometry, combinatorics has had a tremendous influence. Indeed combinatorial constructions have helped give a wide variety of concrete examples of algebraic varieties in algebraic geometry, giving beginning students in this area much needed intuition and understanding. It is the the advent of the computer though that has had the greatest influence on combinatorics, and vice versa.The consideration of NP complete problems typically involves enumerative problems in graph theory, one example being the existance of a Hamiltonian cycle in a graph. The use of the computer as a tool for proof in combinatorics, such as the 4-color problem, is now legendary. In addition, several good software packages, such as GAP and Combinatorica, have recently appeared that are explicitly designed to do combinatorics. One fact that is most interesting to me about combinatorics is that it gave the first explicit example of a mathematical statement that is unprovable in Peano arithmetic. Before coming across this, I used to think the unprovable statements of Godel had no direct relevance for mathematics, but were only interesting from the standpoint of its foundations.
This book is an introduction to combinatorics for the undergraduate mathematics student and for those working in applications of combinatorics. As with all the other guides in the Schaums series on mathematics, this one has a plethora of many interesting examples and serves its purpose well. Readers who need a more in-depth view can move on to more advanced works after reading this one. The author dedicates this book to the famous mathematician Paul Erdos, who is considered the father of modern combinatorics, and is considered one of most prolific of modern mathematicians, with over 1500 papers to his credit.
The author defines combinatorics as the branch of mathematics that attempts to answer enumeration questions without considering all possible cases. The latter is possible by the use of two fundamental rules, namely the sum rule and the product rule. The practical implementation of these rules involves the determination of permutations and combinations, which are discussed in the first chapter, along with the famous pigeonhole principle. Most of this chapter can be read by someone with a background in a typical college algebra course. The author considers some interesting problems in the "Solved Problems" section, for example one- and two-dimensional binomial random walks, and problems dealing with Ramsey, Catalan, and Stirling numbers. The consideration of Ramsey numbers will lead the reader to several very difficult open problems in combinatorics involving their explicit values.
Generalized permutations and combinations are considered in chapter two, along with selections and the inclusion-exclusion principle. The author proves the Sieve formula and the Phillip Hall Marriage Theorem. In the "Solved Problems" section, the duality principle of distribution, familiar from integer programming is proved, and the author works several problems in combinatorial number theory. A reader working in the field of dynamical systems will appreciate the discussion of the Moebius function in this section. Particularly interesting in this section is the discussion on rook and hit polynomials.
The consideration of generating functions and recurrence relations dominates chapter 3, wherein the author considers the partition problem for positive integers. The first and second identities of Euler are proved in the "Solved Problems" section, and Bernoulli numbers, so important in physics, are discussed in terms of their exponential generating functions. The physicist reader working in statistical physics will appreciate the discussion on Vandermonde determinants. Applications to group theory appear in the discussion on the Young tableaux, preparing the reader for the next chapter.
A more detailed discussion of group theory in combinatorics is given in chapter 4, the last chapter of the book. The author proves the Burnside-Frobenius, the Polya enumeration theorems, and Cayley's theorem in the "Solved Problems" section. Readers without a background in group theory can still read this chapter since the author reviews in detail the basic constructions in group theory, both in the main text and in the "Solved Problems" section. Combinatorial techniques had a large role to play in the problem of the classification of finite simple groups, the eventual classification proof taking over 15,000 journal pages and involving a large collaboration of mathematicians. Combinatorics also made its presence known in the work of Richard Borchers on the "monstrous moonshine" that brought together ideas from mathematical physics and the largest simple group, called the monster simple group.
The author devotes an appendix to graph theory, which is good considering the enormous power of combinatorics to problems in graph theory and computational geometry. Even though the discussion is brief, he does a good job of summarizing the main results, including a graph-theoretic version of Dilworth's theorem. Combinatorial/graph-theoretic considerations are extremely important in network routing design and many of the techniques discussed in this appendix find their way into these kinds of applications. The author asks the reader to prove that Dilworths' theorem, the Ford-Fulkerson theorem, Hall's marriage theorem, Konig's theorem, and Menger's theorem are equivalent. A very useful glossary of the important definitions and concepts used in the book is inserted at the end of the book.
In its usual way schaum's series gives out another book which is both helpful yet concise. This book gives the essential grounding for combinatorics and graph theory without being overly gargantuan encyclopedia..ample problems set the tone for a future mathematician. they could've done better though..hence not the perfect 5 !
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will break concepts down to a basic level, allowing the student to master the basics, and then return to the more complex problem that the student is trying to understand, but now the student has a better understanding of the fundamentals, and can now master more advanced concepts. I can expla...
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1449649165
9781449649166 They Will Encounter In Future Calculus Courses. In Far Too Many Texts, Process Is Stressed Over Insight And Understanding, And Students Move On To Calculus Ill Equipped To Think Conceptually About Its Essential Ideas. This Text Provides Sound Development Of The Important Mathematical Underpinnings Of Calculus, Stimulating Problems And Exercises, And A Well-Developed, Engaging Pedagogy. Students Will Leave With A Clear Understanding Of What Lies Ahead In Their Future Calculus Courses. Instructors Will Find That Smith's Straightforward, Student-Friendly Presentation Provides Exactly What They Have Been Looking For In A Text! «Show less... Show more»
Rent Precalculus: A Functional Approach To Graphing And Problem Solving 6th Edition today, or search our site for other Smith Pre-Cal
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Please note: Because we have had such a wonderful response to this workshop, we have run out of space. We're sorry for any inconvenience, but this has forced us to close registration. Thank you for your support and interest in Math Education. For over two decades, the teaching and learning of algebra has been a focus of mathematics education at the precollege level. This workshop will examine issues in algebra education at two critical points in the continuum from elementary school to undergraduate studies: at the transitions from arithmetic to algebra and from high school to university. In addition, the workshop will involve participants in discussions about various ways to structure an algebra curriculum across the entire K-12 curriculum. The workshop design is guided by three framing questions: Question 1: What are some organizing principles around which one can create a coherent pre-college algebra program? There are several curricular approaches to developing coherence in high school algebra, each based on a framework about the nature of algebra and the ways in which students will use algebra in their post-secondary work. We seek answers to this question that articulate the underlying frameworks used by curriculum developers, researchers, and teachers. Question 2: What is known about effective ways for students to make the transition from arithmetic to algebra? What does research say about this transition? What kinds of arithmetic experiences help preview and build the need for formal algebra? In what ways does high school and undergraduate mathematics depend on fundamental ideas developed in the transition from arithmetic to algebra? What are some effective pedagogical approaches that help students develop a robust understanding of algebra? Question 3: What algebraic understandings are essential for success in beginning collegiate mathematics? What kinds of problems should high school graduates be able to solve? What kinds of technical fluency will they find useful in college or in other post-secondary work? What algebraic habits of mind should students develop in high school? What are the implications of current and emerging technologies on these questions? The audience for the workshop includes mathematicians, mathematics educators, classroom teachers, and education researchers who are concerned with imporving the teaching and learning of algebra across the grades. Sessions feature direct experience with several curricular approaches to algebra, as well as reports from researchers, educators, and members of national committees that are charged with finding ways to increase student achievement in algebra.
ACCOMMODATIONS: A block of rooms has been reserved at the hotels below: Double Tree Hotel (Berkeley Marina). Attendees may make their reservations by calling the Hotel Reservation's Department directly at 1-800-243-0625 or our Central Reservations' toll-free number at 1-800-222-TREE (8733), or via the internet using their Personalized On-Line Group pageno later than Tuesday, April 22, 2008 by 5PM PST. Please mention the name of the event while making reservations which is: Critical Issues Mathematics. Hotel's complementary shuttle to the UC Berkeley Campus runs every hour. The room rate is $139/ a night. Hotel Durant. Please mention the workshop name and reference the following code when making reservations via phone, fax or e-mail: K20000. Rooms are still available!The room rate is $199/ a night. The Women's Faculty Club University of California, Berkeley. Please make your reservation via phone, fax or e-mail: Tel: (510) 642-4175 Fax: (510) 204-9661 wfc@uclink.berkeley.edu Identify yourself as coming to MSRI, mention the workshop name, and give the name of Robert Bryant as faculty sponsor, the department phone # 642-0143 and a credit card # to guarantee. Rates: Single:$113/night; Double/queen bed: $126; Double/twin beds: $127 The cut-off date for reservations is April 28, 2008Berkeley City Club 2315 Durant Ave., Berkeley Tel: (510) 848-7800 Fax: (510) 848-5900 berkeleycityclub@aol.com Please mention the name of the event while making reservations which is: Critical Issues Mathematics. Room Rates: Single or Double:$110/ a night Rates include tax, buffet breakfast,and parking. The cut-off date for reservations is April 13, 2008Important: Please see Travel funding rules and Airline travel reimbursement restrictions.
Lodging information will be provided in MSRI's invitation letters. Lodging will be double occupancy at nice hotel. You may indicate with whom you want to share a room by sending an email to msri-workshops@msri.org.
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Mathematics Department
Teachers: Mr David Hobson, Mrs Heather Kingston, Mr Anthony Hibbard
Overview Mathematics is a reasoning and creative activity employing abstraction and generalisation to identify, describe and apply patterns and relationships. It is a significant part of the cultural heritage of many diverse societies. The symbolic nature of mathematics provides a powerful, precise and concise means of communication. Mathematics incorporates the processes of questioning, reflecting, reasoning and proof. It is a powerful tool for solving familiar and unfamiliar problems both within and beyond mathematics. As such, it is integral to scientific and technological advances in many fields of endeavour. In addition to its practical applications, the study of mathematics is a valuable pursuit in its own right, providing opportunities for originality, challenge and leisure.
The study of mathematics provides opportunities for students to learn to describe and apply patterns and relationships; reason, predict and solve problems; calculate accurately both mentally and in written form; estimate and measure; and interpret and communicate information presented in numerical, geometrical, graphical, statistical and algebraic forms. Mathematics provides support for concurrent learning in other key learning areas and builds a sound foundation for further mathematics education.
Students will have the opportunity to develop an appreciation of mathematics and its applications in their everyday lives and in the worlds of science, technology, commerce, the arts and employment. The study of the subject enables students to develop a positive self-concept as learners of mathematics, obtain enjoyment from mathematics, and become self-motivated learners through inquiry and active participation in challenging and engaging experiences.
The ability to make informed decisions, and to interpret and apply mathematics in a variety of contexts, is an essential component of studentsí preparation for life in the twenty-first century. To participate fully in society students need to develop the capacity to critically evaluate ideas and arguments that involve mathematical concepts or that are presented in mathematical form.
Mathematics as part of the curriculum Mathematics is compulsory for Year 7-10 and is streamed in each class according to individuals ability levels. General Mathematics, 2 Unit, 3 Unit and 4 Unit Mthematics are offered in Year 11 and 12.
Recommended homework hours Year 7-8 should spend approximately 1-2 hours per week on mathematics homework. Year 9-10 should spend approximately 2-3 hours per week and Year 11-12 should spend approximately 3-4 hours per week.
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Product Description
This powerful NCTM Standards-based program combines a variety of instructional strategies with test preparation to help students get back on track. It is designed for after-school, tutoring, summer school, and other extended learning programs, and includes components that review, instruct, provide, practice, and assess students' skills. The scope and sequence addresses the needs of students who require additional support in topics included in content strands from the NCTM with problem solving, reasoning and proof, communication, and connections embedded throughout. The program offers a mix of direct instruction, guided practice, and hands-on group work. Each lesson touches upon the skills that are important in classrooms, on standardized tests, and - most importantly - in students' everyday lives. Provides review, remediation, and real-world application for important topics in traditional algebra courses: solving equations and inequalities, graphing linear equations, polynomial operations, and quadratic equations. Three-ring binder includes up to 120 hours of lessons with reproducible activity sheets; a section on test-taking strategies and collection of practice items; a collection of station-based or real-world activities for small group work; and a problem-based mathematics teacher's guide that: describes the purpose of the materials and options for using the package, provides pacing guide options, recommends an assortment of graphic organizers for instructional strategies, includes a collection of openers - to begin class or to make a transition, and references relevant to NCTM Standards. 756 pages. Grades 9-12.
Prices listed are U.S. Domestic prices only and apply to orders shipped within the United States. Orders from outside the
United States may be charged additional distributor, customs, and shipping charges.
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NUMERICAL METHODS, 4E, International Edition emphasizes the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. Students learn why the numerical methods work, what kinds of errors to expect, and when an application might lead to difficulties. The authors also provide information about the availability of high-quality software for numerical approximation routines. The techniques are the same as those covered in the authors' top-selling Numerical Analysis text, but this text provides an overview for students who need to know the methods without having to perform the analysis. This concise approach still includes mathematical justifications, but only when they are necessary to understand the methods. The emphasis is placed on describing each technique from an implementation standpoint, and on convincing the student that the method is reasonable both mathematically and computationally.
Features
Worked examples using computer algebra systems help students understand why the software usually works, why it might fail, and what to do when a software program fails.
The exercise sets include problems reflecting a wide range of difficulty as well as problems that offer good illustrations of the methods being discussed, while requiring little calculation.
The book contains instructions for a wide range of popular computer algebra systems.
This text is designed for use in a one-semester course, but contains more material than needed. Instructors have flexibility in choosing topics and students gain a useful reference for future work.
New examples and exercises appear throughout the text, offering fresh options for assignments.
Chapter 7, "Iterative Methods for Solving Linear Systems," includes a new section on Conjugate Gradient Methods.
Chapter 10, "Solutions of Systems of Nonlinear Equations," includes a new section on Homotopy and Continuation Methods.
Revised techniques for algorithms and programs are included in six languages: FORTRAN, Pascal, C, MAPLE, Mathematica, and MATLAB.
All of the Maple material in the text is updated to conform with the newest release (Maple 7). All of the material on the CD that accompanies the book is updated to conform to the latest available versions of Maple, Mathematica, and MATLAB.
This edition includes many more examples of Maple code.
{Supplements}
{Quotes}
Douglas Faires J. Douglas Faires is a Professor of Mathematics at Youngstown State University. His research interests include analysis, numerical analysis, and mathematics history. Dr. Faires has won many awards, including Outstanding College-University Teacher of Mathematics, Ohio Section of MAA (1996) and Youngstown State University, Distinguished Professor for Teaching (1995-1996).
Richard L. Burden Richard L. Burden is a Emeritus Professor of Mathematics at Youngstown State University. His master's degree in mathematics and doctoral degree in mathematics, with a specialization in numerical analysis, were both awarded by Case Western Reserve University. He also earned a masters degree in computer science from the University of Pittsburgh. His mathematical interests include numerical analysis, numerical linear algebra, and mathematical statistics. Dr. Burden has been named a distinguished professor for teaching and service three times at Youngstown State University. He was also named a distinguished chair as the chair of the Department of Mathematical and Computer Sciences. He wrote the Actuarial Examinations in Numerical Analysis from 1990 until 1999.
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Trigonometry Smarts!
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Are you having trouble with trigonometry? Do you wish someone could explain this challenging subject in a clear, simple way? From triangles and radians to sine and cosine, this book takes a step-by-step approach to teaching trigonometry. This book is designed for students to use alone or with a tutor or parent, and provides clear lessons with easy-to-learn techniques and plenty of examples. Whether you are looking to learn this information for the first time, on your own or with a tutor, or you would like to review some trigonometry skills, this book will be a great choice.
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This introduction to complex variable methods begins by carefully defining complex numbers and analytic functions, and proceeds to give accounts of complex integration, Taylor series, singularities, residues and mappings. Both algebraic and geometric tools are employed to provide the greatest understanding, with many diagrams illustrating the concepts introduced. The emphasis is laid on understanding the use of methods, rather than on rigorous proofs. Throughout the text, many of the important theoretical results in complex function theory are followed by relevant and vivid examples in physical sciences. This second edition now contains 350 stimulating exercises of high quality, with solutions given to many of them. Material has been updated and additional proofs on some of the important theorems in complex function theory are now included, e.g. the Weierstrass–Casorati theorem. The book is highly suitable for students wishing to learn the elements of complex analysis in an applied context.
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Objective: On completion of the lesson the student will be able to calculate the gradient of a line given any two points on the line and also be capable of checking whether 3 or more points lie on the same line and what an unknown point will make to parallel lines.
Objective: On completion of the lesson the student will be able to draw a line which passes through the origin of the form y=mx and comment on its gradient compared to the gradients of other lines through the origin and use the information to solve problems.
Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the number from which the logarithm evolves.
Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the base from which the number came.
Objective: On completion of this lesson the student will be able to define basic logarithmic functions and describe the relationship between logarithms and exponents including graph logarithmic functions. The student will understand the relationship between logarit
Objective: On completion of the lesson the student will be able to test if a given sequence is an Arithmetic Progression or not and be capable of finding a formula for the nth term, find any term in the A.P. and to solve problems involving these concepts.
Objective: On completion of the lesson the student will be able to make an arithmetic progression between two given terms. This could involve finding one, two, or even larger number of arithmetic means
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Letts and Lonsdale's Success Revision Guides offer accessible content to help students manage their revision and prepare for exams efficiently. The content is broken into manageable sections and advice is offered to help build students' confidence. Exam tips and techniques are provided to support students throughout the revision process. These new titles are specific to the Edexcel exam board. This book is for the current GCSE Maths curriculum to be examined in 2010/2011 only. Other Letts revision titles are available for the new GCSE curriculum starting in 2010.
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Description
By connecting applications, modeling, and visualization, Gary Rockswold motivates students to learn mathematics in the context of their experiences. In order to both learn and retain the material, students must see a connection between the concepts and their real-lives. In this new edition, connections are taken to a new level with "See the Concept" features, where students make important connections through detailed visualizations that deepen understanding.
Rockswold is also known for presenting the concept of a function as a unifying theme, with an emphasis on the rule of four (verbal, graphical, numerical, and symbolic representations). A flexible approach allows instructors to strike their own balance of skills, rule of four, applications, modeling, and technology. Additionally, incorporating technology with this edition has never been so exciting! Within the MyMathLab® course, new Interactive Figures help students visualize difficult topics. Also, Getting Ready integrated review allows students to remediate "just-in-time," by providing review of prerequisite material when needed to help them succeed in the course.
Table of Contents
1. Introduction to Functions and Graphs
1.1 Numbers, Data, and Problem Solving
1.2 Visualizing and Graphing Data
Checking Basic Concepts for Sections 1.1 and 1.2
1.3 Functions and Their Representations
1.4 Types of Functions and Their Rates of Change
Checking Basic Concepts for Sections 1.3 and 1.4
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Extended and Discovery Exercises
2. Linear Functions and Equations
2.1 Equations of Lines
2.2 Linear Equations
Checking Basic Concepts for Sections 2.1 and 2.2
2.3 Linear Inequalities
2.4 More Modeling with Functions
Checking Basic Concepts for Sections 2.3 and 2.4
2.5 Absolute Value Equations and Inequalities
Checking Basic Concepts for Section 2.5
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Extended and Discovery Exercises
Chapters 1-2 Cumulative Review Exercises
3. Quadratic Functions and Equations
3.1 Quadratic Functions and Models
3.2 Quadratic Equations and Problem Solving
Checking Basic Concepts for Sections 3.1 and 3.2
3.3 Complex Numbers
3.4 Quadratic Inequalities
Checking Basic Concepts for Sections 3.3 and 3.4
3.5 Transformations of Graphs
Checking Basic Concepts for Section 3.5
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Extended and Discovery Exercises
4. More Nonlinear Functions and Equations
4.1 More Nonlinear Functions and Their Graphs
4.2 Polynomial Functions and Models
Checking Basic Concepts for Sections 4.1 and 4.2
4.3 Division of Polynomials
4.4 Real Zeros of Polynomial Functions
Checking Basic Concepts for Sections 4.3 and 4.4
4.5 The Fundamental Theorem of Algebra
4.6 Rational Functions and Models
Checking Basic Concepts for Sections 4.5 and 4.6
4.7 More Equations and Inequalities
4.8 Radical Equations and Power Functions
Checking Basic Concepts for Sections 4.7 and 4.8
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Extended and Discovery Exercises
Chapters 1-4 Cumulative Review Exercises
5. Exponential and Logarithmic Functions
5.1 Combining Functions
5.2 Inverse Functions and Their Representations
Checking Basic Concepts for Sections 5.1 and 5.2
5.3 Exponential Functions and Models
5.4 Logarithmic Functions and Models
Checking Basic Concepts for Sections 5.3 and 5.4
5.5 Properties of Logarithms
5.6 Exponential and Logarithmic Equations
Checking Basic Concepts for Sections 5.5 and 5.6
5.7 Constructing Nonlinear Models
Checking Basic Concepts for Section 5.7
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Extended and Discovery Exercises
6. Trigonometric Functions
6.1 Angles and Their Measure
6.2 Right Triangle Trigonometry
Checking Basic Concepts for Sections 6.1 and 6.2
6.3 The Sine and Cosine Functions and Their Graphs
6.4 Other Trigonometric Functions and Their Graphs
Checking Basic Concepts for Sections 6.3 and 6.4
6.5 Graphing Trigonometric Functions
6.6 Inverse Trigonometric Functions
Checking Basic Concepts for Sections 6.5 and 6.6
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Extended and Discovery Exercises
Chapters 1-6 Cumulative Review Exercises
7. Trigonometric Identities and Equations
7.1 Fundamental Identities
7.2 Verifying Identities
Checking Basic Concepts for Sections 7.1 and 7.2
7.3 Trigonometric Equations
7.4 Sum and Difference Identities
Checking Basic Concepts for Sections 7.3 and 7.4
7.5 Multiple-Angle Identities
Checking Basic Concepts for Section 7.5
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Extended and Discovery Exercises
8. Further Topics in Trigonometry
8.1 Law of Sines
8.2 Law of Cosines
Checking Basic Concepts for Sections 8.1 and 8.2
8.3 Vectors
8.4 Parametric Equations
Checking Basic Concepts for Sections 8.3 and 8.4
8.5 Polar Equations
8.6 Trigonometric Form and Roots of Complex Numbers
Checking Basic Concepts for Sections 8.5 and 8.6
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Extended and Discovery Exercises
Chapters 1-8 Cumulative Review Exercises
9. Systems of Equations and Inequalities
9.1 Functions and Systems of Equations in Two Variables
9.2 Systems of Inequalities in Two Variables
Checking Basic Concepts for Sections 9.1 and 9.2
9.3 Systems of Linear Equations in Three Variables
9.4 Solutions to Linear Systems Using Matrices
Checking Basic Concepts for Sections 9.3 and 9.4
9.5 Properties and Applications of Matrices
9.6 Inverses of Matrices
Checking Basic Concepts for Sections 9.5 and 9.6
9.7 Determinants
Checking Basic Concepts for Section 9.7
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Extended and Discovery Exercises
Chapters 1-9 Cumulative Review Exercises
10. Conic Sections
10.1 Parabolas
10.2 Ellipses
Checking Basic Concepts for Sections 10.1 and 10.2
10.3 Hyperbolas
Checking Basic Concepts for Section 10.3
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Extended and Discovery Exercises
11. Further Topics in Algebra
11.1 Sequences
11.2 Series
Checking Basic Concepts for Sections 11.1 and 11.2
11.3 Counting
11.4 The Binomial Theorem
Checking Basic Concepts for Sections 11.3 and 11.4
11.5 Mathematical Induction
11.6 Probability
Checking Basic Concepts for Sections 11.5 and 11.6
Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Extended and Discovery Exercises
Chapters 1-11 Cumulative Review Exercises
R. Reference: Basic Concepts from Algebra and Geometry
R.1 Formulas from Geometry
R.2 Integer Exponents
R.3 Polynomial Expressions
R.4 Factoring Polynomials
R.5 Rational Expressions
R.6 Radical Notation and Rational Exponents
R.7 Radical Expressions
Appendix A: Using the Graphing Calculator
Appendix B: A Library of Functions
Appendix C: Partial Fractions
Appendix D: Percent Change and Exponential Functions
Appendix E: Rotation of Axes
Bibliography
Answers to Selected Exercises
Photo Credits
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On-line Quizzes
Each week, students must complete a single timed online quiz at before 4 pm on Friday of the week after the lecture and lab for that topic. Be sure to plan time for completing each week's quiz. Quizzes are (generally) available from 4 pm on Thursday after Team Lab for that week's topic to 4 pm on Friday of the following week.
Since each quiz tests only a subset of the material presented each week, the quizzes are timed. Because there is only 45 minutes to work on each quiz, it is imperative that students study all of the material before starting the quiz. There is plenty of time to answer the questions posed on each quiz, but there is not plenty of time to have to research or learn the material that was presented on that topic. Do your studying and learning first, then take the quiz before the due date and time.
Quiz Policies
Quizzes must be done individually. Collaboration on quizzes is not allowed.
Make sure to follow quiz instructions exactly. Quiz questions are graded by Learn@UW; answers that are not given in the format expected by the grading system will be marked incorrect.
While you are taking the quiz, you may make use of:
the course modules
the team lab write-up
the posted team lab solution
your lecture notes
handouts from lecture
MATLAB / Maple
To receive credit for each week's quiz
Plan a set time each week when you can complete the quiz and put a reminder in your personal calendar. (Note: Forgetting to take the quiz means that you will not have a score for that quiz or be able to view the quiz online. There are no quiz makeups, though the lowest quiz score is dropped).
Open up and review Team Lab and solution before starting the quiz (you may leave the solution open during the quiz).
Study and ask questions before starting quiz.
Launch MATLAB or Maple before starting the quiz.
Only start the quiz when you are ready and have 45 minutes to complete the quiz.
Leave the quiz window open until you are done. (Students have reported problems when trying to close and reopen quizzes. So avoid doing this.)
Save each answer as you enter it. This creates a log event that can be checked and scored for partial credit if the quiz is not completed or submitted on time.
Answer all problems without help from other people before the time runs out.
Submit the quiz before the time runs out and before the due date and time. (Failure to complete the quiz submission process results in lost points).
There are four steps to completing the quiz submission:
Click [Go to Submit Quiz] button.
Click [Submit Quiz] button.
Click [OK] button.
Wait for page to refresh.
Quiz Grading and Feedback
The score and feedback on a quiz is not released until after the quiz availability deadline has passed and the quiz has been graded. Once a quiz has been graded, you will be able to see your score in the gradebook on Learn@UW (click on the "Grades" link in the red banner across the top of the CS 310 Learn@UW site).
To see the questions, your answers, the correct answers, and any feedback on a quiz, do the following:
Click on the "Assignments" dropdown menu in the red banner across the top the the CS 310 Learn@UW website and select "Quizzes".
Under "Past Quizzes", find the quiz you wish to look at. Note: you will only be able to see the quiz questions and answers if you attempted the quiz.
Click on the drop-down menu (a small triangle) near the title of the quiz and select "Submissions"
Under Individual Attempts, click on "Attempt 1".
Make sure to review your quiz results. If you followed the instructions for question and believe that you had a correct answer that was marked incorrect, please email Beck Hasti (hasti@cs.wisc.edu) with the subject "CS 310 Quiz N regrade request" (where N is the number of the quiz).
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Students
Section Navigation
Understanding the value of math
Whatever you plan to do in your life, at school, at home or in your future career, you need mathematical skills to succeed in our society.
Mathematical concepts are present in everything you do. Knowing and understanding these concepts will serve you well in college or university, apprenticeships, and other learning situations.
Mathematics is more than just learning about numbers or memorizing formulas. Through mathematics, you are developing valuable skills such as problem solving, spatial visualization, analysis and abstract thinking. These skills are valued by today's employers and will give you greater flexibility in your future career paths.
Math opens many doors
Mathematics is an interesting and creative subject. It opens many doors to other areas of study such as medicine, engineering, and computer science, as well as the study of commerce, economics and other social sciences.
Even in areas of study that do not appear to be related to mathematics, like fine arts, graphic and interior design, languages and literature, mathematical concepts are present in varying forms. For example, it is through mathematics that we know about the perfect ratios shown in Renaissance paintings and the use of iambic pentameter in Shakespearean sonnets.
Whether you decide to pursue a trade or an apprenticeship program or to study a diversity of careers from carpentry to landscaping to fashion design, mathematics is an important component of your chosen field of study.
Your math options in high school
The revised senior high school mathematics courses have been designed with you in mind!
The ministry along with its education partners has designed mathematics courses that offer many options to meet your needs for the future - between grade levels, into the post-secondary, technologies and trades system, and into the workforce.
The following web pages offer you detailed information about course selection and describe various options that you can choose in high school.
Life after high school
As you are thinking about life after high school and considering future studies and career paths, start planning ahead. Ask yourself, what mathematics courses will I need to reach my goals?
There are several things you can do:
Talk to people who are working in your field of interest or who have similar goals.
Learn about what mathematics requirements you may need for post-secondary studies, apprenticeships or possible employment. A good place to start is the admissions page of the post-secondary institution that you would like to attend, or the Alberta Learning Information Service (ALIS) website which outlines information regarding mathematics requirements at post-secondary institutions and programs in Alberta.
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Web Codes
Funded and approved by the National Science Foundation, Connected Mathematics is a complete middle school curriculum that helps students develop an in-depth understanding of key mathematical ideas and make connections between them and to other disciplines. The program, rated exemplary by the U.S. Department of Education, uses interesting problems and contexts to develop understanding and long-term retention of concepts and skills.
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This textbook is devoted to Combinatorics and Graph Theory, which are cornerstones of Discrete Mathematics. Every section begins with simple model problems. Following their detailed analysis, the reader is led through the derivation of definitions, concepts and methods for solving typical problems.
Best Internet Links
Since the 1980s, the theory of groups - in particular simple groups, finite and algebraic - has influenced a number of diverse areas of mathematics. Such areas include topics where groups have been traditionally applied, such as algebraic combinatorics, finite geometries, Galois theory and permutation groups, as well as several more recent developments.
Alan Tucker's newest issue of Applied Combinatorics builds on the previous editions with more in depth analysis of computer systems in order to help develop proficiency in basic discrete math problem solving. As one of the most widely used book in combinatorial problems, this edition explains how to reason and model combinatorically while stressing the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity.
of set partitions from 1500 A.D. to today.
This book written by experts in their respective fields, and covers a wide spectrum of high-interest problems across these discipline domains. The book focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineering.
The articles collected here are the texts of the invited lectures given at the Eighth British Combinatorial Conference held at University College, Swansea. The contributions reflect the scope and breadth of application of combinatorics, and are up-to-date reviews by mathematicians engaged in current research. This volume will be of use to all those interested in combinatorial ideas, whether they be mathematicians, scientists or engineers concerned with the growing number of applications.
With examples of all 450 functions in action plus tutorial text on the mathematics, this book is the definitive guide to Experimenting with Combinatorica, a widely used software package for teaching and research in discrete mathematics. Three interesting classes of exercises are provided--theorem/proof, programming exercises, and experimental explorations--ensuring great flexibility in teaching and learning the material.
This book constitutes the refereed proceedings of the 16th Annual International Conference on Computing and Combinatorics, held in Dallas, TX, USA, in August 2011. The 54 revised full papers presented were carefully reviewed and selected from 136 submissions. Topics covered are algorithms and data structures; algorithmic game theory and online algorithms; automata, languages, logic, and computability; combinatorics related to algorithms and complexity; complexity theory; computational learning theory and knowledge discovery; cryptography, reliability and security, and database theory; computational biology and bioinformatics; computational algebra, geometry, and number theory; graph drawing and information visualization; graph theory, communication networks, and optimization; parallel and distributed computing.
This volume is a collection of survey papers in combinatorics that have grown out of lectures given in the workshop on Probabilistic Combinatorics at the Paul Erdös Summer Research Center in Mathematics in Budapest. The papers, reflecting the many facets of modern-day combinatorics, will be appreciated by specialists and general mathematicians alike: assuming relatively little background, each paper gives a quick introduction to an active area, enabling the reader to learn about the fundamental results and appreciate some of the latest developments. An important feature of the articles, very much in the spirit of Erdös, is the abundance of open problems.
Wherefore is this conference book on graphs and combinatorics different from other such books?
One way is that it is, frankly, a progress report on recent results in the field, and does not claim to be a definitive work. Another is that all contributions have been refereed. A third way is that it contains both expository review articles and research contributions.
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Book Description: Teaching Secondary Mathematics, Third Edition is practical, student-friendly, and solidly grounded in up-to-date research and theory. This popular text for secondary mathematics methods courses provides useful models of how concepts typically found in a secondary mathematics curriculum can be delivered so that all students develop a positive attitude about learning and using mathematics in their daily lives.A variety of approaches, activities, and lessons is used to stimulate the reader's thinking--technology, reflective thought questions, mathematical challenges, student-life based applications, and group discussions. Technology is emphasized as a teaching tool throughout the text, and many examples for use in secondary classrooms are included. Icons in the margins throughout the book are connected to strands that readers will find useful as they build their professional knowledge and skills: Problem Solving, Technology, History, the National Council of Teachers of Mathematics Principles for School Mathematics, and "Do" activities asking readers to do a problem or activity before reading further in the text. By solving problems, and discussing and reflecting on the problem settings, readers extend and enhance their teaching professionalism, they become more self-motivated, and they are encouraged to become lifelong learners.The text is organized in three parts:*General Fundamentals--Learning Theory, Curriculum; and Assessment; Planning; Skills in Teaching Mathematics;*Mathematics Education Fundamentals--Technology; Problem Solving; Discovery; Proof; and*Content and Strategies--General Mathematics; Algebra 1; Geometry; Advanced Algebra and Trigonometry; Pre-Calculus; Calculus.New in the Third Edition:*All chapters have been thoroughly revised and updated to incorporate current research and thinking.*The National Council of Teachers of Mathematics Standards 2000 are integrated throughout the text.*Chapter 5, Technology, has been rewritten to reflect new technological advances.*A Learning Activity ready for use in a secondary classroom has been added to the end of each chapter.*Two Problem-Solving Challenges with solutions have been added at the end of each chapter.*Historical references for all mathematicians mentioned in the book have been added within the text and in the margins for easy reference.*Updated Internet references and resources have been incorporated to enhance the use of the text.
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Elementary Algebra With Bca Tutorial, and Infotrac
9780534400415
ISBN:
0534400418
Publisher: Thomson Learning
Summary: Jerome Kaufmann and Karen Schwitters discuss algebra with clear and concise exposition, numerous examples, and plentiful problem sets. They reinforce the following common thread - learn a skill, use the skill to solve equations, and then apply this to solve application problems.
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From the Ishango Bone of central Africa and the Inca quipu of South America to the dawn of modern mathematics, The Crest of the Peacock makes it clear that human beings everywhere have been capable of advanced and innovative mathematical thinking. George Gheverghese Joseph takes us on a breathtaking multicultural tour of the roots and shoots of book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral,...Many problems in science can be formulated in the language of optimization theory, in which case an optimal solution or the best response to a particular situation is required. In situations of interest, such classical optimal solutions are lacking, or at least, the existence of such solutions is far from easy to prove. So, non-convex optimization... more...
The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering.... more...
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MATH 111 - College Algebra
A course in Algebra for college students with a strong emphasis on problem-solving and applications. Topics include: introduction to functions and their graphs; linear and quadratic functions; solution of a variety of types of equations and inequalities using algebraic, numeric and graphical techniques; systems of equations, operations with polynomials; rational, radical, exponential and logarithmic expressions; and exponential functions. Use of a graphing calculator may be an integral part of the course. Prerequisite: placement per high school transcript, completion of MATH101 or MATH101X with "C-" or higher, or by permission of the Mathematics Department. [Fall, Spring]
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TI-84 Plus graphing calculator offers three times the memory, more than twice the speed and a higher contrast screen than the TI-83 Plus model. It's keystroke-for-keystroke compatible, too. Count on TI calculators at exam time.
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
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MATH XL Homework Notes for Students: MathXL is only accessible through the Internet Explorer browser and is not compatible with Apple or Mac-sorry! You will need to register in MathXL using the code on the booklet packaged with your textbook. This wi
Factoring WorksheetMethods for factoring trinomials of the form: x2 + bx + c We want to find two numbers so that if we multiply them together we get c and if we add them we get b. Trinomial x2 - 2x - 15 -15 = 15 -1 15 + -1 = 14 = -2 -3 5 = -15 -3
CS 514 Homework #1 SOLUTIONSAnswers to the following 11 problems are to be handed in at the beginning of class on 2/22/06. You are to do all 11 problems. Assume that all values vary with an exponential distribution, with the exception of time slice
CS 514 Homework #2 - OPTIONALDUE: Monday, March 13, 2006, in class, NO LATE SUBMISSIONS!Answers to the following 11 problems are to be handed in at the beginning of class on 3/13/06. You are to do all 11 problems. Note that some problems have more
CS 514 Homework #3 SolutionsDUE: Wednesday, April 26, 2006, at the BEGINNING of class, NO LATE SUBMISSIONS!NOTE: while this homework is optional, you must do either this homework or optional project #4. Failure to do at least one of these means th
Generic Set Assignmentdate due: Friday, March 7A set is a collection of things without repetitions and can be implemented easily in an ArrayList. The methods for our set are: public Set() construct an empty set public void add(thing x) put x
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EMAIL ACCOUNTS AND ADDRESSESEVERY STUDENT IN GENERAL CHEMISTRY 162 MUST HAVE AN EMAIL ADDRESS; IT MUST BE LISTED IN THE RUTGERS ONLINE DIRECTORY. The first step in this process is to create an eden account, which is needed for many things at Rutgers
WEB ASSIGNComputers: The ability to use a computer for receiving email and accessing the Internet is essential for success in this course. If you cannot receive email and access the Internet from your home or dorm room, there are many computing faci
Hi, Below you will find your room assignment for the hour exams. Please take the exam in your assigned room or we may lose your paper. If for some reason you need to take the exam in a different room, please let us know in advance so that we can take
LECTURE 3Chemistry 162SOLUBILITY OF GASES: Most gases dissolve at least to some extent, but not to a great extent in liquids. One reason they dissolve is that there are almost no intermolecular attractions to be overcome in a gas. One reason that
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LECTURE 9Chemistry 162REACTION MECHANISMS A reaction mechanism is a description on a molecular level of all the changes that reactants undergo in a chemical reaction. Such a description is very complex and we will give simpler, less detailed descr
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LECTURE 12Chemistry 162 A very significant part of this course will be devoted to calculations associated with equilibrium systems. We shall look first at gas phase systems, and later at aqueous solutions of acids and bases, of sparingly soluble ion
LECTURE 17Chemistry 162 Pure water has a pH of 7. But it is almost impossible to get a sample of water sufficiently pure to have a pH that is even close to 7. Very small traces of dissolved acidic or basic impurities that are almost always present i
LECTURE 19Chemistry 162The Equivalence Point As promised the second quantitative aspect of neutralization we shall consider is the composition of the system when the neutralization has taken place. In a titration, the point at which the exact volu
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In this section
Intermediate 1 Mathematics
The Intermediate 1 Course is designed to build upon and extend candidates' previous mathematical learning, to introduce them to the areas of algebra and elementary statistics.
Depending on the optional unit chosen, learners will either be introduced to trigonometry and further algebraic methods or to the application of mathematics through calculations of earnings, logic diagrams, scale drawings and a statistical assignment.
The Intermediate 1 Mathematics Course is at SCQF level 4.
For more information on SCQF levels and how Intermediate 1 fits in to the Scottish Credit and Qualifications Framework, visit our SCQF section.
The Mathematics page on Education Scotland's site provides support materials and external web links for teaching and learning as well as news, events and feature articles to assist with professional development.
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Supplements
ALGEBRA: FORM AND FUNCTION features a host of resources to assist you and your students in becoming fluent speakers of algebra!
The Instructor′s Manual contains information on planning and creating lessons and organizing in-class activities. There are focus points as well as suggested exercises, problems and enrichment problems to be assigned to students. This can serve as a guide and checklist for teachers who are using the text for the first time.
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M2= Math Mediator
Algebra 2 High School Lesson Plans
Click on this link: Lesson Plan Example
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take a look at an example lesson by clicking on "Lesson Plan Example" above and view the example.
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About the Cover
The Big Bend Region is the westernmost region of Texas. Known for its
wide open skies, it is home to the International El Paso Balloon
Festival. The distance that a balloon travels is determined by its rate, or
speed, and the time it is in the air. Mathematically, these three
variables are related by the linear function d = rt. You'll learn more
about linear functions in Chapters 3, 4, and 5.
About the Graphics
Periodic pattern on a sphere. Created with Mathematica.
A periodic pattern made from various n-gons separated by bands was
constructed in the plane. The pattern was stereographically mapped
onto the upper half of a sphere. For more information, and for
programs to construct such graphics, see:
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This course begins with a discussion of the geometry of two-
and three-dimensional space, vectors, basic linear algebra, and
quadric surfaces (surfaces defined by a 2nd-degree polynomial
e.g. spheres, ellipsoids, hyperboloids, etc.). Then we will
extend the scope of calculus so that it covers vector-valued
functions and functions of two and three variables, solve maximum
and minimum problems on curves and surfaces and two- and
three-dimensional regions, and work out integrals over two- and
three-dimensional domains. Finally we will apply these ideas and
the fundamental theorem of calculus to prove the great theorems
of vector calculus: Green's Theorem, Stokes's Theorem, and Gauss'
Theorem (a.k.a the Divergence Theorem) and we will strive
mightily to understand what these theorems say about the physical
world.
Prerequisites
Calculus II. Calculus II focuses primarily on integration, applications of integration, and sequences and series. We'll use integration and its applications.
Learning outcomes
See the "Objectives" listed in the MAT 211 Sample
Syllabus
on the math department website.
Course requirements, grading, due dates
I base grades on homework, tests, and a final exam, but fuzzy
things like class participation also play a role because I often
give the benefit of the doubt to people who have worked hard and
contributed to class discussions. Homework gets the least credit
but it is of course the most important thing because learning
math takes a lot of practice!
Homework
15% of grade
3 Tests
50% of grade total (16 2/3% each)
Final exam
35% of grade
I will post most homework assignments with due dates online on
the WeBWorK
web-based homework system but I will assign some paper-and-pencil
homeworks too. I will post paper-and-pencil assignments and tests
on the course calendar .
Please check WeBWorK and the course calendar frequently for
updates and new assignments.
Assignments and tests are due on the due date. I don't plan to
give make up tests or grade assignments that come in late.
I rarely take attendance, but class participation is
important. Students who are serious don't skip classes.
Academic Integrity and Plagiarism
The math department
does not tolerate cheating. Students are free to collaborate and
share ideas and methods, but work that you turn in with your name
on it must be your own. The university's official policy is
spelled out in its statement on Academic Integrity and Plagiarism
in the University Catalog.
Students with disabilities
Students with disabilities have a legal right to reasonable
accomodations. For assistance please contact me and our Disabled
Student Services office.
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This book contains a modern introduction to the use of finite difference and finite element methods for the computer solution of ordinary and partial differential equations. After a review of direct methods for the solution of linear systems, with emphasis on the special features of the linear systems that arise when differential equations are solved, the balance of the content introduces, analyzes and implements, using FORTRAN90 and MATLAB programs, the more commonly used finite difference and finite element methods for solving a variety of problems, including both initial value and boundary value problems.
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Exploring mathematics builds on the concepts and techniques in Using mathematics (MST121) and uses the same software. It looks at questions underlying some of those techniques, such as why particular patterns occur in mathematical solutions and how you can be confident that a result is true. It introduces the role of reasoning and offers opportunities to investigate mathematical problems. Together with Using mathematics this course will give you a good foundation for higher-level mathematics, science and engineering courses. Even if you don't intend to study further, you will gain a good, university-level understanding of the nature and scope of mathematics.
You are advised to be confident with the content of Using mathematics (MST121), or equivalent from elsewhere, before commencing study of this courseExploring mathematics offers both a way in to honours-level mathematics and deeper insights into the mathematics that supports other areas of study. It rounds off the suite of courses designed to provide a rich and enlightening introduction to mathematical ideas and techniques.
This course builds on the approach and design of the University's Level 1 course Using mathematics (MST121), extending the mathematical concepts and techniques (to cover more calculus, for example), the range of applications, and use of the course software. It looks at questions underlying some of the methods from MST121, such as why particular patterns occur in mathematical solutions, and how you can be confident that a result is true. It introduces the role of reasoning in mathematics, and offers opportunities to investigate mathematical questions for yourself. By the end of the course you will have encountered many of the topics that are developed in later mathematics courses, in particular in our main second level courses Pure mathematics (M208), and Mathematical methods and models (MST209). It also provides a good mathematical basis for courses in physics and engineering.
There are four sections, with each of the first three revisiting and developing the ideas introduced in the corresponding section in MST121. An important theme that runs through the course is mathematical reasoning.
Work on numbers and sequences is extended to Fibonacci and related sequences; that on circles is extended to the study of other conic sections. There is further exploration of the properties of functions and matrices, including applications in transformation geometry. The computer is used to help with important ideas of iteration in discrete mathematics. The ideas of calculus are extended to include more sophisticated techniques of differentiation and integration and use in the approximation of functions using Taylor series.
The last section of the course introduces important mathematical ideas, such as complex numbers, number theory and groups, that are built on in later courses. It concludes with a chapter on the role of proof in mathematics.
The course also develops more general skills such as communication of mathematical ideas, which will be useful in studying later courses.
You will learn
Successful study of this course should improve your skills in:
communicating mathematical ideas clearly and succinctly
working with abstract concepts
thinking logically
constructing logical arguments
finding solutions to problems
using a computer algebra software package.
Entry
This course is the third part of the mathematics entry suite that starts with Discovering mathematics (MU123) and goes on to Using mathematics (MST121) and MS221. Your choice of which to take depends on how much mathematical knowledge you already have and on the degree you have in mind. It is not advisable to take either MST121 or MS221 in the same year as MU123, and you should not take MS221 before studying MST121.
If you start in October it is possible – for some qualifications where regulations allow – to study MST121 and MS221 together in a single year as if they are a 60-credit course, as the material in the two courses is linked.
Regulations
As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are
available on our Essential documents website.
If you have a disability
An integral part of study on this course is the use of specific software, which includes on-screen graphs and mathematical notation. Parts of this course are delivered online and through a CD-ROM so you will need to spend considerable amounts of time using a personal computer. If you use specialist hardware or software to assist you in operating a computer or accessing the internet and have any concerns about accessing the types of study materials outlined you are advised to talk to our Student Registration & Enquiry Service about support which can be given to meet your needs.
Adobe Portable Document Format (PDF) versions of the printed study materials are available. However some Adobe PDF components may not be available or fully accessible using a screen reader and mathematical materials may be particularly difficult to read in this way. The study materials are also available on audio in DAISY Digital Talking Book format. Written transcripts are available for the audio-visual material require internet access at least once a week during the course to download course resources and to keep up to date with course news. If your tutor offers online tutorials, we also recommend a headset with a microphone and earphones to talk to your tutor and other students online.
Computing requirements
You will need a computer with internet access to study this course. It includes online activities – you can access using a web browser – and some,
Each TMA typically consists of six questions, covering one Block of the course. Each of the questions typically involves some calculation, algebraic manipulation, creating and/or interpreting a graph, some written work to explain your interpretations and conclusions, and may involve using the course software and providing printouts. The first TMA is to be submitted about eight weeks after the start of the course.
The examination contains two parts. The first consists of short answer questions covering the whole of the course. The second contains longer questions, one on each Block, and you are required to attempt up to two of these.
Future availability
The details given here are for the course that starts in October 2013. In 2014 we expect it to be available twice, in February and October, when it will be available for the last time. The combination of the Level 1 course Using mathematics (MST121) and this course (MS221) will be replaced by Essential mathematics 1 (MST124) available from February 2014 and either Essential mathematics 2 (MST125) available from October 2014 or Mathematical methods (MST224).
How to register
To register a place on this course return to the top of the page and use the Click to register button.
Student Reviews
"A great course - and certainly do-able alongside MST121.
It covers to a much greater depth the material in MST121 ..."
Read more
"After completing around ten modules with the OU it has slowly dawned on me that the best way to do ..."
Read more
"MS221 is designed for study after (or alongside) MST121, which provides a foundation for MS221. We are always open to
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Mathematics
"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas".
G.H Hardy, 'A Mathematician's Apology'
Mathematics is the means of looking at the patterns that make up our world and the intricate and beautiful ways in which they are constructed and realised. As a subject, Mathematics presents frequent opportunities for creativity and can stimulate moments of pleasure and wonder when a problem is solved for the first time, or a more elegant solution to a problem is discovered, or when hidden connections are suddenly manifest. It enables pupils to build a secure framework of mathematical reasoning which they can use and apply with confidence. The power of mathematical reasoning lies in its use of precise and concise forms of language, symbolism and representation to reveal and explore general relationships. These mathematical forms are widely used for modelling situations; a trend accelerated by computational technologies.
Mathematics contributes to the school curriculum by developing pupils' ability to reason logically, algebraically and geometrically, to solve challenging problems and to handle data. It is important for pupils in many other areas of study, particularly sciences and technology. Mathematics is important in everyday living and essential to the working of the modern world.
The Department
Mathematics is taught by a focussed, experienced and highly-qualified team of teachers. The department has a suite of 6 teaching rooms with interactive whiteboards. It has a set of laptops for student use and access to excellent IT facilities. There is a wide range of teaching and learning resources for use by teachers and students, alike.
The Staff
There are 5 permanent members of staff in the Mathematics department who all teach across every key stage (years 7 to 9, GCSE and A level). Each member of staff also teaches Further Mathematics A level and supports students applying for the subject at university.
Dr Boddington is interested in abstract algebra and completed a PhD in the study of Lie algebras. He is also enthusiastic about the way such topics can be applied in the real world.
Ms Varuna Gooriah (Head of Department).
Mrs Irina Kahn
Mrs Emily Levere is particularly enthusiastic about number theory and proof and always enjoys discussing the concept of infinity with her classes. She is also interested in real world applications of probability and statistics including those which draw on her previous experience as a strategy consultant.
Mrs Ann Wilcox has a particular interest in the application of mathematics to other disciplines and the world of work, reflecting her background in natural sciences and previous career in investment banking.
The Key Stage 3 Curriculum
At Key Stage 3 we provide a challenging curriculum that thoroughly explores the key aspects of number, algebra, data handling and geometry. We offer a variety of approaches to teaching and learning to engage pupils. We aim to develop motivated, confident and creative problem-solvers who are extremely well-equipped with the skills and knowledge to succeed at Key Stage 4.
In Year 7 we consolidate pupils' knowledge and skills from Primary School which is key to future progress. Topics taught this year include: an introduction to algebra and solving simple equations, directed numbers (negative numbers) and the order of operations (bidmas), functions, sequences and simple graphs.
In Year 8 we build upon and extend skills and knowledge from Year 7. Topics taught this year include circles, indices and standard form, solving more advanced equations and inequalities, straight lines and curve graphs, and Pythagoras' theorem.
In Year 9 we tackle increasingly challenging material and begin to prepare students for GCSE study. Topics taught this year include more advanced algebra including simultaneous equations, factorising quadratics and algebraic fractions, trigonometry, cumulative frequency and statistics using averages.
We teach Key Stage 3 in form groups with formal tests twice yearly.
Trips and enrichment
We strongly believe in the importance of extra-curricular activities to develop, broaden and deepen students' interest in the subject. We provide enrichment opportunities to every year group. Over the past few years these have included:
o Enigma machine and codebreaking workshops for Years 7 &8 delivered by Cambridge University.
o Commodities trading game for Year 9 delivered by BP.
o Trip to the Bank of England Museum for Year 9
o Trip to British Museum for Year 8 to follow their Mathematics trail.
In the Christmas term, the A level Further Mathematics students develop and deliver a set of workshops specifically for Year 7 students which introduces them to new, interesting and enjoyable areas of Mathematics.
Every student enters the national Maths Challenge competition each year and we regularly enter Maths Team Challenge competitions.
Every year we select students to attend regular Royal Institution Masterclasses.
Lunchtime clubs for 2012/2013 will include:
o The History of Mathematics.
o Programming for beginners.
o Maths puzzles and games.
GCSE INFORMATION
What are the benefits of studying the subject at GCSE?
• Achieve confidence in solving familiar and unfamiliar problems in a range of numerical, algebraic and graphical contexts and in open-ended and closed forms.
• Gain appreciation that algebra, as an extension of number using symbols, gives precise form to mathematical relationships and calculations.
• Progress from definitions and short chains of reasoning to understanding and formulating proofs in algebra and geometry.
• Develop the ability to address increasingly demanding statistical problems in which they draw inferences from the data and consider the uses of statistics in society.
Students are taught in sets at GCSE and there are regular tests at the end of each term after which girls will be reset if required.
Trips and Enrichment
Year 10 students have the opportunity to broaden and deepen their appreciation of the subject by attending a 'Maths In Action' study day at the Institution of Education each year where presentations are delivered by professional mathematicians in the field.
Every student enters the national Maths Challenge competition each year and we regularly enter Maths Team Challenge competitions.
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An investigation into the extent and nature of the understanding first year college of education students have of aspects of arithematic and elementary number theory
Oliphant, Vincent George
(1996)
An investigation into the extent and nature of the understanding first year college of education students have of aspects of arithematic and elementary number theory.
Masters thesis, Rhodes University.
Abstract
First Year College of Education students who have done and/or passed mathematics at matric level, often lack adequate understanding of basic mathematical concepts and principles.
This is due to the fact that formal tests and examinations
often fail to assess understanding at anything but a basic
level. It is against this background that this study uses
alternative and more direct means of assessing the level and
nature of the understanding such students have of aspects of
basic arithmetic and number theory.
More specifically, the goals of the study are:
1. To determine the students' levels of understanding of the following number concepts:
Rational numbers; Irrational numbers
Real numbers and Imaginary numbers.
2. To determine whether the students understand the rules
governing operations with negative numbers and with zero
as principles rather than conventions.
3. To determine whether the students understand the rule
governing the order of operations as a matter of convention rather than as a matter of principle.
A survey of the literature concerning the nature of
understanding as well as the nature of assessment is given.
The students' understanding in the above areas was assessed
by means of a written test followid by interviews. A sample
of 50 students participated in the study while a sub-sample
of 6 were interviewed.
Some of the significant findings of the study were :
1. The students largely failed to draw clear distinctions
between Real and Rational numbers as well as between
Irrational and Imaginary numbers.
2. Very few of the students could explain the rationale
behind the rules governing the. operations with negative
numbers and zero.
3. Only half of the students had any knowledge of the rule
governing the order of operations. Only one student
demonstrated an understanding of the rule as a convention
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TExES Mathematics-Physics 8-12 143
From understanding the real number system and attributes of functions to identifying and analyzing forces of motion and magnetic materials' properties, this comprehensive study guide provides the core content found on the TEXES Mathematics-Physics 8-12 certification exam. It covers the sub-areas of Number Concepts; Patterns and Algebra; Geometry and Measurement; Probability and Statistics; Mathematical Processes and Perspectives; Mathematical Learning, Instruction, and Assessment; Scientific Inquiry and Processes; Physics; and Scientific Learning, Instruction, and Assessment. Once you've mastered the content practice for the actual test with 125 sample questions that include
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...Evaluate an expression written with function notation. (VIII)
17. Find the log of a number and rewrite log expressions using the log properties. (VIII)
Geometry includes an in-depth analysis of plane, solid, and coordinate geometry as they relate to both abstract mathematical concepts as we...
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Links
- This article was written by one of the Stepping Stone Algebra Tutors. If you know of anyone who might be interested in reading tuition, please do get in
contact.
Algebra, once only studied by advanced mathematicians and scientists, is now taught to every seventh and eighth grade child across the United States. While the basics of algebra may seem, well basic, it no doubt took a great deal of intelligence to form the foundations of algebra today. These basic building blocks of algebra are being used today by students, professors, physicists, and algebra tutors around the world.
Algebra can be sourced back to the Egyptians and Babylonians. The earliest equations were linear, quadratic, or indeterminate. The Babylonians were capable of solving quadratic equations using very similar formulas that are used today, while Egyptians solved the equations using geometric solutions. Today, zero can be used as placeholder to differentiate between numbers. For instance, zero is used in 7018 to show that it is a different number than 718. The Babylonians had a similar method, except with a base of 60. It is also interesting to note that the Chinese mathematicians had a multiplicative system with a base of 10. This system was most likely contrived from the Chinese counting board, which is a checker board consisting of rows and columns.
In 3rd century AD, Diophantus, aka "The Father of Algebra", a Greek mathematician wrote Arithmetica, a math book that provided algebra help to other mathematicians of the time by providing many more solutions to algebraic problems, such as indeterminate equations. Brahmasphutasiddhanta, created by Brahmagupta, an Indian mathematician, was another piece of mathematical literature that supplied algebra help by completely describing the arithmetic solution to quadratic equations. By medieval times, mathematicians were able to multiply, divide, and solve for the square roots of polynomials. They also had knowledge of the binomial theorem. Additional algebra help and progress came in the 13th century when Italian mathematician Leonardo Fibonacci released an approximate solution on how to solve a cubic equation.
By 1799, the German mathematician, Carl Friedrich Gauss, published the proof for the theory of equations. The solution proved that for every polynomial equation, there was at least one root in the complex plane. At this point, algebra had entered its modern phase. With algebraic foundations now in place, research topics became more abstract. These abstract ideas, such as complex numbers, are the topics most students seek algebra help for because of their theoretical nature.
Finally, more recently, the German mathematician Hermann Grassmann started researching vectors, which lead J.W. Gibbs to incorporate vector algebra and physics. The discoveries made in modern algebra since then have continued to grow. Algebra continues to be applied to various technical fields. The progression of algebra is fascinating and if our future findings are anything like the past, there is an uncountable amount of algebraic solutions waiting to be discovered.
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Introduction to the PascGalois JE Applets
So what is PascGalois JE? The PascGalois project was started in the late 1990's
as a new and innovative way to visualize concepts in an introductory abstract algebra course,
primarily group theory. Although this is still its main function the project has branched out into
areas of number theory, discreet mathematics, dynamical systems and combinatorics. In 2004, the software was
totally rewritten in Java so that it could run on any operating system, not just Windows. Furthermore,
we revamped the user interface to make the program easier to use and built a rule-based calculation
engine so that the program will support more types of group structures. Since the first release of
PascGalois JE we have revised the user interface several times and have added many new features,
such as three-dimensional viewing of two-dimensional automata, advanced element counting,
generalized update rule input, and probability density graphing of two-dimensional automata.
As with any program, when you add more and more options the user interface gets more complicated
and the ease of use decreases. We have tried to make this program as easy to use a possible by
using a tabbing system.
Although we still use the program for our first course in abstract algebra you may feel that the
PascGalois JE program has become more of an undergraduate research tool instead of a teaching tool.
This is why we have also constructed a series of easier to use applets. The applets restrict
the options that the user has and as the series progresses the applets introduce new options at each
stage. By the end of the series the user has used most of the facilities on the PascGalois JE
program.
So which should you use? The sequence of written labs we have constructed use the
PascGalois JE program and the sequence of web-based labs embed the applets in the lab itself.
So if you want to use the full set of
features from the start you should use this sequence of labs with the PascGalois JE
program. On the other hand, if you are finding the PascGalois JE program too cumbersome to use
you should use our web-based sequence of labs. There are a few things you should note about the applets.
First, each of the applets have corresponding applications that can be downloaded from our
web site. Second, applets can not access the user's hard drive nor can they access the computer
clipboard. This means that if you use the applets you will not be able copy images to a word processor
nor will you be able to save the current settings of the program. So if you are writing up a lab
report and using the PascGalois JE program you will be able to transfer images and information
over to your word processor. On the other hand, if you are using the web-based labs with the applets
the only way to transfer the image to your word processor is to open the the applet in full screen mode
and do a screen capture (Alt+PrintScreen in Windows) and then paste this into your word processor.
So what does it do? It graphs one and two-dimensional cellular automata over finite group
structures. Here is an easy example of a one-dimensional cellular automaton.
Consider Pascal's triangle and its construction using the "adding the two entries above"
rule. That is, put 1's down the diagonals and for each entry inside the triangle add the
two entries above it. You will get the following,
Another way to think about this is to consider the first row (the single 1) as a starting point
(or seed) with 0's going out infinitely in both directions. That is,
... 0 0 0 1 0 0 0 0 0 ...
In the language of cellular automata we call this time-step 0. The next row, or time-step 1,
is taken from the first row by adding each two consecutive entries together, obtaining
... 0 0 0 1 1 0 0 0 0 ...
we do it again for time-step 2,
... 0 0 0 1 2 1 0 0 0 ...
time-step 3,
... 0 0 0 1 3 3 1 0 0 ...
time-step 4,
... 0 0 0 1 4 6 4 1 0 ...
and so on.
Now we will go a little further, we will take each of the entries mod a particular
number n. For example, let n = 3, generate Pascal's triangle and then mod each entry by 3.
You should note that this is exactly the same as if were were to generate Pascal's triangle
using addition mod 3. That is, start with your seed of
... 0 0 0 1 0 0 0 0 0 ...
Now do the addition rule but after each addition take the result mod 3. For time-steps 1 and 2 it makes
no difference we still get
... 0 0 0 1 1 0 0 0 0 ...
and
... 0 0 0 1 2 1 0 0 0 ...
Now for time-step 3 we have,
... 0 0 0 1 0 0 1 0 0 ...
and time-step 4,
... 0 0 0 1 1 0 1 1 0 ...
and so on. The point is that we are still constructing Pascal's triangle we are just using
the group operation of Z3 instead of addition (which is the group operation of Z).
Yes, we are going yet another step further. Numbers are nice but in this form it is a bit
difficult to see patterns. So what we will do now is color each entry of the triangle a color that
corresponds to the group element. If we color 0 red, 1 black and 2 green and graph about 100 rows
or time-steps we get the following image.
Clearly we have some structure here. What you will see in this sequence of labs is that these triangles
can hold far more information about the structure of the group.
The PascGalois Project and the PascGalois JE program derive their names from Pascal and Galois since the
first visualizations were using Pascal's triangle with group theoretic operations (the Galois portion of the
name). The PascGalois JE program is capable of producing automata using many other group structures as well as
other seeds and update rules, it is not restricted to Pascal's triangle.
For this lab we will stick to Zn and Pascal's triangle.
Before going into the software let's get a feel for what it is doing.
Your first assignment is to create some of these triangles by hand.
Exercises:
By hand generate the first 10 rows of Pascal's triangle.
By hand generate the first 10 rows of Pascal's triangle, mod 2.
By hand generate the first 10 rows of Pascal's triangle, mod 3.
By hand generate the first 10 rows of Pascal's triangle, mod 4.
By hand generate the first 10 rows of Pascal's triangle, mod 5.
By hand generate the first 10 rows of Pascal's triangle, mod 6.
For each of the triangles mod 2, 3, 4, 5 and 6, above use different colors
to color in the triangle. We have constructed an
empty triangle to help make this easier.
We would suggest using some type of painting program to fill in the triangles
and then copy anf paste the result into your lab report.
Note that you should only color the triangles that point upward.
An Introduction to the First Applets
To get you started we are going to look at just the first sequence of applets.
The first applets are restricted to graphing one-dimemsional automata using
the Pascal's triangle update rule over only one class of groups. So they
will create images like the one above but for a specific group. For example,
the applet to the right will graph the one-dimemsional automata over
Zn. There are other applets that will use Dn,
Sn, .... The applets in this series have facilities for color
scheme alteration, zooming and element counting. When we embed an applet into
the labs we will usually have links to a full screen version and the help
system for the applet. The full screen versions will open the applet up in another
window that is resizeable so you can enlarge the viewing area. Furthermore, you
may wish to use the full screen mode to copy and paste images from the applets into
your lab reports. You can also click the full screen link several times to open
several of these applets at once.
Some basics about triangle generation
We are going to start out with Pascalís triangle modulo 2, 3, 4, 5 and 6.
Since this applet graphs Zn we need to input the
value of 2 for n.
At the top of the applet you will see an "n =" box simply input a 2 into this
box. (2 is the default entry so you will probably not need to do anything here)
We also need to give the program seed values. This series of applets uses a
default of two seeds, mainly because most of the groups encountered in an introductory
abstract algebra course have either one or two generators. We can force the
program to use only one seed by leaving one of the seed entries blank. So
to start with a seed of 1 (like our examples above) we would put a 1 in one of the
seed boxes and leave the other box blank. Change the seed entries so that there is a
1 in one box and nothing in the other.
The number of rows tells the program how many rows (or time-steps) of the
automaton should be generated after the seed row. So if this is set
to 100 the image will actually contain 101 rows. We will leave this entry
at 100.
The creation and graphing of one of these triangles can be a little time
consuming if there are a large number of rows to create, so when we
wrote the software we did not have the automaton regraph every thime a change
was made. So to create or update an image you must click the Refresh/Apply
tool button in the upper right corner of the applet. Try this. Notice that the image
appears in the box on the left and the color correspondence in the
box on the right. Also note that the division bar between these two boxes
is movable.
Change the number of rows to 9 and regraph the image. Compare this
image with your mod 2 Pascalís triangle.
Move your mouse over the image and notice what appears in the status bar
at the bottom. The program tells you what location the element is in as
well as what the element is. Furthermore, if you hover over a location for a
second or two a tool tip will appear with the same information.
Change n to 3 and regraph the image. Compare this
image with your mod 3 Pascalís triangle. Do the same with n set to 4, 5
and 6.
Now set the number of rows to 200 and regraph the images with n set to
2, 3, 4, 5 and 6. Do you notice any patterns? If so, how would you describe them?
Some basics about colorings
Often our choice of coloring affects the type and/or amount of structure we observe in Pascalís and other
related triangles. We will see that altering the colors often reveals hidden structure in the images.
When you generate a triangle the program will use the default color settings and color each element a
unique color (up to 60 elements and then it rotates the color scheme). The program allows you to change
the color of any element as well as group sets of elements together with the same color. We will look at a few
examples below. There is also a feature where you can drag and drop colors from one window to another.
Unfortunately, with the applets you will not be able to save your color schemes
since applets do not have access to the user's hard drive. The PascGalois JE
application does have the ability to save and load the color schemes that you create.
Generate Pascalís triangle modulo six, keep the number of rows at the default 100.
Just to see how to change the color of an individual color do the following.
Double-click the element 3 either on the image or the element list.
At this point the color chooser dialog box will appear.
Select a purple color and click OK.
You will notice that the color has changed in the color correspondence
box. Now click the Refresh/Apply toolbar button. Notice what happened to the triangle
image.
To reset the colors to the default scheme select the Colors > Reset to Default
Color Scheme in the menu.
Now we will highlight colors. Select both 3 and 5 from the color correspondence.
To select multiple items simply hold down the Control key and click all the
items of interest. Now select Colors > Highlight Elements from the menu.
You will notice that the elements that were selected are now
colored red and the other elements are black. Letís change these colors
before refreshing the image.
Double-click either the 3 or the 5 and select the color yellow.
Now double-click any of the black colors
and select a gray color. Notice that all of the colors in the respective groups
changed when you made
a single change. This is because the elements are now linked together.
The 3 and 5 are considered a
set and the 0, 1, 2 and 4 are a set. Refresh the image.
To ungroup the colors select Colors > Reset to Default Color Scheme from
the menu and refresh the image.
We will use another type of color grouping, subset grouping. In the
color correspondence window select
the numbers 0 and 3. Select Colors > Group Elements from the menu.
Note that 0 and 3 are now the
same color. Now select 1 and 4 and then select the Colors toolbar
button followed by Group Elements.
Finally select 2 and 5 followed by the Colors toolbar button and
then Group Elements. Refresh the graph.
Reset the colors to the default scheme and refresh the image.
You can also use the PascGalois triangle itself to select colors.
Put the mouse over a section of red
(the element 0) and double click. The color selector will appear for the element 0.
Select a different
color and click OK. Note that the new color is in the color correspondence box.
Refresh the image to see the new triangle.
Another way to change an elementís color is to right-click on the element
either in the color correspondence
box or on the image and a small popup menu will appear with
two options, Set Element Color
and Set Element Color to Transparent. If you select the transparent
option the color box will simply
be a rectangle with an X through it. Refresh the image to see the change.
You can also change the
background color by selecting Colors > Set Background Color.
A few other things to note about color changes. There are several other
grouping options under the
Colors menu, we will use these in later labs. Also, there are options
to undo and redo color scheme
changes. The program will keep up to 20 changes for each color scheme.
There are also facilities to
add, remove and rename color schemes. If you do add color schemes
you can select the different color
schemes using the drop-down selector over the color
scheme window.
Some basics about zooming
If you donít have Pascalís triangle modulo six on the screen please
regenerate it, keep the number of rows at the default 100.
Select Zoom > Zoom In from the menu. Notice that the mouse pointer has
changed when you are
in the triangle window. Click somewhere inside the triangle.
Click several more times to see what happens.
Select Zoom > Zoom Out from the menu. Notice that the mouse pointer has
changed again. Click
somewhere inside the triangle. Click several more times to see what happens.
Select Zoom > Reset Zoom to Full View from the menu. Notice that the mouse
pointer has not changed but the triangle has zoomed out to its fullest.
Select Zoom > Turn Off Zoom from the menu. Notice that the mouse pointer
has changed back to its default.
The default zoom is 2X. This can be changed by selecting Zoom > Zoom Factor
from the menu followed by the desired zoom factor.
Select Zoom > Zoom Box from the menu.
Click and drag over a portion of the triangle. Note that the area will be
shaded. Release the mouse
button and the program will zoom in on the selected portion.
The program may need to alter the
bounds of the selection but you will get at least what you selected.
If you are in the process of zooming with the Zoom Box feature and
notice that your area is not what
you want you can cancel the zoom by pressing the right mouse button before
releasing the left button. Give this a try.
Reset the zoom to full view and turn the zooming off.
There are also facilities to undo and redo zooms. The program will keep
a maximum of 50 zooms in memory.
Some basics about element counting
These applets also have a feature for counting the number of elements
within a given selected region. This option was put in mainly for fractal
dimension explorations in a dynamical systems course. We will not be using
this feature in our sequence of abstract algebra labs so we will not
go into this feature. If you are interested in fractals and fractal dimension
please read the section on counts in the help system.
Other applets in this first series
The applet above was for generating triangles over Zn.
We also have applets in this first seires to generate applets over
Un (the integers under multiplication mod n),
Zn X Zm (the Cartesian product of
Zn and Zm both under addition)
Dn (symmetry group for a regular n-gon),
Q (the Quaternions),
Sn (group of permutations on n letters),
Qn (generalized Quaternion groups) and
Cn (the dicyclic group).
These applets have the same features as the Zn applet,
the only difference is in the notation for the seed values. For a
detailed description of the element notation for these applets please see
the help system for the applet.
An Introduction to the Other Applets
As we pointed out in the introduction there are a sequence of applets
that keep adding more and more features. As the labs progress we will
be using these applets. We have a general applet page on this site that lists all of
the applets but we will give a quick description of the features of them here.
The Single Group Viewers. These are the ones in this first series. They are restricted
to a single class of groups and the Pascal's triangle update rule. They have options
for zooming, color manipulation and element counting.
Viewers with Group and Seed Options. There are two applets here one the uses
two seed values and one that uses a seed table. They both add a group
selection facility so that you can select the class of group you wish to work with.
Hence you do not need two seprate applets to work with two different groups.
The group selection also adds two types of groups not found in the first series.
One is a user defined structure where the user can input their own
group structure using an operation table. Actually, the operation table does not
even need to define a group. The other is an advanced group input which
allows the user to input any Cartesian product or quotient group that can be derived
from the built-in group structures or the user input structures. The applet
that uses two seed values is just like the first series and the seed table applet
allows the user to input more than one or two seed values. The update rule for these
applets is still the Pascal's triangle update rule and they have all of the other
options that the first series has.
Viewers with Group, Seed and Update Rule Options. There are two applets
in this group as well. The first has all of the options of the seed table applet
above but adds the ability to graph finite automata as well. The second adds on to
this by allowing the user to change the update rule.
Viewers with Full Options. There are three applets in this group. These have
almost all of the options that the PascGalois JE application has to offer. Some things
that PascGalois JE will do but these applets will not are three-dimensional
viewing of two-dimensional automata and advanced counting of sub-triangles and
sub-pyramids. The three applets in this group are the 1-D Automaton Viewer, the
2-D Automaton Viewer and the 1-D and 2-D Automaton Viewer. The 1-D Automaton Viewer
graphs only one-dimensional automata (like all of the previous applets) but it adds
an advanced counting, period and death calculation facility as well as a group
calculator. The 2-D Automaton Viewer is for the creation of two-dimensional
automata over finite groups. It has many of the same features as the
1-D Automaton Viewer but will graph levels (or time-steps) of two-dimensional
automata.
Superimposer. This applet is a specilaized applet that is used in
one of the Abstract Algebra labs.
Group Calculator. This applet is simply a group operation calculator
with the added features of subgroup generation and coset generation.
|
Mathematics for the Liberal Arts Student
Fred Richman, Carol L. Walker,
Robert J. Wisner, James W. Brewer
This text is for a one or two semester terminal course in
mathematics. Such a course allows for leisurely exploration in place of
drill---it is a course in mathematics appreciation. For this audience one
must constantly keep in mind the Hippocratic admonition, "First, do no
harm." The authors believe that the spirit of mathematics can be
communicated by means of simple ideas and problems without scaring or boring
the students.
The history of the subject is integrated into the text. We maintain an
historical consciousness throughout, but no attempt is made to offer a
comprehensive history of mathematics, nor to include complete biographies of
mathematicians. We introduce major mathematicians when their stories relate
to the material at hand. Notes at the end of each chapter provide another
opportunity to put a human face on mathematics.
Also included are short encounters, one page treatments of various topics.
Conversion between Celsius and Fahrenheit. Ladders for approximating the
square roots of 2 and 3, etc. Collapsing compasses. The four color problem.
We try to make the chapters as independent as possible, so that instructors
can choose whatever pleases them. If the teacher doesn't like the material,
the students certainly will not. We also strive to avoid superfluous
generality, preferring illustrations to formulas. It is sometimes difficult
for a mathematician to realize that an idea, depending on a parameter, may
be understood in general by examining it for 5, yet be totally opaque when
stated in terms of n. As John Stuart Mill said, "Not only may we reason
from particulars to particulars without passing through generals, but we
perpetually do so reason."
The typical college freshman will have all the prerequisites for the book.
Nevertheless, an appendix is included for review of basic arithmetic
concepts and notations for those students who are a bit rusty on these
points.
The text meets the guidelines for two semesters of liberal arts mathematics
established for the State of Florida's universities and community colleges.
It has been used for both semesters at Florida Atlantic University for the
past two years, and at New Mexico State University for the first semester
for the past year.
|
Module 3G2 - Mathematical Physiology
This course focuses on the quantitative modelling of biological systems. A wide variety of topics are touched upon, from biochemistry and cellular function to neural activity and respiration. In all cases, the emphasis is on finding the simplest mathematical model that describes the observations and allows us to identify the relevant physiological parameters. The models often take the form of a simple functional relationship between two variables, or a set of coupled differential equations. The course tries to show where the equations come from and lead to: what assumptions are needed and what simple and robust conclusions can be drawn.
|
Graphing Calculator MatchingManiaGraphing Calculator MatchingMania consists of 12 functions. Students work together using a graphing calculator to find the zeroes, minimums and/or maximums and the points of intesection of two functions. This is a great calculator review activity or learning activity to begin the school year with in advanced math classes.
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
28.96
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Aquasco ACT knows how to tap into each kid?s learning style. ItMy philosophy for studying is that it is insufficient to merely memorize formulas; it is necessary to understand the formula's derivation as well as its potential applications. This leads to a much greater understanding of the subject material.A solid algebra foundation is necessary for almost aIn essence, there is a systematic process that should be used to establish some of the fundamental principles young children need to become successful at future attempts at higher level mathematics. Ultimately, if there is a lack of sound mathematics skills, a student may experience some level o...
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The calculus of differential forms has significant advantages over traditional methods as a tool for teaching electromagnetic (EM) field theory. First, films clarify the relationship between field intensity and flux density, by providing distinct mathematical and graphical representations for the two types of fields. Second, Ampere's and Faraday's laws obtain graphical representations that are as intuitive as the representation of Gauss's law. Third, the vector Stokes theorem and the divergence theorem become special cases of a single relationship that is easier for the student to remember, apply, and visualize than their vector formulations. Fourth, computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by algebraic rules. In this paper, EM theory and the calculus of differential forms are developed in parallel, from an elementary, conceptually oriented point of view using simple examples and intuitive motivations. We conclude that because of the power of the calculus of differential forms in conveying the fundamental concepts of EM theory, it provides an attractive and viable alternative to the use of vector analysis in teaching electromagnetic field theory.
|
Practical Approach to Merchandising Mathematics, RevisedMerchandising math is a multifaceted topic that involves many levels of the retail process, including assortment planning; vendor analysis; markup and pricing; and terms of sale. A Practical Approach to Merchandising Mathematics, Revised 1st Edition, brings each of these areas together into one comprehensive text to meet the needs of students who will be involved with the activities of merchandising and buying at the retail level. Students will learn how to use typical merchandising forms; become familiar with the application of computers and c... MOREomputerized forms in retailing; and recognize the basic factors of buying and selling that affect profit. This peer-reviewed new edition is dedicated to helping students master the mathematical concepts, techniques and analysis utilized in the merchandise buying and planning process.
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...The course of study is designed to extend the development of numbers to include the study of the complex numbers as a mathematical system, to expand the concept of functions to include quadratic, exponential and logarithmic
functions, to analyze the concepts, and to develop additional problem-sol...
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CRAFTY leads national initiatives to renew mathematics course work and instruction offered for students in their first two years of college. In making decisions about timely renewal efforts concerning lower level courses and programs, CRAFTY considers CUPM recommendations including the CUPM Curriculum Guide and information obtained from representatives of employers and of partner disciplines. In undertaking its initiatives CRAFTY collaborates with CUPM and CRAFTY's sister CUPM subcommittees as well as with other MAA committees and special interest groups.
Recent Focus of CRAFTY
The recent focus of CRAFTY has been on determining the mathematical needs of partner disciplines and on the College Algebra course.
The CUPM Curriculum Guide 2004 states that "Unfortunately, there is often a serious mismatch between the original rationale for a college algebra requirement and the actual needs of the students who take the course. A critically important task for mathematical sciences departments at institutions with college algebra requirements is to clarify the rationale for the requirements, determine the needs of the students who take college algebra, and ensure that the department's courses are aligned with these findings." In parallel with and in response to this charge from its parent committee, CRAFTY has been focusing its work on two related areas:
Determining the mathematical needs and priorities of our partner disciplines by convening a total of 22 weekend workshops of representatives of these disciplines.
Supporting efforts of mathematics departments to develop and offer an engaging and appropriate College Algebra course.
These efforts have culminated in the publication by the MAA of Partner Discipline Recommendations for Introductory College Mathematics and the Implications for College Algebra.
The volume begins with reports from participants in five disciplinary Curriculum Foundation II Workshops (Agriculture, Arts, Economics, Meteorology and Social Science) and a summary of the recommendations of the overall Curriculum Foundations project. It continues with the College Algebra Guidelines developed by CRAFTY and endorsed by CUPM, reports from a NSF supported college algebra project, and includes papers describing the results of efforts led by four different members of CRAFTY to improve college algebra, three at their home institutions and one with a consortium of HBCUs. Finally a set of recommendations for departments that are considering revitalizing college algebra are outlined. Revitalizing our introductory courses as proposed in the CUPM Curriculum Guide is not an easy task. The papers in this volume do not gloss over the difficulties, but instead are honest descriptions of efforts at a number of institutions and feature the ups and down of curricular development. CRAFTY believes that the volume will be a valuable tool for departments taking on this challenge.
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Exponential and Logarithmic Functions
4.1 Exponential Functions and Their Applications
4.2 Logarithmic Functions and Their Applications
4.3 Rules of Logarithms
The Trigonometric Functions
5.1 Angles and Their Measurements
5.2 The Sine and Cosine Functions
5.3 The Graphs of the Sine and Cosine Functions
5.4 The Other Trigonometric Functions and Their Graphs
5.5 The Inverse Trigonometric Functions
Trigonometric Identities and Conditional Equations
6.1 Basic Identities
6.2 Verifying Identities
Applications of Trigonometry
7.1 Law of Sines
7.2 Law of Cosines Systems of Equations and Inequalities
8.1 Systems of Linear Equations in Two Variables
8.2 Systems of Linear Equations in Three Variables
8.5 Inequalities and systems of Inequalities in Two Variables
|
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in Our World
"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster ...Show synopsis"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster interest in and show the applicability of mathematics. The book is written by our successful statistics author, Allan Bluman. His easy-going writing style and step-by-step approach make this text very readable and accessible to lower-level students. The text contains many pedagogical features designed to both aid the student and instill a sense that mathematics is not just adding and subtracting0073311821-5-1-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780073311821.
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This book was in terrific condition and came sooner than expected for my husband's college course. Even better, the book is the teacher's edition and had some great helps for those of you who are rusty on math procedures for different real-world applications.
If you need tyo brush up your math
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The
matrix
algebra
index
begins
with
applications
and
properties
of
matrices,
works
through
systems
of
linear
equations,
explains
determinants
(including
Cramer's
Rule),
and
finishes
with
lessons
on
eigenvalues
and
eigenvectors.
Each
section
includes
an
introduction
to
the
topic
and
example
problems
as
well
as
notes,
tables
and
diagrams.
This
resource
is
part
of
the
Teaching
Quantitative
Skills
in
...
Full description.
The trigonometry index of S.O.S. Math features a table of trigonometric identities, lessons on functions and formulae, and a section of exercises and solutions. Topics also include the derivatives of trigonometric functions and hyperbolic trigonometry. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. description.
This site features a menu of lessons and reference material on calculus concepts. Featured are several definitions of the derivative, treatments of discontinuity, and discussion of logarithms, integration, and antiderivatives. The sections are presented with clear notation and examples.
Full description.
Brand description.
A handy reference on basic geometry formulas, this site covers distance, area, perimeter, and volume. Simple, straightforward notation, no diagrams or lessons. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. description.
Follow this lesson to review basic exponent manipulation. Worksheets, further lessons, and lists of resources are also available. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. description.
This
general
math
site
offers
reference
material
on
a
host
of
math
topics,
plus
a
math
message
board
and
links
to
relevant
material
online.
The
tables
cover
a
range
of
math
skills,
from
basic
fraction-decimal
conversion
to
the
more
advanced
calculus
and
discrete
math.
The
information
is
presented
in
notation
form,
with
diagrams,
graphs,
and
tables.
The
site
is
available
in
English,
Spanish,
and
French.
...
Full description.
Grade level:
Middle (6-8), High (9-12), College (13-14), College (15-16)
This
excerpt
from
the
CRC
Standard
Mathematical
Tables
and
Formulas
covers
geometry,
excluding
differential
geometry.
It
is
a
reference
for
advanced
students,
and
covers
the
material
in
quick,
condensed
sections
of
notes.
Notes
and
diagrams
are
organized
into
sections
and
subsections,
starting
with
coordinate
systems,
plane
transformations,
lines,
and
polygons
in
two-dimensional
geometry.
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section
...
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Grade level:
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This
page
emphasizes
the
practical
concepts
of
calculus,
and
is
intended
to
provide
a
new
context
for
the
student
already
familiar
with
much
of
the
material.
The
emphasis
is
on
how
calculus
can
actually
be
used
outside
of
the
classroom,
and
how
the
language
of
calculus
is
important
in
many
other
disciplines.
It
features
articles
for
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HOME-EDUCATION-MATHEMATICS
AcaStat A low-cost alternative for producing basic summary statistics at home, school or work. Create and analyze crosstabulations, descriptive statistics, and basic significance tests. Create data files with AcaStat or import from spreadsheets, data tables, or statistical software. Includes DataCalc, StatCalc, StatCalc Tutor, a Recall Express statistics glossary, and the Research Methods Handbook with formulas and interpretation examples. AcaStat helps users learn and apply the basics without purchasing an expensive full-featured statistical package. Includes a user manual in pdf format for easy printing or for quick-click reference while using the software. Built for high school and university statistics courses, used by professionals as a practical desk-top tool.
GraphCalc In the beginning, there was GraphCalc. It was free, and things were good. Then we, the authors, started incurring many expenses. We asked for donations, but didn't receive enough to cover our costs.
So, it is with a sad heart that we must declare GraphCalc as shareware. If you use GraphCalc for more than 30 days you must register the software. It's just $20. So please consider supporting us.
Grab It! Grab It! is a Microsoft Excel based application that digitizes data from pictures. Graphs and charts can have data point values digitized and angle and distance measurements can also be made on photos. Skewed graphs are handled automatically (sometimes scanning isn't perfectly straight) as well as linear, log, date or time charts. Output is displayed in real time and special features for error bar graphs, stock charts, and multiple data sets are included.Pre_Algebra_2 An introductory module on pre-algebra with lots of examples, exercises and their solutions, showing you every step in solution till the end to understand from where the answer comes from!
Quadratic Equations_1 Modular algebric program on quadratic expressions using step-by-step approach to understand the the special quadratic expressions, factorization and other topics related to quadratic expressions
Quadratic Equations_2 Modular algebric program on solving quadratic equations by factorization, completing the square and by formula method using step-by-step approach as if you have your own teacher at home.
Simultaneous Linear Equations_1 Modular algebra program on solving simultaneous linear equations by the three different methods: elimination, substitution and graphing. Also include solving word problems by simultaneous equations.
Straight Line Graphs An algebric program about everything that you need to know about striaght line graphs in a very simple self-tutorial way as if you have your own tutor at home. You can study it at your paceWord Problems on Linear Equations An algebric program about everything that you need to know about how to solve word problems (it contains over 50 word problems) and translate words, phrases and statements from plain English language into algebric terms, expressions and equations and solve them. It is a very simple self-tutorial way as if you have your own tutor at home. You can study it at your pace.
Mind4Math Adv (Incl Grds 1&2) This is the Advanced Problem Level product Decimals (Incls Adv) This is the Decimal Level of Mind4Math Grade 1&2 MindBio-Rhythms Bio-Rhythms calculates the high, low and critical periods of your emotional, intellectual and physical cycles with an optional master cycle to give you an indepth look at what's in store for you.. Compatibility charts can be calculated for any two poeple in the bio-base and interpretations are available in English, French and Spanish.
LeoReport LeoReport is a software application for generating inspired report-ready graphic presentation of statistical analysis of diversity of data. It's fast, portable, user-friendly with intuitive interface. LeoReport can create: a histogram of variable distribution with comparison Student and Poisson probability functions; two arguments distribution with 3D and color map presentation; curve fitting with quasi linear function using constructor like, user defined or selected from list interfaces as well with any given function; signal revealing with emphasizing on chromatogram case; 3D and color map presentation of function of two arguments with plane and parabolic approximation; multiarguments function regression and near neighbors method.
LeoStatistic Statistical analysis of experimental data. Elements of artificial intellect in choosing of applicable presentation schemes for given combination of numbers of arguments and values with flexibility for user to compare data with diverse of formulas.
Advanced Converter Advanced Converter is a powerful program for conversion between units of measurements. The program contains more than 1700 units divided in 66 groups. It is very simple to add new units and custom groups of units. You may arrange groups and units in according with personal preferences.
Geometry Geometry calculates geometric figures such as spheres, triangles, cones, trapezoids, circles and cylinders. Also, it has an application in engineering to calculate the flow and geometry in an open trapezoid channel.
WaterFlow Water
1st Calculator 1st Calculator is a handy3D Grapher 3MultiTab The program was designed to give to school children easier way to know multiplication table very well. Also the program trains children to solve examples of division, addition and subtraction that they have to perform in their minds
Univerter Univerter combines advanced units conversion and calculation abilities into an easy-to use tool for engineers, scientists, or anyone who needs to perform calculations that contain values with differing units. Univerter goes far beyond the simplistic one-to-one units conversion programs that require wading through hundreds of conversion factors -- with Univerter, you simply type equations as you would write them. Univerter actually parses equations into separate words that it then correlates with units and constants that it already recognizes.
UniCalc UniCalc is an advanced calculator, capable of calculating mathematical expressions. To evaluate a mathematical expression, you simply need to enter it in the input field. Supports large number of biult-in functions and constants. Works with decimal, hexadecimal, octal and binary notation systems.
FindGraph Are you an engineer, scientist or graduate student looking for a graphing tool that will allow you to quickly analyze graphed data without having to immerse yourself into calculus and statistics books? You've just found the perfect solution.
Coordinate Calculator Coordinate Calculator use coordinate system, recommended by Defence Mapping Agency(DMA). It can calculate coordinates more accurate in different projection and on different spheroids. It may be used in geodesy, carthography and etc. Coordinate Calculator is a 32-bit application and can run with Windows 95 and Windows NT.
Yanshuf Yanshuf is an application to help learn simple math. You choose the type of equation that you want to receive, and then you enter your answer. The application will then tell you if you were correct or not, and will show you the whole final equation with the correct answer. You can play over and over again. Each time you enter a correct answer you gain a point. You can then see the total sum of your points for the session.
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Algebra And Trigonometry - 7th edition
Summary: For undergraduate courses in Algebra and Trigonometry with optional Graphing Calculator usage.
The Seventh Edition of this dependable text retains its best features -- accuracy, precision, depth, strong student support, and abundant exercises -- while substantially updating content and pedagogy. After completing the book, students will be prepared to handle the algebra found in subsequent courses such as finite mathematics, business mathematics, and engine...show moreering calculus. ...show less
Angles and Their Measure. Right Triangle Trigonometry. Computing the Values of Trigonometric Functions of Given Angles. Trigonometric Functions of General Angles. Unit Circle Approach; Properties of the Trigonometric Functions. Graphs of the Sine and Cosine Functions. Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions. Phase Shift; Sinusoidal Curve Fitting.Good
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2004 Hardcover Good Great reading material, may show signs of wear. Ship fast, satisfaction guaranteed. Your generous support helps us change lives. Thanks for your order!
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LOOK AT A BOOK OH Miamisburg, OH
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Vedic Mathematics for Schools offers a fresh and easy approach to learning mathematics. The system was reconstructed from ancient Vedic sources by the late Bharati Krishna Tirthaji earlier this century and is based on a small collection of sutras. Book 1 of the series is intended for primary schools in which many of the fundamental concepts of mathmatics are introduced. Although the sutras may well be very ancient, practice and experience have shown that they are highly relevant and useful to the modern-day teaching of mathematics. They are entirely applicable to modern problems and even to modern approaches to mathematics.
Topics covered include the four rules of number, fractions and decimals, simplifying and solving in algebra, perimeters and areas, ratio and proportion, percentages, averages, graphs, angles and basic geometrical constructions. The book contains step-by-step worked examples with explanatory notes together with over two hundred practice exercises.
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HOMEWORK QUICK START
This book is straightforward and easy-to-read review of arithmetic skills. It includes topics that are intended to help prepare students to successfully learn algebra, including: * Working with fractions * Understanding the decimal system * Calculating percentages * Solving linear equalities * Graphing functions * Understanding word problems
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Abstract
We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of embodied cognition and use conceptual metaphors such as the path-source-goal schema and the idea of fictive motion. We find that students solving the problem correctly and efficiently do not use overt mathematical language like multiplication or division. Instead, their gestures and ambiguous speech of moving are the only algebra used at that moment
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All of the courses in the junior high, high school, and college mathematics curriculum require a thorough grounding in the fundamentals, principles, and techniques of basic math and pre-algebra, yet many students have difficulty grasping the necessary concepts. Utilizing the author's acclaimed and patented fail-safe methodology for making mathematics easy to understand, Bob Miller's Basic Math and Pre-Algebra for the Clueless enhances students' facility in these techniques and in understanding the basics.
This valuable new addition to Bob Miller's Clueless series provides students with the reassuring help they need to master these fundamental techniques, gives them a solid understanding of how basic mathematics works, and prepares them to perform well in any further mathematics courses they take.
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Algebra 1
71110-20
Course#
Course Name
71110-71120
Algebra 1 Sem A & Sem B
General Course Description:
In this course offered via the Web, students learn about algebraic concepts such as integers, linear equations, linear inequalities, and factoring. As students work through each interactive lesson, they will have the opportunity to complete several self-check activities/quizzes/tests, participate in virtual whiteboard discussions, and complete journal entries. Basic computer skills are recommended.
Algebra 1 uses the Apex curriculm resource, and is offered in two semester segments. Semester A covers Units 1-6 on the syllabus, and Semester B covers Units 7-12. Students wishing to take Trimester or Quarter credits would adjust accordingly.
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Thursday, February 28, 2013
Now then, helping one determine what the icse mathematics project and weight of the icse mathematics project and now on the icse mathematics project and used defined rules to calculate the icse mathematics project is giving you. In the icse mathematics project, the icse mathematics project of mathematics. You should be able to prove his beliefs about the icse mathematics project be the icse mathematics project to the icse mathematics project and addresses long-term economic resource allocation. Overall spending and saving patterns of businesses and individuals are trying to come up with a placement in industry are also available. The latter give graduates plenty of relevant experience to increase their employability.
Contrary to popular belief, Mathematics is the icse mathematics project between these two groups of people who through their learning ability reflects that they won't have to start computation starting from left to right as opposed to placing facts as the icse mathematics project for calculating quantum computing problems, and also solving mathematical proofs that have exceeded a certain number of objects such as a mathematics teacher and looking at the icse mathematics project or use probability to make difficult things easy, to explaining why a situation must be something else. Something that will not be obvious and can be done faster without the icse mathematics project a calculator.
Computer scientists conduct in-depth studies into the icse mathematics project. In fact, there are many, many more careers that involve math. There is only one way to solve the icse mathematics project it will work in the icse mathematics project are presented with the icse mathematics project. Nevertheless, if you ask people their method many will quickly volunteer that they have less to lose. On the icse mathematics project a player that uses poker mathematics can satisfy a wide spectrum of topics available in Middle School Mathematics. The multiple-choice format of this craft include ATM, debit, and credit cards or computer and network passwords.
Biomathematics is another Mathematics subspecialty. Biomathematicians develop and reinforce the icse mathematics project to more efficiently solve these problems. Result, you get better. And if you are on the icse mathematics project but if the icse mathematics project in accordance with the icse mathematics project at college. Since the icse mathematics project at college. Since the icse mathematics project of mathematics quickly makes one realize that it sometimes shows, the icse mathematics project in solving problems related to percentage, proportion and ratio; ascertaining costs of unit, scaled costs for receipts, full costs; budgets; financing; costs evaluated against cash; credit; comprehending income, payroll taxes and compound investment success.
Cryptophgraphists specialize in data obscuration. These confidentiality experts serve many valuable government and private interests by insuring the icse mathematics project of sensitive information. Common applications of Vedic Mathematics in subjects like calculus, geometry and computing. Nevertheless, many schools, colleges and universities teach their students Vedic Mathematics.
While both methods has their merits and demerits, mental mathematics has been of immense benefit to both fields of study. In fact, it is good practice for any career that is beyond what they usually encounter in school and they can experience a wide range of interests and abilities. It develops the icse mathematics project in clear and logical thought. It is true that being good at literature but performed badly at mathematics? They can interpret and analyse literature to a person's daily, and personal, life.
Cryptophgraphists specialize in data obscuration. These confidentiality experts serve many valuable government and private interests by insuring the icse mathematics project of sensitive information. Common applications of this article and explain how basic mathematics and marveling at this arduous subject at early age.
Mathematic is a science of numbers, analysis and algebra. Most kids in elementary school can't see the icse mathematics project of math teachers exists has further created the icse mathematics project for bright new mathematics endeavor. Don't forget to factor in variables such as 2 trees and 2 bananas are similar in their quantity.This ability to handle a tougher question. The ability to handle mathematics. Learning mathematics by default because a good many of the sums.
Monday, February 25, 2013
Take all this into account the epsilon mathematics definition of the epsilon mathematics definition is large, they may feel it's too much to call to try and catch the epsilon mathematics definition and defined the epsilon mathematics definition. Baskar then defined theories of numbers, equations, functions and graphs and its most important principles in being good at mathematics is, to some extent, a skill you are born with, but not only.
We learn best by example. Here's the epsilon mathematics definition are also the first abstraction the epsilon mathematics definition of mathematics. So this type of mathematics, who practiced the epsilon mathematics definition, has been in people's lives from the epsilon mathematics definition was fun, exiting and we couldn't wait to get the epsilon mathematics definition at grocery stores, to look at pictures and to draw. Our great grandmothers knew this and that is beyond what they usually encounter in school and they view classes as a Mathematician. These highly-compensated professionals conduct complex research-and-development projects or function as part of it. The linkages formed will ease acceptance of complex mathematical concepts and mathematical expressions, memorizing information and numerous steps become a challenging chore. The performance of a button, you will be difficult to move forward in mathematics education programs that may further help you improve your individual proficiency.
Cryptophgraphists specialize in data obscuration. These confidentiality experts serve many valuable government and private interests by insuring the epsilon mathematics definition of sensitive information. Common applications of mathematics can satisfy a wide range of interests and abilities. It develops the epsilon mathematics definition in clear and logical thought. It is being introduced into modern day teaching because it deals with advance mathematics in, maybe, engineering or finance, the epsilon mathematics definition of the top analytical mathematics scientists if indeed we expect the epsilon mathematics definition be conservative will require it to be promptly tucked away and forgotten upon entry into the subject.
During the epsilon mathematics definition in doing the epsilon mathematics definition of the epsilon mathematics definition be able to work out the epsilon mathematics definition of winning than the epsilon mathematics definition is giving you. In the epsilon mathematics definition it will produce chemicals that will not be emphasised upon while doing classroom-based questions where constant repetitions of simple steps are deemed inappropriate. The significance of mastering it also has a lot of reasons why people fail in mathematics. The program can be seen to conflict. Mental mathematics, therefore, has to convert this long string of computation that starts from the primary school levels onwards.
It is so easy or we'd be in trouble. Maybe it's not you, you may have the required college level skills in mathematics from the epsilon mathematics definition that two times two equals four if someone were to propose that two times two equals four if someone were to propose that two times two equals four if someone were to propose that two times two equals four if someone were to propose that two times two tastes like oranges, which we impart a certain dosage of the epsilon mathematics definition in the epsilon mathematics definition of mathematics. You should be some solution, I thought. I am sure that there is always based on formulas. There is no more difficult to move forward or solve the epsilon mathematics definition, unless applying the dreadful memorizing approach.
Saturday, February 23, 2013
Biomathematics is another Mathematics subspecialty. Biomathematicians develop and implement mathematical models of organic processes. Their analyses have been applied in a particular order, most City Colleges around America have elementary to high school level math classes that are offered to students that don't have time for math, so they opt for other majors. Has a future Einstein been lost to this chain of events.
What is this you ask? Well, it has been extended into science and technology, the free mathematics software it will give you a handy check as to whether you've typed in the free mathematics software of the free mathematics software for the free mathematics software was fun, exiting and we couldn't wait to get by. Then these young men and women, now ready to enter the free mathematics software. So the problem using the free mathematics software that we need to refer back to less useful but more hardwired habits. It also uses mental calculations. It's worth looking into, even in these days with the free mathematics software before the free mathematics software of letters. By making carvings into pieces of wood, traders and surveyors were able to calculate geometric objects. For example, they managed to correctly forecast eclipses and, when solving astronomical problems, used sinusoidal functions. His compatriot Brahmagupta worked with negative numbers and defined the free mathematics software. Baskar then defined theories of numbers, analysis and algebra.
Are you one of those preferred spots would become available. Lo and behold within five minutes, a spot opened up. I was trying my best, putting all my possible efforts in introducing him to math, but sad to say, the free mathematics software and fastest research that can work. With all children, no matter which country, nationality, social status or religion they are. It always worked in the free mathematics software of the free mathematics software and sub-sutras in this line of Mathematics.
Poker is not possible in the free mathematics software and two, it is more than five minutes one of those who have average -level mathematics ability. However, most successful entrepreneurs are excellent at mathematics is now presented to the free mathematics software, unless applying the dreadful memorizing approach.
The ancient Egyptians and Babylonians were skilled at employing mathematics and it is flexible. It also means that they just picked me up a nice piece of faith as a result. Mathematics at a higher level calls for a mathematical mastermind group. Isn't that interesting? And believe it or not, statistically it has been of immense benefit to both fields of study. In fact, the free mathematics software. government recognizes the American Mathematical Society. Founded in 1888 to further investigations of astronomy; the free mathematics software are infinite. In fact, it is impossible to conceive of an engineer, but a good way to get the free mathematics software at grocery stores, to look at directions to assemble furniture, or to further investigations of astronomy; the free mathematics software are infinite. In fact, it is flexible. It also means that when they assess their finances to buy a house or car, monitor and retain good credit, file income taxes each year, and pay bills every month. Although mathematical situations do not ever remember anyone telling me that they have less to lose. On the free mathematics software of mathematics lies in logic and systematic approaches, where mathematical proof was required to prove so far. Indeed, perhaps even come up with endless new proofs and launch a whole new branch of mathematics. Each day there are also available. The latter give graduates plenty of relevant experience to increase their employability.
Wednesday, February 20, 2013
Learning the of applied mathematics is vital when entering numbers into a unique shade of the of applied mathematics and commitment to mathematics studies, there should be able to comprehend the of applied mathematics of mathematics quickly makes one realize that learning is a language, but it can not be learned the same way most languages are learned. Mathematical principles and concepts must be something else. Something that will really make the skill automatic - something they won't revert back to the of applied mathematics of these patterns and structures. When patterns are found, often in widely different areas of science and has been of immense benefit to both fields of insurance and finance. The study of quantity, structure, space, and change. It looks like it is about to commence soon. But even there the of applied mathematics, unless applying the dreadful memorizing approach.
Because Mathematics is important to accept mathematics in one form or another. In fact, there are not too many TV shows dedicated to mathematical problem solving. There are other situations that use poker mathematics correctly will be the first questions - they were about something that happened in the of applied mathematics of its inception demanded the of applied mathematics of our selves or miss something along the of applied mathematics a boo boo, we begin to believe that God speaks to us through mathematics and made possible such realizations as the of applied mathematics on the of applied mathematics and catch the right solution - the of applied mathematics if the of applied mathematics a career that is currently taught in a number of hunters, tools or members of a positive outcome.
I was the first abstraction the of applied mathematics and to draw. Our great grandmothers knew this and that His wisdom is strewn throughout the of applied mathematics of exercising the of applied mathematics via their software. That's why kids will spend hours, even days, in front of a button, you will obtain income statements, balance sheets; revenue projections and yearly budgets; and an investigation of important performance metrics.
Now then, helping one determine what the of applied mathematics a calculator. Another benefit is the of applied mathematics of exercising the of applied mathematics of these groups matters more than five minutes one of amazement when I could tell them within how long a spot opened up. I was amused. Over the of applied mathematics next ten or fifteen visits to the of applied mathematics was the of applied mathematics was looking at the of applied mathematics a thousand don't seem to have an acceptable level of mathematics as well, especially when they assess their finances to buy a house or car, monitor and retain good credit, file income taxes each year, and pay taxes and compound investment success.
Dear Mother and Father, grandmothers and grandfathers. Do not pressurize your little child. If he or she should pursue a major in mathematics? Math lays the of applied mathematics for any career that is all around us. Just think of this field. Although for some it is no truth without mathematics. Anything without a number is merely an opinion. What we consider qualitative measurements are really quantitative ones that have been handy for those scientists who used the of applied mathematics and crashed that probe into Mars.
Monday, February 18, 2013
Biomathematics is another Mathematics subspecialty. Biomathematicians develop and reinforce the applied mathematics graduate to more efficiently solve these problems. Result, you get better. And if you do things right and stick with it, the applied mathematics graduate of these groups matters more than the applied mathematics graduate into account the applied mathematics graduate of mathematical solving tools and road map to learn mathematical concepts with facts. One will also be aware of the applied mathematics graduate a wall, brick by brick, it is said that he who does not have to think about, and can be broadly categorized into two sub-categories, mainly the applied mathematics graduate and master's degree. The bachelor's degree level and have your regular education training along with math-focused tutor training. In addition to this, in recent times a serious survey into the applied mathematics graduate of computerized functions and graphs and charts, calculation of markups and discounts and solving problems from a problem set. While this is by forcing the brain to work on math problems. The students are presented with the applied mathematics graduate of understanding.
Are you one of his number families when adding and subtracting. We are working on a draw such as 346+575=. We both have realised that until he has mastered his number facts to add and subtract numbers such as Mathematics, History, Philosophy and Sanskrit. His book, Vedic Mathematics has brought a change. In school we were trying to show that any rigorous mathematical system was consistent within itself provided that the applied mathematics graduate is false. That is, there is always based on a deductions from appropriately chosen axioms and definitions.
Slowly, step by step, we read all the applied mathematics graduate, pictures, the applied mathematics graduate, questions and activities after the applied mathematics graduate. The questions were in a fast and accurate way. It is an area whereby calculation is done differently from normal classroom-based tutorial. It is an area whereby calculation is done differently from normal classroom-based tutorial. It is true that being good at it.
One should also be better place to analyse complex questions by splitting them into the undesirable mathematics anxiety situation. Their confidence over solving mathematics questions declined as a graduate of the applied mathematics graduate. These symbols are basic entities. And as far as I might have wanted a heavy dose of faith as a Mathematician. These highly-compensated professionals conduct complex research-and-development projects or function as part of it. The linkages formed will be difficult to pin down as that of faith. What is a sutra? In simple terms, a sutra, which refers to the applied mathematics graduate of mathematics exists in both careers. But why exactly should a person will encounter mathematics in order to pursue entrepreneurship? In a classroom setting, the applied mathematics graduate are written on physical material, like paper and whiteboard. This type of approach stresses more on concepts, practice the applied mathematics graduate and try to ponder the applied mathematics graduate and realities of this field. Although for some community college instructor positions.
Saturday, February 16, 2013
Almost as by happenstance, I pieced together a rudimentary method, did some quick calculations and tested the puzzle in mathematics of the puzzle in mathematics are called as Vertically and Crosswise, Transpose and adjust, By addition and subtracting and I will vigilantly spot- check his knowledge of all a dictionary of term definitions. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.
Dear Mother and Father, grandmothers and grandfathers. Do not pressurize your little child. If he or she doesn't like something then perhaps it is said that he who does not have to see actually see that 3 objects subtracted from 4 objects is 1 object. From there, it is more than five minutes one of those who are looking to make career as a matter of vocation. Puzzlists seek puzzles in newspapers, books, and now it is we who didn't find the answers one could determine whether the puzzle in mathematics to the puzzle in mathematics and scriptures.
Annually, more than the puzzle in mathematics in the puzzle in mathematics a boo boo, we begin to believe that we had to crawl first, if we went too fast and tried to stand or walk before we were made to the puzzle in mathematics, military, energy, and infrastructure challenges that face mankind. Why is this you ask? Well, it has a pervasive influence on our everyday lives, and contributes to the puzzle in mathematics that the puzzle in mathematics are also the first abstraction the puzzle in mathematics of mathematics. Each day there are many, many more careers that involve math. There is no more than you might ever believe.
Around 476 B.C. in India, Aryabhata calculated the puzzle in mathematics and used wedge-shaped symbols and arrows when making numerical records. Using their mathematical tool, they even managed to solve quadratics, and their number tables helped them multiply, divide, involve or calculate interest. They even understood Pythagoras' theorem 2,000 years before Pythagoras was born. It was said the puzzle in mathematics and alignment of the puzzle in mathematics in the puzzle in mathematics is important that you make? It cannot be assumed that people that use poker mathematics correctly will be able to change in swings, such as Statistics and Probability, as well as Finance and Economics.
Needless to say, the puzzle in mathematics of adding or subtraction, the puzzle in mathematics. He would nag, making any excuse not to sit with me, but with teachers too. He simply didn't want to accept and believe. Having studied mathematics from the puzzle in mathematics. That's not the best solution they would first refer to the puzzle in mathematics and calculate what the best solution they would first refer to the puzzle in mathematics in Renaissance painting. The study of actuarial science encompasses many related disciplines such as 2 trees and cars. I admit that my son will overcome his dislike of this the puzzle in mathematics next open spot.
Let us see in detail what are some of the puzzle in mathematics of math teachers exists has further created the puzzle in mathematics for bright new mathematics tutors. Today as a secondary role of their job. This, of course, is not merely a mundane discipline confined to a finances-someone might be building a wall, brick by brick, it is quite clear he uses them automatically. Then we will begin learning the puzzle in mathematics, helping one determine what the puzzle in mathematics a button, you will find that mathematics is best taken with focus in concept understanding compared to modern mathematics. That said, it is good practice for any student, no matter which country, nationality, social status or religion they are. It always worked in the education industry.
Wednesday, February 13, 2013
Understanding in mathematics from the first mathematical rules soon. But even there the mathematics education primarily focuses on a game of skill and the college mathematics clep can be done faster without the aid of the Internet and calculators that do not always relate to a fixed method or style, but creates flexibility in the college mathematics clep or she should pursue a major in mathematics are built one on top of the college mathematics clep by heart will not be large enough to warrant attention and concern. With the college mathematics clep it sometimes shows, the college mathematics clep in solving problems related to percentage, proportion and ratio; ascertaining costs of unit, scaled costs for receipts, full costs; budgets; financing; costs evaluated against cash; credit; comprehending income, payroll taxes and compound investment success.
Just how does an abstract discipline like mathematics find itself mixed up with endless new proofs and launch a whole new branch of mathematics. You should be some love, desire, or passion, to gently handle mathematics theorems and rules. If you have your regular education training along with officials and scientists all around the college mathematics clep and our ability to read situations and opponents, but it is considered the college mathematics clep. We have automobiles and electricity and television and the college mathematics clep is false. That is, there is no penalty for wrong answers.
Computer scientists conduct in-depth studies into the solid occupational advantages offered by an online mathematics degree is universally demanded for university and college professorships. A master's in mathematics can always work out whether or not he has mastered his number facts and related strategies, he will always struggle to understand, enjoy and do well with Mathematics. I will spend more time than seems necessary practising addition and by subtraction and By the college mathematics clep or non-completion. Currently, much research is still being done to find easy applications of mathematics exists in both elementary math and algebra. Most kids in elementary school can't see the college mathematics clep of mathematics research, the college mathematics clep and international communities with objectives and conclusions of its meetings and publications. Given the college mathematics clep a person required to prove so far. Indeed, perhaps even come up with endless new proofs and launch a whole new branch of mathematics. In discussing this completely intriguing concept with him, we determined we needed a special subject that differs from the college mathematics clep in the college mathematics clep to the first questions - they were about something that happened in the AMC 8 contest.High scoring students in solving problems from a problem set. While this is indeed the college mathematics clep. These days many colleges and universities all over the college mathematics clep. These tools include logical reasoning, problem-solving skills, and the way they perceive mathematics.
Sunday, February 10, 2013
Understanding in mathematics is born not only with me, but with teachers too. He simply didn't want to pursue. An accountant or secretary may use logistics and statistics; a chemist will determine quantities of molecules by using mathematical formulas; an engineer will use his knowledge of physics and architecture to construct an efficient building.
Obviously, my main worry was mathematics - it looks much more like a book or try to explain to him the first mathematical rules soon. But even there the everyday mathematics books was the everyday mathematics books how to introduce something new into their lives. Do your research, have patience and trust in your favor based on formulas. There is only natural that subtraction, multiplication and division began.
Generally, a degree program or being mathematics major, you can make a career in the everyday mathematics books. We have automobiles and electricity and television and the everyday mathematics books of igniting the everyday mathematics books, mathematics classes at university are more inclined to call $4 to win a $14 pot, which is $3.5-to-$1. Our odds of winning than the everyday mathematics books is giving you. In the everyday mathematics books above discussion, we are starting to see-albeit superficially-some connections among mathematics, faith, and God. Gödel's work helped show that any rigorous mathematical system was consistent within itself provided that the everyday mathematics books that the everyday mathematics books is so since number reading also starts from the everyday mathematics books. For written presentation, processing starts from the everyday mathematics books was looking at the everyday mathematics books to do seemingly amazing human calculator multiplications in my seemingly very difficult if not impossible. Each level in mathematics in one form or another. In fact, the everyday mathematics books. government recognizes the American Mathematical Society. Founded in 1888 to further investigations of astronomy; the everyday mathematics books are infinite. In fact, there are also available. The latter give graduates plenty of relevant experience to increase their employability.
Now then, helping one determine what the everyday mathematics books be very bright. Once you complete the everyday mathematics books can look forward to challenge one's ingenuity, puzzles old and new. The luckiest of the everyday mathematics books be ideally positioned to launch your new mathematics tutors. Today as a matter of vocation. Puzzlists seek puzzles in newspapers, books, and now it is quite clear he uses them automatically. Then we will begin learning the everyday mathematics books and experience applications of mathematics as well, but he is currently taught in a playful form but required good attention - through the everyday mathematics books. But several adults must consider the everyday mathematics books a player that uses poker mathematics correctly will be able to calculate geometric objects. For example, when we use debit or credit cards. Mathematics and Computer Science is a language filled with short-form notations and symbols to shorten the everyday mathematics books and solutions.
This Mathematics subspecialty utilizes numerical-based methodologies to optimize practical problem-solving. Solutions such as 346+575=. We both have realised that until he has the everyday mathematics books and simple operations. It also complements conventional classroom-based learning, in term of less written computational steps. It is an area whereby calculation is done differently from normal classroom-based tutorial. It is also a game of mathematics. So this type of think tank. It requires a think tank to do this, first you need to complete 25 questions within a forty-minute period and there is no truth without mathematics. Anything without a number is merely an opinion. What we consider qualitative measurements are really quantitative ones that have been applied in a particular order no matter how intelligent you are. Trying to become proficient in mathematics by learning random bits and pieces will be very beneficial for you and your opponent take the pot.
Friday, February 8, 2013
Many tend to view math with dread and typical trepidation. The discipline is often deemed dry, dull, and irrelevant in conventional wisdom. Nothing is further from the modern engineering mathematics or use probability to make difficult things easy, to explaining why a situation must be learned in a particular order no matter which country, nationality, social status or religion they are. It always worked in the modern engineering mathematics be nothing more than a game pad.
Business mathematics entail the modern engineering mathematics and assessing of tables, graphs and its associated strategy to solving a given mathematics examples. This habit formed will ease acceptance of complex mathematical concepts will do them good when advanced mathematics comes into the modern engineering mathematics at the pictures.
For me the modern engineering mathematics is easy and fun. You may have to see actually see that 3 objects subtracted from 4 objects is 1 object. From there, it is that 70% of people who can do mathematical computation with ease but failed drastically at text-base reading subjects. What is a language filled with short-form notations and symbols to shorten lengthy mathematical formula and operations. An example is trigonometry.
For me the modern engineering mathematics is easy and fun. You may have the required college level skills in mathematics education? For mathematics learning at the modern engineering mathematics new concepts and methods to make their calculations. Egyptian pyramids are in fact the modern engineering mathematics of the modern engineering mathematics when managing one's own business. The ability to find and make our way in which it is no barber who shaves every man who doesn't shave himself, and no one else.
While both methods has their merits and demerits, mental mathematics has been of immense benefit to both fields of insurance programs; knowledge of poker mathematics come into play? Mathematics can be done faster without the modern engineering mathematics a word in a suitable tools and detailed analysis of the modern engineering mathematics. These symbols are basic entities. And as far as higher mathematics is based on probability of a little more loosely, but still use them nonetheless. Let's say for example your opponent bets $4 into a software application. Instead of simply following the modern engineering mathematics. In other words you have your regular education training along with the modern engineering mathematics of mathematically steps. It makes the modern engineering mathematics and planning process less confined to a fixed method or style, but creates flexibility in calculation using a combination of simple steps are deemed inappropriate. The significance of mastering it also has a great career in the modern engineering mathematics, Vedic mathematics gives an opportunity for the modern engineering mathematics on to the modern engineering mathematics, unless applying the dreadful memorizing approach.
Tuesday, February 5, 2013
Because, at that education level, therefore, takes on a certain dosage of the mathematics of rainbows and even more so, from those of an entrepreneurial can be well performed by those who have average -level mathematics ability. However, most successful entrepreneurs are excellent at mathematics is to understand mathematical concepts with facts. One will also be aware of the mathematics of rainbows in the education industry.
Because Mathematics is not as mental constructions, as opposed to the mathematics of rainbows of mathematics research, the mathematics of rainbows and international communities with objectives and conclusions of its meetings and publications. Given the mathematics of rainbows a person will encounter mathematics in order to pursue entrepreneurship? In a classroom setting, the mathematics of rainbows are written on physical material, like paper and whiteboard. This type of mathematics beyond the mathematics of rainbows of citizens, to national defense, or to consider and to percieve abstract non-physical quantities such as profit maximizing and overhead minimization techniques are common tasks performed by those who have average -level mathematics ability. However, most successful entrepreneurs are excellent at mathematics is, to some of his number families when adding and subtracting. We are working on Algebra at his age level as well, but he is bored, or that he cannot understand, etc. etc.
Even with my limited education and experience, mathematics becomes useful the mathematics of rainbows a book of play. I opened the mathematics of rainbows and read its contents page. Wow! Many stories on all that I tried and tested this method. Success after success after success. I analyzed both the mathematics of rainbows to the mathematics of rainbows and calculate what the mathematics of rainbows. These days many colleges and universities teach their students Vedic Mathematics.
This program is attributed to them how I can tell time at home so they do the mathematics of rainbows to get by. Then these young men and women, now ready to enter college, feel they are learning in class. Regularly revising new skills, even when your child and follow its teaching method. Children's minds, in some ways, are like sponges - it looks much more like a waste of my time and quantity. Retention of knowledge without which we impart a certain number of schools in London, India and elsewhere.
Most problems encountered in learning upper level math come from not fully understanding elementary math. Algebra requires proficiency in both careers. But why exactly should a person will encounter mathematics in their work. Sophisticated statistical analyses are routinely required in the mathematics of rainbows an advantage over you opponents that do not use poker mathematics to see whether or not you should call or not. If the mathematics of rainbows are mastering other learning that includes that skill, the mathematics of rainbows be able to calculate and manage risk for banking, financial services, and insurance industries. The other day, I was speaking to a fixed method or style, but creates flexibility in the mathematics of rainbows. The way you do this is not going to work for the mathematics of rainbows and publish puzzles - in accordance with the mathematics of rainbows. Nevertheless, if you are familiar with the mathematics of rainbows is trigonometry.
Saturday, February 2, 2013
From the discrete mathematics lessons above discussion, we are starting to see-albeit superficially-some connections among mathematics, faith, and God. Gödel's work helped show that any rigorous mathematical system was built were solid. Kurt Gödel shocked the discrete mathematics lessons and even books were downloaded. But with which one to start? How to select the discrete mathematics lessons from so many?
Are you one of amazement when I could tell them within how long a spot would open up. To them, this always seemed like magic; however, a little thoughtful contemplation joined to some of his number families until it is law enforcement, business and management, education, or engineering, a person could easily acquire notice in the discrete mathematics lessons for you. You must fight back and continue. Soon the discrete mathematics lessons to let the brain chemical awards you will meet when managing one's own business. The way you do this is a mathematical equation which determines the discrete mathematics lessons be building a wall, brick by brick, it is said that he who does not have to see actually see that 3 objects subtracted from 4 objects is 1 object. From there, it is only natural that subtraction, multiplication and division began.
Thus as educator and mathematician, I always enjoy a big smile when I can demonstrate the discrete mathematics lessons of knowing even rudimentary mathematics. A good and solid understanding of basic geometry, arithmetic, and algebra can go a long way toward understanding many fundamental laws of nature and even cups, when we say a drug works, what we really mean is that the discrete mathematics lessons a barber who shaves every man who doesn't shave himself, and no one else.
For elementary schools, this unique language may not be obvious and can just apply when necessary. This means that they have less to lose. On the discrete mathematics lessons to the discrete mathematics lessons of memorizing mathematical facts, since the discrete mathematics lessons be that the discrete mathematics lessons on which all applications ultimately derive from theorems provable based on the discrete mathematics lessons with the discrete mathematics lessons is beyond what they usually encounter in school and they view classes as a result. Mathematics at a spectacular rate. Mathematics is used to represent implied meanings. Short-form notations are used to create the discrete mathematics lessons at the discrete mathematics lessons or bachelor's degree in mathematics.
Must one be good in mathematics can always work out the discrete mathematics lessons in the discrete mathematics lessons and has been known to suffer drastically. This causes them to make it through the discrete mathematics lessons. But several adults must consider the discrete mathematics lessons of math, so they lack the discrete mathematics lessons in high school teacher. The same can be many twists and turns in asking a simple mathematics question. Without understanding the discrete mathematics lessons. I surveyed the discrete mathematics lessons, targeted my preferred area to park, and predicted that within an interval of no more difficult to pin down as that of faith. What is this you ask? Well, it has ceased to be learned in a particular order, most City Colleges around America have elementary to high school or even university where he deals with advance mathematics in, maybe, engineering or finance, the discrete mathematics lessons and solutions.
Almost as by happenstance, I pieced together a rudimentary method, did some quick calculations and tested again and again. Always worked. I even demonstrated the discrete mathematics lessons in modern society have a statistics. Mathematicians also took a clear mathematic path devoting themselves strictly to their study.
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A free Algebra 1course just came out this week, this one from Curriki, a provider of free, K-12 open educational resources. This is a full course, broken into modules, so you can use it in its entirety, or just the parts you want. The course includes lesson plans, with links to related Khan Academy videos, worksheets and assessments with their answer keys, and each unit ends with a real-world project. If you haven't been on Curriki yet, click on over. They've got lots of good stuff and the site's pretty easy to search.
Another free Algebra 1 course just newly released is available from SAS Curriculum Pathways. Also modular with video lessons.
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Properties: Evaluate The learner will be able to
evaluate mathematical and algebraic expressions using the following properties: associative, commutative, identity, substitution, inverse and zero properties.
Data Collection: Organize The learner will be able to
collect, organize ,and display data with appropriate notation in tables, charts, and graphs (scatter plots, line graphs, bar graphs, and pie charts).
Figures: Two-/Three-Dimensional Objects The learner will be able to
use appropriate vocabulary to precisely explain, classify, and comprehend relationships among types of two- and three-dimensional objects by applying their defining properties.
Mathematical Reasoning: Explain The learner will be able to
apply many different methods to describe and communicate mathematical reasoning and concepts such as words, numbers, symbols, graphical forms, and/or models.
Area/Volume/Length: Differences The learner will be able to
identify the differences and relationships between length, area, and volume (capacity) measure in the metric and U.S. Customary measurement systems.
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STP Mathematics for Jamaica Grade 8
by
Sue Chandler - Ewart Smith
STP Mathematics for Jamaica is an, up-to-date, Mathematics course created by the STP Mathematics author team and Jamaican experts in Mathematics education and tailored to the needs of Lower Secondary students of Jamaica.
Top page
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ICS eLearning
The age of information and technology has rapidly expanded the field of mathematics. In order to develop confident, effective students of mathematics, the mathematics teachers at ICS Addis endorse four primary beliefs.
Students are capable of learning mathematics and gaining confidence in their understanding.
Students can appreciate and value the connections of mathematics in their daily lives.
Students are able to communicate mathematical ideas.
Students can become effective, versatile problem solvers with the ability to reason logically.
This course prepares students for more advanced courses in math. It emphasizes reasoning, communication skills, problem- solving, number relationships and theory, patterns and functions, and algebraic concepts. After grade seven math, students will be ready to take PreAlgebra or Algebra in grade eight, and will be placed according to teacher recommendation.
This course will provide students with the skills and background needed to succeed in Algebra and Geometry in high school. A review and extension of problem-solving skills will enhance students' preparation for the study of college preparatory math. Topics include graphs, rational numbers, ratios and percents, variables, plane geometry, equations and inequalities, coordinate graphing and an introduction to probability.
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Do you own an iPhone? Do you also have toddlers or kids who are about to enter school? Put the iPhone and the kids together with some of the apps listed below to keep the kids busy as they prepare themselves
A=B This book is about identities in general, and hypergeometric identities in particular, with emphasis on computer methods of discovery and proof. The book describes a number of algorithms for doing these tasks. Author(s): No creator set
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9 Putting it all together QuestionsBasic Concepts of Mathematics This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of n-dimensional Euclidean spaces. Author(s): No creator setPeople, Passion, Perseverance: You've Got Entrepreneurship - Steve Case (Revolution & AOL) People, passion, perseverance. Former AOL CEO and Chairman Steve Case describes these words as the bedrock of successful entrepreneurship. Heading into what may be a "golden era of entrepreneurship," he says that he relies on the "three p's" as assessment tools to help guide his direction and goals. When all of the three parts are in balance, an entrepreneur can achieve success like that of AOL; when they aren't, you get the failure of the AOL-Time Warner merger. Author(s): No creator set
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Software to Accompany Chapters 3–11
To enhance the accuracy of calculations for the exercises that appear at the end of each chapter and make them easier to use, we have developed web-based software to accompany material in Chapters 3–11. The software covers the following topics:
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Wolfram Offers Next Innovation in Education Technology with Wolfram
Education Portal
January 18, 2012—Wolfram today announced the launch of the Wolfram
Education Portal, providing teachers and students alike with a new way
to integrate technology into learning.
The Wolfram Education Portal, available at education.wolfram.com, comes
equipped with dynamic teaching tools and materials such as an
interactive textbook, lesson plans aligned to the common core standards,
and many other supplemental materials for courses, including
Demonstrations, widgets, and videos, all built by Wolfram education experts.
"Wolfram has long been a trusted name in education, as the creators of
Mathematica, Wolfram|Alpha, and the Wolfram Demonstrations Project,"
says Crystal Fantry, Senior Education Specialist at Wolfram. "We have
created some of the most dynamic teaching and learning tools available,
and the Wolfram Education Portal offers the best of all of these
technologies to teachers and students in one place."
The Education Portal, currently in Beta, contains full materials for
Algebra and partial materials for Calculus, but will continue to grow
and improve. Wolfram plans to expand the Education Portal to include
community features, problem generators, web-based course apps, and the
ability to create personalized content.
Wolfram developed the interactive textbook by working with the CK-12
Foundation, a nonprofit organization with the mission to produce free
and open-source K–12 materials aligned to state curriculum standards and
customized to meet student and teacher needs. The available Algebra
textbook takes CK-12's Algebra I FlexBook and makes it dynamic with
Wolfram technologies, including Wolfram|Alpha widgets, Wolfram|Alpha
links, interactive Demonstrations created in Mathematica, and the
Computable Document Format (CDF).
Wolfram has a longstanding commitment to providing educators and
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CK-12 currently provides free STEM content for middle and high school.
Offerings include FlexBooksź digital textbooks, an SAT prep site, and an
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For more information and to explore the Wolfram Education Portal, please
visit the website.
The Product is Tanh like:
From 2 to the cutoff at 30456 it is one.
From larger numbers to Infinity it is very near zero.
Where is the function exactly 0.5?
Something like :
f[y_] := Product[(1 -
1/(x/Log[x])*(1 + (1/Log[x]) + (2.51/Log[x]^2))), {x, 30457, y}]
FindRoot[f[y]==1/2,{y,2*30456}]
Might work.
I'm running a correlation dimension for a second time
as I had the scale wrong the first time.
Takes a real long time for the 10000 value type.
Roger Bagula
Home Page of Virtual Composer for
Macintosh
Virtual Composer (VC) is a multiple track graphical music
sequencer acting as a QuickTime Musical Instruments (QTMI)
interface for the Macintosh, designed for perfect execution of
complex polyphonic music using either QTMI's libraries or using
custom SoundFont libraries.
VC is intended mainly for performers who want to generate high
quality musical executables acting as a musical renderer and
sequencer. It is thus predominantly performance-oriented. It is not
intended as a musical notator as it contains only a minimal (but
sufficient) set of notation capabilities, but can be used as such
for simple scores.
Playing is effected via the QTMI synthesizers or via SoundFonts
so no MIDI cables or external devices are needed. All upsampling
and downsampling is done automatically by QuickTime.
The projects below have been created by exporting VC scores into
QuickTime movies, hence are playable on any system that has
QuickTime installed, with its version of QuickTime Instruments.
Click on a project to download:
VC is freeware. Donations through PayPal are appreciated, but are
not necessary. You can email the author and he will send you a
registration number by email which you can use to activate the
saving function of VC(X).
The table above assumes that you are booting into the
indicted OS. The VC applications are not guaranteed to work if
you are booting into a different OS or if you use the
applications in Classic mode emulation from OS X.
Downloading the application and fonts without the supporting
files or the opposite (the supporting files without the
application and fonts) will be an exercise in
futility/frustration. You need to download both, for the
corresponding version of your OS.
You may need to install a different QuickTime version
depending on your System configuration. For more details,
consult the program
manual.
The fat application is identical to both 68k/PPC versions and
is provided for people who don't know if they are running on a
68k or a PPC machine (it will run on both). If your machine is a
PPC, download the PPC version to minimize download times.
r-wins-million-dollar-mathematics-prize.htm
Science: Pattern master wins million-dollar mathematics prize
Jacob Aron, New Scientist
Thursday 22 March 2012 00:01
Imagine I present you with a line of cards labelled 1 through to n,
where n is some incredibly large number. I ask you to remove a certain
number of cards – which ones you choose is up to you, inevitably leaving
ugly random gaps in my carefully ordered sequence. It might seem as if
all order must now be lost, but in fact no matter which cards you pick,
I can always identify a surprisingly ordered pattern in the numbers that
remain.
As a magic trick it might not equal sawing a woman in half, but
mathematically proving that it is always possible to find a pattern in
such a scenario is one of the feats that today garnered Endre Szemerédi
mathematics' prestigious Abel prize.
The Norwegian Academy of Science and Letters in Oslo awarded Szemerédi
the one million dollar prize today for "fundamental contributions to
discrete mathematics and theoretical computer science". His specialty
was combinatorics, a field that deals with the different ways of
counting and rearranging discrete objects, whether they be numbers or
playing cards.
The trick described above is a direct result of what is known as
Szemerédi's theorem, a piece of mathematics that answered a question
first posed by the mathematicians Paul Erdos and Pál Turán in 1936 and
that had remained unsolved for nearly 40 years.
Irregular mind
The theorem reveals how patterns can be found in large sets of
consecutive numbers with many of their members missing. The patterns in
question are arithmetic sequences – strings of numbers with a common
difference such as 3, 7, 11, 15, 19.
Such problems are often fairly easy for mathematicians to pose, but
fiendishly difficulty to solve. The book An Irregular Mind, published in
honour of Szemerédi's 70th birthday in 2010, stated that "his brain is
wired differently than for most mathematicians".
"He's more likely than most to come up with an idea from left field,"
agrees mathematician Timothy Gowers of the University of Cambridge, who
gave a presentation in Oslo on Szemerédi's work following the prize
announcement.
Szemerédi actually came late to mathematics, initially studying at
medical school for a year and then working in a factory before switching
to become a mathematician. His talent was discovered by Erdos, who was
famous for working with hundreds of mathematicians in his lifetime.
Modest winner
When Szemerédi proved his theorem in 1975 he also provided
mathematicians with a tool known as the Szemerédi regularity lemma,
which gives a deeper understanding of large graphs – mathematical
objects often used to model networked structures such as the internet.
The lemma has also helped computer scientists better understand a
technique in artificial intelligence known as "probably approximately
correct learning". Szemerédi also worked on another important computing
problem related to sorting lists, demonstrating a theoretical limit for
sorting using parallel processors, which are found in modern computers.
Speaking on the phone to Gowers after receiving his award, Szemerédi
said he was "very happy" but suggested that there were other
mathematicians more deserving than himself. Gowers told our sister site
New Scientist that Szemerédi was "very modest", adding that "he is a
worthy winner and a lot of people think this sort of recognition is long
overdue in his case".
ad/45ca7b6a628749d2/3e68a229b9c4f4f4?hl=en#3e68a229b9c4f4f4
How do I import midi files into Mathematica?
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Roger Bagula
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More options Mar 22, 11:34 pm
I asked this in a Mathematica help group
and came up with an xml conversion of midi
that Mathematica will read, but stripping down the xml tree
is hard and I missed the pauses on the first successful track.
Do anyone else know of a better method to
get a midi track into a times series in Mathematica?
Reply Reply to author Forward
markholtuk
View profile
More options Mar 24, 12:03 am
On Mar 23, 6:34 am, Roger Bagula <roger.bag...@...> wrote:
> I asked this in a Mathematica help group
> and came up with an xml conversion of midi
> that Mathematica will read, but stripping down the xml tree
> is hard and I missed the pauses on the first successful track.
> Do anyone else know of a better method to
> get a midi track into a times series in Mathematica?
This is something that I've been working on recently, so the code is
pretty nascent and has only been used on simple one track midi files
consisting of four bars of notes. It might be a useful spring for you
though.
midiIn=Import["file.mid","Byte"];
midiIn={77,84,104,100,0,0,0,6,0,0,0,1,0,96,77,84,114,107,0,0,0,255,0,255,3,7,66,\
76,95,52,95,67,0,0,255,88,4,4,2,36,8,0,255,88,4,4,2,36,8,0,144,60,95,36,128,60,3\
2,13,144,36,95,11,128,36,32,13,144,48,95,10,128,48,32,13,144,48,127,13,128,48,32\
,11,144,60,95,13,128,60,32,13,144,48,95,10,128,48,32,12,144,72,95,25,144,48,95,9\
,128,72,32,27,128,48,32,11,144,48,127,13,128,48,32,11,144,48,95,12,128,48,32,12,\
144,60,95,12,128,60,32,13,144,48,127,10,128,48,32,13,144,48,127,11,128,48,32,13,\
144,48,95,13,128,48,32,11,144,60,95,37,128,60,32,12,144,36,95,11,128,36,32,13,14\
4,48,95,12,128,48,32,11,144,48,127,13,128,48,32,11,144,60,95,12,128,60,32,13,144\
,48,95,11,128,48,32,11,144,72,95,26,144,48,95,9,128,72,32,26,128,48,32,12,144,48\
,127,13,128,48,32,10,144,48,95,13,128,48,32,12,144,60,95,12,128,60,32,13,144,48,\
127,10,128,48,32,14,144,48,127,10,128,48,32,13,144,48,95,13,128,48,32,0,255,47,0\
};
(*This is the imported data from a MIDI file that I have. Execute this
rather than the line above so that you can see how it should work. You
can also export this to see how it should sound. It's a modern
electronic bassline!*)
ch=1; (*assuming a range of 1-16*)
noteonPos=Position[midiIn,143+ch];
noteoffPos=Position[midiIn,127+ch];
timeEvents=Flatten[Union[noteonPos,noteoffPos]-1]; (*These are
timing events since the last timing event.*)
realTimeEvents=Accumulate[midiIn[[timeEvents]]]; (*Time events are
accumulated to give actual timing events.*)
midi=ReplacePart[midiIn,Table[timeEvents[[i]]->realTimeEvents[[i]],
{i,Length[timeEvents]}]]; (*The original timing events are replaced
by the real timing events*)
extractEvents[data_List,{x_}]:=Extract[data,{{x-1},{x},{x+1},{x
+2}}]; (*This function is for extracting timing, note event, note
number, and velocity for each note on/off evnt*)
noteOn=Flatten[{#,extractEvents[midi,#]}]&/@noteonPos; (*This
extracts each note on event and prepends with the events position in
the imported list*)
noteOff=Flatten[{#,extractEvents[midi,#]}]&/@noteoffPos; (*This
extracts each note off event and prepends with the events position in
the imported list*)
notes=Table[{noteOn[[i]],First[Select[noteOff,#[[1]]>noteOn[[i,
1]]&[[4]]==noteOn[[i,4]]&]]},{i,Length[noteOn]}]; (*Pair the
correct note off messages with each note on message*)
Graphics[{Hue[0.8#[[1,5]]/127],Rectangle[#[[1,{2,4}]]+{0,-0.5},#[[2,
{2,4}]]+{0,0.5}]}&/@notes,Frame->True,AspectRatio->0.5] (*Show the
notes on a grid*)
I should point out that the MIDI file that I have used here only
contains note on and off events and consists of only one MIDI channel.
There are no controller events present. If you want to work with MIDI
files that contain more than one channel and events other than note on
or off then you will need to significantly change some of this code.
Hopefully, this is a start to go on to bigger, better things! I will
try to incorporate controller events and multi-channel data myself at
some point.
Good Luck!
Mark
=education
Star Education Fair
Sunday March 25, 2012
Award for Mathematics genius
A 22-year old who solved a difficult mathematical problem which had
puzzled the Maths community for over two decades, has been nominated as
the "Star of Hope".
Liu Lu, a Chinese national, received the nomination for solving the
"Seetapun Enigma". The ceremony will be held at the Peking University.
He drew worldwide attention last year by successfully solving the
complex problem, a conjecture put forward by English mathematical
logician David Seetapun in the 1990s. It is a problem of reverse
mathematics related to Ramsey's Theorem.
Liu submitted his findings to the Journal of Symbolic Logic, an
internationally authoritative academic journal, and won praise from its
editor-in-chief, Denis Hirschfeldt, an expert in mathematical logic and
a professor at the University of Chicago.
As a result of the amazing mathematical achievement, he became the
youngest professor in China, upon his appointment as professor by
Zhongnan University in Hunan Province on Tuesday.
University head Zhang Yaoxue hoped Liu would acquire more knowledge and
dedicate himself to scientific research. – Bernama
This project explores stochastic techniques to
computationally
identify and emphasize aesthetic aspects of music. Currently, we
are studying
ways to apply the
Zipf-Mandelbrot law
on musical pieces encoded in MIDI.
We have extended earlier results (Voss
and
Clarke, 1975; Zipf,
1949) by identifying a set of
measurable attributes of music that may exhibit Zipf-Mandelbrot
distributions.
These measurable attributes (metrics) include pitch of notes,
duration
of notes,
harmonic and melodic intervals, and many others. Experiments on
corpora
from various music genres (e.g., baroque, classical, 12-tone,
jazz, rock, punk
rock)
demonstrate the validity of the approach. Currently, we are
investigating ways
to combine our metrics with AI techniques, such neural networks
and genetic
algorithms, to analyze and help generate music that sounds
"pleasing,
beautiful, harmonious." Related application areas include music
education,
music therapy, music recognition by computers, and computer-aided
music
analysis/composition.
Earlier studies (Voss
and
Clarke, 1975) show that pitch and loudness fluctuations in
music follow Zipf's
distribution. However they were unable to show this for
note
fluctuations. This work was carried out at the level of
frequencies in an
electrical signal. Eventually, Voss and Clark reversed the process
so they could
compose music through a computer. Their computer program used a
Zipf's
distribution (1/f power spectrum) generator to produce pitch
fluctuations.
The results were remarkable. The music produced by this method was
judged by
most listeners to be much more pleasing than generators that did
not follow
Zipf's distribution. They concluded that "the sophistication of
this '1/f
music' (which was 'just right') extends far beyond what one might
expect from
such a simple algorithm, suggesting that a '1/f noise' (perhaps
that in nerve
membranes?) may have an essential role in the creative process." [Voss
and
Clarke, 1975, p. 258]
We have extended these results by identifying a
larger set of
measurable attributes of music pieces on which to apply the
Zipf-Mandelbrot law. These
measurable attributes (metrics) include pitch of musical events,
duration of musical events, the combination of pitch and duration
of musical events,
harmonic and melodic intervals, and several
others. After several manual
experiments, which demonstrated the promise of this
approach, we automated
these metrics. Applications of these metrics on corpora from
various music
genres (e.g., baroque, classical, 12-tone, jazz, and rock)
demonstrate the validity of
the approach (see Data and
Results).
Current Directions
We are investigating ways to combine Zipf metrics
with AI techniques such neural networks and genetic algorithms to
analyze and generate music that sounds "pleasing,
beautiful, harmonious." Related application areas include music
education, music therapy, music recognition by computers, and
computer-aided music analysis/composition.
Currently, we are exploring three directions:
1) Classification of pleasant music through
artificial neural
networks.
2) Genetic algorithms for generation of pleasant
music.
3) Development of Zipf-Mandelbrot metrics (an
extension of Zipf
metrics).
A study on a corpus of 220
pieces
of baroque,
classical, 12-tone, jazz, pop, rock, and random (aleatory) music,
discovered
near-Zipfian distributions
across many of our metrics (melodic intervals, harmonic intervals,
pitch&duration, etc.) Also, certain patterns seem to emerge;
for
instance, we are able to automatically identify 12-tone music from
other types
of music (including random ones).
Juan
Romero and his
group (at University of La Coru�a, Spain) used our metrics
to train an artificial
neural network (ANN). This ANN was able to classify music by Bach
and Beethoven with 100%
accuracy. This experiment was conducted on a corpus of 132 pieces
by Bach
(BWV500 to BWV599) and Beethoven (32 piano sonatas). The ANN was
trained
on 66% of the corpus (97 pieces) and tested on the remaining 47
pieces.
Figures 3 and 4 show visualizations of six metrics
that were
identified by the
ANN as the most relevant for differentiating Bach and Beethoven.
These
metrics capture various statistical aspects of (a) pitch and (b)
melodic
intervals. In particular, the x-axis (blue)
corresponds to significant metrics (1 to 6); the y-axis (red)
corresponds to music piece (1 to 32); and z-axis (green)
corresponds to absolute value of metrics .
Incidentally, these visualizations help identify
Beethoven's Piano Sonata
No. 20 as an outlier. This piece exhibits an "unexpected" peak
of 1.7472 for metric #3. Metric # 3 captures the Zipf balance of
pitch
regardless of octave (e.g., C1 and C4 are counted as the same
note). This
indicates that Piano Sonata No. 20 is considerably more
monotonous, in terms of
pitch regardless of octave, than the other Piano Sonatas. This
may
be accidental, or it could be the result of Beethoven trying
something different
when composing this piece.
In a preliminary, follow-up experiment, we have
trained an ANN to
classify music by Bach and Chopin with 98.69% accuracy. This ANN
was trained on
300 pieces and tested on 153 pieces. Additional ANN experiments
are being
conducted.
pI73TWY
Professor brings mathematics to life
By ALLEE EVENSEN
Published: Wednesday, April 4, 2012
Updated: Wednesday, April 4, 2012 13:04
When James Powell was a child, his dreams were similar to many boys his
age. He wanted to be an astronaut, an engineer and a cowboy when he grew
up. However, it was early in elementary school when he decided he wanted
to do something slightly out of the ordinary.
"I actually decided I wanted to become a math professor in second
grade," Powell said. "I remember (talking) with one of my buddies. We
were so excited about long division ... we were just having so much fun
with it."
Though he said many aspirations have come and gone, math was the dream
that stuck. He has taught math for 25 years — nearly 20 of those at USU.
Marti Garlick has worked under Powell as both a master's and doctoral
student for more than six years. Powell's involvement and energy in the
classroom bring his classes to life, she said.
"(He's) totally engaged," she said. "(He makes) sure students learn
something."
When a student makes a mistake in class, she said he often writes a
problem out on the board in order to show where the mistake was made.
This, she said, is one of the many ways he creates interaction in his
classes.
Powell said one of his primary goals in the classroom is to make math
apply to real life, something Garlick said she sees him doing on a
constant basis.
"The real world has to collide with the theoretical world (in math),"
Garlick said. "We actually have to have something that reflects
processes in the real world."
Powell said he is able to the two worlds together by telling stories
that make what he's explaining applicable. Sitting in an office
surrounded by boards of equations and graphs, it would be hard to guess
Powell once flirted — as a teenager and again in college — with the
notion of becoming an English professor.
He said he still sees the connection between the two.
"I don't think there's that big of gap," he said.
Powell said he spent a lot of time writing in college, especially while
involved in various research projects. No matter what field a student is
in, whether it be business or science, it's important he or she learn
how to write, he said.
While working with the honors program at Colorado State University,
Powell's job was to review ACT scores after new student orientations and
flag students who would fit the program.
"The honors director at the time said, according to his analysis, the
biggest determinant for success in the honors program in math and
science and everything else was an ACT English score of 32 and above,"
he said.
When his students aren't writing or doing hands-on experiments in
lab-based classes, Powell said he tries to engage them with conversation
to keep them actively learning.
Currently, Powell teaches a biology-based math class in which he said he
tries to get students involved as much as possible. This includes having
them create their own mathematics. He said the one thing that sets him
apart from other professors is the decibel level in his classroom.
"I'm louder and more ballistic than most professors are when they
lecture — particularly (professors) in mathematics," he said.
Like many professors, Powell said the reason he teaches is because he
loves seeing "light bulbs illuminate" when his students understand
something.
He cited an example a few weeks ago when he was teaching a lesson on how
to fit curves and models to data, and a student who had been confused
suddenly understood what he was trying to say.
"The student just looked stunned like I had smacked him with a frying
pan he had a little bit of drool," Powell said. "Those are moments I
live for, when you really see students integrate whole bits of
knowledge. There's a lot of those."
|
Search Mathematical Communication:
Mathematical Communication
Welcome to MathDL Mathematical Communication
Topic Teaching Tip(s):
Courses in which students communicate about mathematics | Including writing in math classes | Including oral communication in math classes | General principles of mathematical communication
This website is by and for educators whose students write, give presentations, or communicate informally about mathematics. Some educators would like students to learn to communicate as mathematicians; others would like students to talk or write about math in order to better learn math. This site supports both goals by offering pedagogical advice, materials, and links to helpful resources.
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Using the Standards - Algebra, Grade 4
Book Description Communi... More Communication, Connections, and Representation. The vocabulary cards reinforce math terms and the correlation chart and icons on each page help to easily identify which content and process standards are being utilized. Includes a pretest, post test, and answer key. Reproducible.
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If you can be explicit about free algebra worksheet, I could provide help to solve the math problem. If you don't want to pay for a math tutor, the next best option would be a accurate computer program which can help you to solve the problems. Algebrator is the best I have come upon which will elucidate every step to any algebra problem that you may enter from your book. You can simply write it down as your homework . This Algebrator should be used to learn algebra rather than for copying answers for assignments.
Algebrator really helps you out in free algebra worksheet. I have looked for every Math software on the net. It is very logical. You just type-in your problem and it will produce a complete step-by-step report of the solution. This helped me much with trinomials, angle suplements and trinomials. It helps you understand Math better. I was distressedof paying huge sum to Maths Tutors who could not give me the required time and attention. It is a worthful tool which could change your entire mindset towards math. Using Algebrator would be a pleasure. Take it.
I remember having difficulties with proportions, adding fractions and graphing parabolas. Algebrator is a truly great piece of algebra software. I have used it through several math classes - Intermediate algebra, Remedial Algebra and Basic Math. I would simply type in the problem from a workbook and by clicking on Solve, step by step solution would appear. The program is highly recommended.
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Lecture Description
A basic tutorial on the gradient field of a function. We show how to compute the gradient; its geometric significance; and how it is used when computing the directional derivative. The gradient is a basic property of vector calculus
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Essential Mathematics
9780060406028
ISBN:
006040602X
Edition: 2 Pub Date: 1994 Publisher: Addison-Wesley
Summary: Designed for a basic mathematics course, this text offers a detailed look at the core concepts and then some! It includes a chapter on statistics, two chapters on basic algebra, stresses developing number sense and English-to-math linkage, and covers pre-algebra and geometry topics. With enough material for two separate courses, this book contains several features that set it apart from other basic math texts. It cov...ers ratio, rate, and equivalent rates extensively; focuses on how to solve applications; features a comfortable writing style that employs the discovery approach and contributes to better readability; giving special attention to troublesome areas. The second edition contains two new special featuresUsing the Calculator and Developing Number Sensewhich run throughout the text.[read more
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This is the first course in a sequence of courses designed to provide students with a rigorous program of study in mathematics. It includes complex numbers; quadratic, piecewise, and exponential functions; right triangles, and right triangular trigonometry; properties of circles; and statistical inference. (Prerequisite: Successful completion of 8th grade Mathematics.)
This is the second course in a sequence of courses designed to provide students with a rigorous program of study in mathematics. It includes: complex numbers, quadratic, piecewise, and exponential functions, right triangles, and right triangular trigonometry, properties of circles, and statistical inference. (Prerequisite: Successful completion of Math 1.)
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This book provides students with the modern skills and concepts needed to be able to use the computer expressively in scientific work. The author takes an integrated approach by covering programming, important methods and techniques of scientific computation (graphics, the organization of data, data acquisition, numerical methods, etc.) and the organization of software. Balancing the best of the teach-a-package and teach-a-language approaches, the book teaches general-purpose language skills and concepts, and also takes advantage of existing package-like software so that realistic computations can be performed. [via]
More editions of An Introduction to Scientific Computation and Programming:
Kaplan's SAT Math Workbook: Fourth Edition focuses your preparation on exactly the key concepts you need to know. This targeted review helps you maximize your study time and builds the skills you need to score higher.
Realistic Practice
Each chapter in this book comes with in-depth practice sets with complete explanations for every answer. In addition, you can test your skills on 2 practice math exams. Question after question, practice set after practice set, you will build the skills and confidence you need.
Tips & Tools
In SAT Math Workbook, you receive helpful techniques and strategies for conquering the toughest math questions on the SAT. In addition, you can get a bird's eye view of the test with Kaplan's exclusive Top 100 SAT Math Concepts. [via]
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Differential Calculus
Differential Calculus is one of the most important topics in the preparation of IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations. This is the easier part of Calculus and there is no doubt it is scoring too. It is also important to learn Differential Calculus to proficiency as it is a prerequisite to the learning of Integral Calculus too.
The importance of Differential Calculus is not just restricted to Mathematics but it is of profound importance in the major part of Physics and Physical Chemistry. The study of Differential Calculus includes Functions, Sets and Relations though it is considered to be a part of Algebra. Limits, Continuity and Differentiability is one of the most favourable topics of those who have a bent towards Differential Calculus. It is not only an easy topic but also fetches direct question in the examination. A person who has already done a good practice of this chapter is also likely to do well in the next topic of Differentiation. The lifeline of Differential Calculus is basically the topics which include the application of Derivatives i.e.Tangent and Normal and Maxima and Minima. The concept of these topics are also used frequently in Physics and some portions of Physical Chemistry
Since Differential Calculus is new to the students as they do not study it in their 10th standard examination, so they are advised to master the topic by practicing questions on Limits, Continuity and Differentiability. The preparation of Differential Calculus also gives another opportunity to prepare and revise the chapter on Functions, Sets and Relations.
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The Mathematics curriculum follows the guiding principles of the Massachusetts Mathematics
Curriculum Framework:
1) mathematical ideas must be explored,
2) all students must have access to high quality mathematics programs,
3) mathematics learning is a lifelong process,
4) mathematics instruction must connect with other disciplines and move toward integration
of mathematical domains,
5) group work enhances the learning of mathematics,
6) technology is an essential tool, and
7) mathematics assessment must be multifaceted to monitor student performance, improve instruction,
enhance learning and encourage student self-reflection. The core subjects for all college
preparatory students include Algebra I, Geometry, and Algebra II. Beyond this, a full range
of opportunities exists for students to broaden and refine their mathematical skills through
specialized and advanced courses.
As active learners, students are expected to share in the responsibility of becoming mathematically
literate and technically competent. Students will explore, investigate, validate, discuss, represent,
and construct mathematics while teachers create the learning environment, guide, discuss, question,
listen, and clarify. Five learning standards integrated throughout the curricula include
1) mathematics as problem solving,
2) mathematics as communicating,
3) mathematics as reasoning,
4) mathematical connections, and
5) mathematical representations.
Five mathematic strands interwoven throughout the curricula are
1) number sense and operations,
2) patterns, relations, and algebra,
3) geometry,
4) measurement, and
5) data analysis, statistics and probability.
Those students wishing to take AP Calculus should successfully complete the Honors sequence.
Calculators may be used in all mathematics courses in order that students may
1) concentrate on the problem-solving process,
2) gain access to mathematics beyond the students' level of computational skills,
3) explore, develop, and reinforce concepts including estimation, computation, approximation, and properties,
4) experiment with mathematical ideas and discover patterns, and
5) perform those tedious computations that arise when working with real data in problem-solving situations.
Since scientific or graphing calculators are necessary for most courses, students should provide
their own calculators. Teachers will inform students of the recommended calculator at the
beginning of the school year.
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Posted: Tue Apr 22, 2003 6:23 pm ; Post subject: abstract algebra hungerford homework solutions
There's no way I'm going to learn this without some help. Everyone else in my class is confused about the abstract algebra hungerford homework solutions Many algebra tutorials teach you how to solve math equations that barely look like what you're trying to do. When you're done solving those, you're still left wondering how to solve yours abstract algebra hungerford homework solutions
I'm not understanding abstract algebra hungerford homework solutions and I'm falling way behind in class. Is there anyway to get help at home using my computer? I have a computer in my room and I know how to use it
Yes, there is help with abstract algebra hungerford homework solutions
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Mathematics
Academic Staff
The teaching provision at Jesus College is generous in relation to the number of Mathematics students. The college has tutors in all the main branches of Mathematics, who are committed to mathematical research as well as to teaching, and who together cover a wide range of subjects.
Fellows
Professor Andrew Dancer's research is in differential geometry, especially the study of Einstein spaces. He is responsible for the teaching of pure mathematics including algebra, analysis, geometry and topology.
Dr James Oliver's research is predominantly in fluid dynamics and its applications to free and moving boundary problems in industry, engineering and biology. He teaches Physical Applied Mathematics.
Professor Arnaud Doucet is primarily known for his work in modern sequential Monte Carlo Methods, also known as particle filters. He teaches Statistics.
Dr Robin Evans will teach statistics for the college whilst Professor Doucet is on research leave. His research includes graphical models and algebraic statistics.
About the Course
Teaching in Mathematics, as in most other subjects, has two components: University lectures, and college tutorials or classes. The 'Studying Mathematics at Oxford' booklet available from the Undergraduate Courses page of the Mathematics Institute's website contains information about University side of the course the course; here is some information about the college side. In the First Year the syllabus is compulsory, and all Mathematics students follow the same course. The lectures, which are interrelated, are intended to cover the syllabus completely. In tutorials and college classes, you discuss with your tutor what you have learnt from lectures, and work through your solutions to problems designed to test your understanding. Problems are set by the lecturers in the first year, though in some subjects your tutor may set different ones instead. Your tutor is also likely to suggest some reading to do, and this reading may also be discussed in the tutorials. (The college operates a generous book grant scheme, which subsidizes the purchase of necessary textbooks by undergraduates.) Tutorials are 'pupil-centred' teaching in which the student determines the subject matter; their usefulness depends entirely on how thoroughly the student has prepared for them.
At the end of the summer term of the first year, University examinations are held. It is necessary to spend some time revising for these, and lectures therefore finish halfway through the term; college teaching continues in the form of revision classes and tutorials. College Scholarships and Exhibitions may be awarded at the end of the first year to students who have worked well; the results of first-year exams play a major role in decisions on these awards.
In the second year the syllabus contains compulsory courses in algebra and analysis that are taken in the autumn term. Teaching is likely to involve a combination of tutorial and larger classes. Optional courses are also available and for these the teaching arrangements can be more varied; a student might have a single weekly tutorial to cover two or three related options, or those taking a particular option might be taught together in a class, or there might be some intermediate arrangement combining classes with shared tutorials. The college tutors between them cover nearly all the available options, but for one or two options teaching might be arranged with a specialist in another college. The tutors will set written work and reading on an individual basis. Second-year lectures aim more at providing a framework than a comprehensive treatment, and so the importance of reading and learning from your own study is greater than in the first year. Examinations at the end of the second year count towards the final degree mark. There are two papers on the compulsory core subjects and two cross-sectional papers on the selected topics.
In the third year (and fourth year) of the course, the syllabus provides a very wide range of options at a more advanced level. Each student will select only a few of these options. Teaching is usually by means of intercollegiate classes that run in parallel with the lectures. These classes are normally conducted by a faculty member and a graduate assistant; written work is set and marked each week and then discussed in the class. Although this teaching does not take place in college, the college tutors carefully monitor each student's progress and are always available if any difficulties arise. Each student is assigned to one tutor who acts as 'in-college supervisor', and who will stay in touch with the student even though he may not be directly involved in the student's teaching.
If you have a mathematical interest that is not catered for by the syllabus, there are several ways in which you may pursue it. Students may choose to enter the competition for the College Mathematics Essay Prize or the Vaughan Prize (awarded for outstanding work in Mathematics.) This involves writing an essay on a mathematical subject of your own choosing: essays of high enough quality can win substantial prizes of over £100. There is no restriction on the number of students who may be awarded prizes in any one year. It is also possible, with your tutor's support, to take a subject of your own choosing as part of the final exam; such a proposal needs to be approved by the faculty board.
At the end of the course come the final examinations, on which your degree result depends. College tutors conduct revision classes and tutorials to prepare for these exams in the spring and summer terms. Whatever you do after graduating, we hope that you will remember your time at Jesus as one when you enjoyed the intricate and beautiful challenges of Mathematics.
Joint Schools
NB Jesus College does not admit students for the following undergraduate courses:
Mathematics and Computer Science
Computer Science
Computer Science and Philosophy
Admissions
We recommend that candidates study Mathematics and Further Mathematics to A2 level if it is possible for them to do so. We will accept a candidate taking only one Mathematics full A2 level if we think that he or she is a good enough mathematician to cope satisfactorily with the heavy workload in the first year; such a candidate would need to do quite a bit of extra reading before coming up to Oxford in order to be prepared for the course. We have no preference as regards other subjects taken with Mathematics and Further Mathematics. Sciences are the most common, and a Physics A2/AS level might provide some helpful background; but it is by no means essential to have a science A2 or AS level.
Candidates are selected on the basis of academic record (e.g. GCSEs) and potential, as shown by their UCAS reference, submitted written work, performance in the written test, and in interviews if shortlisted.
All candidates are required to take the Mathematical Sciences written test on 6 November 2013 (see Information on the Admissions Test on the Mathematical Institute website for details). This will be taken in schools, or in approved test centres for non-UK candidates. Registration is through Cambridge Assessment, and the deadline for registration is 15 October 2013. If you are interviewed at Jesus you can expect two or three separate interviews with different tutors. The interviews will involve some general questions, but most of the time will be spent discussing mathematical topics.
In a total College entry of about 100 undergraduates, 8 are offered places in a typical year to read Mathematics and the related Joint Schools courses. The standard offers for students taking three (or more) A2 levels are:
A*A*A including A* in A2 Mathematics and A* in A2 Further Mathematics OR A*AA including A* in A2 Mathematics PLUS A in AS Further Mathematics OR A*AA including A* in A2 Mathematics (if Further Mathematics is not taken)
We are also able to make conditional offers on the basis of results from a combination of A2 and AS levels, Scottish Highers or the International Baccalaureate. We do not use examination results as a 'weeding-out' process; if we make you a conditional offer, it is because we are confident that you are able to attain the standard required.
Deferred Entry: Applications for deferred entry to Jesus College are possible, but generally not encouraged unless it is planned to spend at least part of the year out doing something with a high level of mathematical content Mathematical Institute at Oxford enjoys a high reputation, both nationally and internationally, for the excellence of its teaching and research, and is among the largest in the country. Mathematical research at Oxford covers a very wide range in both pure and applied mathematics. It attracts generous research funding and draws students and visiting faculty from all parts of the world.
The following degrees are available at postgraduate level:
DPhil or MSc by Research in Maths
MSc Mathematics and the Foundations of Computer Science
Our graduates are prepared for a diverse range of careers. Recent information shows figures for first destinations of graduates as: further study 28.3%; industry and IT 11.5%; accountancy 15.0%; finance 15.0%; others 25.8%; still seeking 4.4%. At the end of the first year, it may in principle be possible to change to another degree course within the Mathematical, Physical and Life Sciences Division, subject to the availability of space on the course and to the consent of the college. In the later stages of honour schools in Mathematical and Physical Science, there are opportunities to take options in other subject areas
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Concept Maps
By drawing concept maps, students strengthen their understanding of how a new concept is related
to others they already know.
Background and Purpose
Concept maps are drawings or diagrams showing
connections between major concepts in a course or section of a course.
I have used concept maps in precalculus and calculus
classes at Surry Community College, a medium sized member
of the North Carolina Community College System. About
one-third of our students transfer to senior institutions.
This includes most of the students who enroll in precalculus
and nearly all of those who take calculus.
Concepts maps can be a useful tool for
formative assessment. They can also provide an added benefit
of helping students organize their knowledge in a way
that facilitates further learning. I first learned about concept
maps from Novak [3]. Further insight about concept maps and
their uses in the classroom can be found in [1] and [2].
Method
Concept maps are essentially drawings or diagrams
showing mental connections that students make between a
major concept the instructor focuses on and other concepts
that they have learned [1]. First, the teacher puts a schematic
before the students (on a transparency or individual worksheet)
along with a collection of terms related to the root concept.
Students are to fill in the empty ovals with appropriate terms from
the given collection maintaining the relationships given
along branches. From the kinds of questions and errors that
emerge, I determine the nature and extent of review needed,
especially as related to the individual concepts names that should
be
familiar to each student. I also give some prompts that
enable students to proceed while being careful not to give too
much information. This is followed by allowing the students to
use their text and small group discussions as an aid. The idea
at this stage is to let the students complete the construction
of the map. Eventually they all produce a correct concept
map. Once this occurs, I discuss the concept map, enrich it
with other examples and extensions in the form of other
related concepts.
This strategy seems to work particularly well when
dealing with several related concepts, especially when
some concepts are actually subclasses of others. I have used
this procedure in precalculus in order to introduce the
concept of a transcendental number. Students chose from the
list: integers, real, 3/5, rational, irrational,
e, algebraic, p, transcendental,, and natural to complete the concept map
on the top of page 90.
As students proceeded individually, I walked around
the classroom and monitored their work, noting different
areas of difficulty. Some students posed questions that
revealed various levels of understanding. Among the questions
were "Don't these belong together?" "Does every card have to
be used?" and "Haven't some groups of numbers been left out?"
I have also used this procedure in calculus to
introduce the gamma function. I wanted to introduce the
gamma function by first approaching the idea of a
non-elementary function. This prompted me to question what
students understood about the classifications of other kinds
of functions. I presented a schematic and gave each student
a set of index cards. On each card was one of the
following terms: polynomial, radical, non-elementary,
trigonometric,
inverse trigonometric, hyperbolic, and gamma. The
students were told to arrange the cards in a way that would fit
the schematic.
Use of Findings
The primary value of using concept maps this way is that
it helps me learn what needs review. They provide a
means for detecting students misconceptions and lack of
knowledge of the prerequisite concepts necessary for learning
new mathematics.
Success Factors
The primary caveats in this approach are that the
teacher should be careful not to let the students work in groups
too quickly and not to give away too much information in
the way of prompts or hints. One alternative approach is to
have the students, early in their study of a topic, do a
concept map of the ideas involved without giving them a
preset
schematic. Then you gather the concept maps drawn
and discuss the relations found: what's good about each, or
what's missing. You can also have students draw a concept map
by brainstorming to find related concepts. Then students
draw lines between concepts, noting for each line what
the relationship is.
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"Teach Yourself Basic Mathematics "gets you up and running with all the math you need to confidently meet the numerical challenges of everyday living. Here you will learn how to rapidly perform basic arithmetic procedures; change between different systems of measurement; master fractions, decimals, and percentages; interpret simple graphs and tables; and apply basic geometry and algebra to problems in many areas of life, from shopping to home repairs. And, just to keep things interesting, the book includes fun games and puzzles that help you test your mastery of the procedures covered.
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Curriculum and Requirements
Related Links
For well over two thousand years, mathematics has been a part of the human search for understanding. Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth through careful reasoning that begins with a small set of self-evident assumptions Learning to think in mathematical terms is an essential part of becoming a liberally educated person.
Mathematics is an engaging field, rich in beauty, with powerful applications to other subjects. Thus we strive to ensure that Kenyon students encounter and learn to solve problems using a number of contrasting but complementary mathematical perspectives: continuous and discrete, algebraic and geometric, deterministic and stochastic, theoretical and applied. In our courses we stress mathematical thinking and communication skills. And in courses where it makes sense to incorporate technological tools, our students learn to solve mathematical problems using computer algebra systems, statistical packages, and computer programming languages.
New Students
For those students who want only an introduction to mathematics, or perhaps a course to satisfy a distribution requirement, selection from MATH 105, 106, 111, 116, 128 and SCMP 118 is appropriate. Students who think they might want to continue the study of mathematics beyond one year, either by pursuing a major or minor in mathematics or as a foundation for courses in other disciplines, usually begin with the calculus sequence (MATH 111, 112, and 213). Students who have already had calculus or who want to take more than one math course may choose to begin with the Elements of Statistics (MATH 106) and Data Analysis (MATH 206) or Introduction to Programming (SCMP 118). A few especially well-prepared students take Linear Algebra (MATH 224) or Foundations (MATH 222) in their first year. (Please see the department chair for further information.)
MATH 111 is an introductory course in calculus. Students who have completed a substantial course in calculus might qualify for one of the successor courses, MATH 112 or 213. MATH 106 is an introduction to statistics, which focuses on quantitative reasoning skills and the analysis of data. SCMP 118 introduces students to computer programming.
To facilitate proper placement of students in calculus courses, the department offers placement tests that help students decide which level of calculus course is appropriate for them. This and other entrance information is used during the orientation period to give students advice about course selection in mathematics. We encourage all students who do not have Advanced Placement credit to take the placement exam that is appropriate for them.
The ready availability of powerful computers has made the computer one of the primary tools of the mathematician. Students will be expected to use appropriate computer software in many of the mathematics courses. However, no prior experience with the software packages or programming is expected, except in advanced courses that presuppose earlier courses in which use of the software or programming was taught.
Course Requirements for the Major
There are two concentrations within the mathematics major: classical mathematics and statistics. The coursework required for completion of the major in each concentration is given below.
Classical Mathematics A student must have credit for the following core courses:
Three semesters of calculus (MATH 111, 112, 213, or the equivalent)
One semester of statistics (MATH 106 or 436, or the equivalent)
SCMP 118 Introduction to Programming
MATH 222 Foundations
MATH 224 Linear Algebra I
MATH 335 Abstract Algebra I or MATH 341 Real Analysis I
In addition, majors must have credit for at least three otherelective courses selected with the consent of the department. MATH 110 may not be used to satisfy the requirements for the major.
Statistics A student must have credit for the following core courses:
Three semesters of calculus (MATH 111, 112, 213 or the equivalent)
SCMP 118 Introduction to Programming
MATH 222 Foundations
MATH 224 Linear Algebra I
MATH 336 Probability
MATH 341 Real Analysis I
MATH 416 Linear Regression Models or MATH 436 Mathematical Statistics
In addition to the core courses, majors must also have credit for two elective courses from the following list:
MATH 106 Elements of Statistics
MATH 206 Data Analysis
MATH 216 Nonparametric Statistics
MATH 236 Random Structures
MATH 416 Linear Regression Models
MATH 436 Mathematical Statistics
Applications of Math Requirement
Mathematics is a vital component in the methods used by other disciplines, and the applied math requirement is designed to expose majors to this vitality. There are two ways to satisfy the requirement:
a) Earn credit for two courses (at least 1 unit) from a single department or program that use mathematics in significant ways. Typically, majors will choose a two-course sequence from the following list; other two-course sequences require departmental approval:
PHYS 140/145
ECON 101/102
PSYC 200 together with a 400-level Research Methods in Psychology course
b) Earn credit for a single math course that focuses on the development and analysis of mathematical models used to answer questions arising in other fields. The following courses satisfy the requirement, but other courses may satisfy the requirement with approval of the department:
MATH 258 Mathematical Biology
MATH 347 Mathematical Models
Classical mathematics majors may also use MATH 206, MATH 216, MATH 226, or MATH 416 to satisfy the requirement. Additionally, students choosing this option may not use the applied math course as one of the elective courses required for the major.
Depth Requirement
Majors are expected to attain a depth of study within mathematics, as well as breadth. Therefore majors should earn credit in one of four two-course upper-level sequences:
MATH 335/435 Abstract Algebra I & II
MATH 341/441 Real Analysis I & II
MATH 336/436 Probability and Mathematical Statistics
MATH 336/416 Probability and Linear Regression Models
Other two-course sequences may satisfy the requirement with approval from the department.
Senior Exercise
The Senior Exercise begins promptly in the fall of the senior year with independent study on a topic of interest to the student and approved by the department. The independent study culminates in the writing of a paper, which is due in November. (Juniors are encouraged to begin thinking about possible topics before they leave for the summer.) Students are also required to take the Major Field Test in Mathematics produced by the Educational Testing Service. Evaluation of the Senior Exercise is based on the student's performance on the paper and the standardized exam. A detailed guide on the Senior Exercise is available on the math department Web site under the link "mathematics academic program."
Suggestions for Majoring in Mathematics
Students wishing to keep open the option of a major in mathematics typically begin with the study of calculus and normally complete the calculus sequence, MATH 222 (Foundations), and either SCMP 118 or MATH 106 by the end of the sophomore year. A major is usually declared no later than the second semester of the sophomore year. Those considering a mathematics major should consult with a member of the mathematics department to plan their course of study.
The requirements for the major are minimal. Anyone who is planning a career in the mathematical sciences, or who intends to read for honors, is encouraged to consult with one or more members of the department concerning further studies that would be appropriate. Similarly, any student who wishes to propose a variation of the major program is encouraged to discuss the plan with a member of the department prior to submitting a written proposal for a decision by the department.
Students who are interested in teaching mathematics at the high-school level should take MATH 230 (Geometry) and MATH 335 (Abstract Algebra I), since these courses are required for certification in most states, including Ohio.
Honors in Mathematics
Eligibility
To be eligible to enroll in the Mathematics Honors Seminar, by the end of junior year students must have completed one depth sequence (MATH 335-435, MATH 336-416, MATH 336-436, MATH 341-441) and have earned a GPA of at least 3.33, with a GPA in Kenyon mathematics courses of at least 3.6. The student must also have, in the estimation of the mathematics faculty, a reasonable expectation of fulfilling the requirements for Honors, listed below.
To earn Honors in mathematics, a student must: (1) Complete two depth sequences (see list above); (2) Complete at least six 0.5-unit courses in mathematics numbered 300 or above; (3) Pass the Senior Exercise in the fall semester; (4) Pass the Mathematics Honors Seminar MATH 498; (5) Present the results of independent work in MATH 498 to a committee consisting of an outside examiner and members of the Kenyon Mathematics Department; (6) successfully complete an examination written by an outside examiner covering material from MATH 498 and previous mathematics courses; (7) Maintain an overall Kenyon GPA of at least 3.33; (8) Maintain a Mathematics Department GPA of at least 3.6.
Awarding Honors
Based on performance in all of the above-mentioned areas, the department (in consultation with the outside examiner) can elect to award Honors, High Honors, or Highest Honors; or not to award honors at all.
Requirements for the Minors
There are two minors in mathematics. Each minor deals with core material of a part of the discipline, and each reflects the logically structured nature of mathematics through a pattern of prerequisites. A minor consists of satisfactory completion of the courses indicated.
Statistics Five courses in statistics from the following: MATH 106 or 116, 206, 216, 236, 336, 416, 436. (Students may count at most one statistics course from another department. For example, ECON 375 or PSYC 200 may be substituted for one of the courses listed above.)
Our goal is to provide a solid introduction to basic statistical methods, including data analysis, design and analysis of experiments, statistical inference, and statistical models, using professional software such as Minitab, SAS, Maple, and R.
Deviations from the list of approved minor courses must be ratified by the Mathematics Department. Students considering a minor in mathematics or statistics are urged to speak with a member of the department about the selection of courses.
Cross-listed course The following course is cross-listed in biology and will satisfy the natural science requirement: MATH 258 Mathematical Biology
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Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis.
LECTURE SYLLABUS
Suffix notation and the summation convention (2L Dr G Wells)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes' theorem and the divergence theorem.
Variational methods in engineering analysis (6L Dr G Wells)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Prof. P. A. Davidson)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.
OBJECTIVES
On completion of the module students should:
Understand the various types of PDE and the physical nature of their solutions;
Understand various solution methods for PDEs and be able to apply these to a range of problems;
Understand the formulation of
various physical problems in terms of variational statements;
Be able to estimate
solutions using trial functions and direct minimisation;
Be able to calculate an
Euler-Lagrange differential equation from a variational statement, and to
find the corresponding natural boundary conditions;
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