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Fri, 24 May 2013 13:00:57 PDT Fri, 24 May 2013 13:00:57 PDT http://www.merlot.org:80/merlot/images/merlot.gif http://www.merlot.org:80/merlot/ 44 34 http://www.merlot.org/merlot/viewMaterial.htm?id=734964 This is a free online course offered by the Saylor Foundation.'The courses included in this program are designed for the high school student preparing for college or the adult learner who needs a refresher course or two in mathematics.Each of the courses in this series includes instructional videos and practice problems from Khan Academy™ (Khan Academy™ is a library of over 3,000 videos covering a range of topics, including math and physics) that will help you master the foundational knowledge necessary for success in College Algebra (MA001: Beginning Algebra) and beyond.These courses focus on the ways in which math relates to common “real world” situations, transactions, and phenomena, such as personal finance, business, and the sciences. This “real world” focus will help you grasp the importance of the mathematical concepts you encounter in these courses and understand why you need quantitative and algebraic skills in order to be successful both in college and in your day-to-day-life.' http://www.merlot.org/merlot/viewMaterial.htm?id=734972 This is a free online course offered by the Saylor Foundation.'This introductory mathematics course is for you if you have a solid foundation in arithmetic (that is, you know how to perform operations with real numbers, including negative numbers, fractions, and decimals). Numbers and basic arithmetic are used often in everyday life in both simple situations, like estimating how much change you will get when making a purchase in a store, as well as in more complicated ones, like figuring out how much time it would take to pay off a loan under interest.The subject of algebra focuses on generalizing these procedures. For example, algebra will enable you to describe how to calculate change without specifying how much money is to be spent on a purchase–it will teach you the basic formulas and steps you need to take no matter what the specific details of the situation are. Likewise, accountants use algebraic formulas to calculate the monthly loan payments for a loan of any size under any interest rate. In this course, you will learn how to work with formulas that are already known from science or business to calculate a given quantity, and you will also learn how to set up your own formulas to describe various situations by translating verbal descriptions to mathematical language. In the later units of this course, you will discover another tool used in mathematics to describe numbers and analyze relationships: graphing. You will learn that any pair of numbers can be represented by a point on a coordinate plane and that a relationship between two quantities can be represented by a line or a curve.Units 6, 7, and 8 may seem more abstract than the earlier ones, as you will deal with expressions that contain mostly variables and not too many numbers. While the procedures you will master in these units might seem to have little practical application, you have to keep in mind that they result in formulas that describe very real situations in business, accounting, and science. Knowing how to perform various operations with algebraic expressions will eventually enable you to solve quadratic and even more complex equations. You will explore a variety of real-world scenarios that can be described by these kinds of equations. For example, if a ball is thrown up in the air, solving a quadratic equation will help you find out when it will hit the ground. As another example, if you know the area of a rectangular garden, then you can use a quadratic equation to find the length of each side.' http://www.merlot.org/merlot/viewMaterial.htm?id=734974 This is a free online course offered by the Saylor Foundation.'“Everything is numbers.” This phrase was uttered by the lead character, Dr. Charlie Epps, on the hit television show “NUMB3RS.” If everything has a mathematical underpinning, then it follows that everything is somehow mathematically connected, even if it is only in some odd, “six degrees of separation (or Kevin Bacon)” kind of way.Geometry is the study of space (for now, mainly two-dimensional, with some three-dimensional thrown in) and the relationships of objects contained inside. It is one of the more relatable math courses, because it often answers that age-old question, “When am I ever going to use this in real life?” Look around you right now. Do you see any triangles? Can you spot any circles? Do you see any books that look like they are twice the size of other books? Does your wall have paint on it?In geometry, you will explore the objects that make up our universe. Most people never give a second thought to how things are constructed, but there are geometric rules at play. Most people never think twice about a rocket launch, but if that rocket is not launched at an exact angle, it will miss its target. A football field has to be measured out to be a rectangle; if you used another shape, such as a trapezoid, that would give an unfair advantage to one team, because that one team would have more space to work with.In this course, you will study the relationships between lines and angles. Have you ever looked at a street map? Believe it or not, there is a lot of geometry on a map, as you will see from this course. You will learn to calculate how much space an object covers, which is useful if you ever have to, say, buy some paint. You will learn to determine how much space is inside of a three-dimensional object, which is useful for those times you are trying to fit four suitcases, three kids, two adults, and a dog into the back of your vehicle.These are just some of the topics you will be learning. As you will quickly see, everything is not just numbers; it is also relationships. Even nature itself knows this. What did the little acorn say when it grew up? “Gee, I’m a tree!”' http://www.merlot.org/merlot/viewMaterial.htm?id=733171 This is a free online course offered by the Saylor Foundation.'Mathematics is about structure, about reasoning, and about modeling. This course braids these three threads together. Mathematical logic began as the study of the reasoning used in mathematics, but it turns out to be useful in describing the mathematical concept of structure and in modeling automated reasoning—that is, modeling computation.The logical approach to structure gives an alternate perspective on such other mathematical subjects as combinatorics and abstract algebra. This, for the most part, is described by the area of model theory, which is the focus of Unit 1.In Unit 2, we will look at modeling computation. The central fact of these models, from a logical standpoint, is that once we can handle a computation as a definable mathematical object, we can prove that certain computations are impossible. The most famous such proof is Gödel’s Incompleteness Theorem, showing that it is impossible to compute truth in a system sufficiently strong to describe natural number arithmetic.Finally, in Unit 3, we turn to proof theory. Just as modeling computations results in new insights, modeling the process of mathematical proof results in a surprising connection: a proof is analogous to a computation.These three often interact. Proofs and computations have natural parallels with the language we use to describe structures. Structures from model theory give natural settings for computation, as in Gödel’s Incompleteness Theorem. After completing this course, you will understand all three.' http://www.merlot.org/merlot/viewMaterial.htm?id=733364 This is a free online course offered by the Saylor Foundation.'This course will introduce you to a number of statistical tools and techniques that are routinely used by modern statisticians for a wide variety of applications. First, we will review basic knowledge and skills that you learned in MA121: Introduction to Statistics. Units 2–5 will introduce you to new ways to design experiments and to test hypotheses, including multiple and nonlinear regression and nonparametric statistics. You will learn to apply these methods to building models to analyze complex, multivariate problems. You will also learn to write scripts to carry out these analyses in R, a powerful statistical programming language. The last unit is designed to give you a grand tour of several advanced topics in applied statistics.' http://www.merlot.org/merlot/viewMaterial.htm?id=733365 This is a free online course offered by the Saylor Foundation.'The objective of this course is to study the basic theory and methods in the toolbox of the core of applied mathematics, with a central scheme that addresses “information processing” and with an emphasis on manipulation of digital image data. Linear algebra in the Saylor Foundation’s MA211 and MA212 are extended to “linear analysis” with applications to principal component analysis (PCA) and data dimensionality reduction (DDR). For data compression, the notion of entropy is introduced to quantify coding efficiency as governed by Shannon’s Noiseless Coding theorem. Discrete Fourier transform (DFT) followed by an efficient computational algorithm, called fast Fourier transform (FFT), as well as a real-valued version of the DFT, called discrete cosine transform (DCT) are discussed, with application to extracting frequency content of the given discrete data set that facilitates reduction of the entropy and thus significant improvement of the coding efficiency. DFT can be viewed as a discrete version of the Fourier series, which will be studied in some depth, with emphasis on orthogonal projection, the property of positive approximate identity of Fejer’s kernels, Parseval’s identity and the concept of completeness. The integral version of the sequence of Fourier coefficients is called the Fourier transform (FT). Analogous to the Fourier series, the formulation of the inverse Fourier transform (IFT) is derived by applying the Gaussian function as a sliding time-window for simultaneous time-frequency localization, with optimality guaranteed by the Uncertainty Principle. Local time-frequency basis functions are also introduced in this course by discretization of the frequency-modulated sliding time-window function at the integer lattice points. Replacing the frequency modulation by modulation with the cosines avoids the Balian-Low stability restriction on the local time-frequency basis functions, with application to elimination of blocky artifact caused by quantization of tiled DCT in image compression. Gaussian convolution filtering also provides the solution of the heat (partial differential) equation with the real-line as the spatial domain. When this spatial domain is replaced by a bounded interval, the method of separation of variables is applied to separate the PDE into two ordinary differential equations (ODEs). Furthermore, when the two end-points of the interval are insulated from heat loss, solution of the spatial ODE is achieved by finding the eigenvalue and eigenvector pairs, with the same eigenvalues to govern the exponential rate of decay of the solution of the time ODE. Superposition of the products of the spatial and time solutions over all eigenvalues solves the heat PDE, when the Fourier coefficients of the initial heat content are used as the coefficients of the terms of the superposition. This method is extended to the two-dimensional rectangular spatial domain, with application to image noise reduction. The method of separation of variables is also applied to solving other typical linear PDEs. Finally, multi-scale data analysis is introduced and compared with the Fourier frequency approach, and the architecture of multiresolution analysis (MRA) is applied to the construction of wavelets and formulation of the multi-scale wavelet decomposition and reconstruction algorithms. The lifting scheme is also introduced to reduce the computational complexity of these algorithms, with applications to digital image manipulation for such tasks as progressive transmission, image edge extraction, and image enhancement.' http://www.merlot.org/merlot/viewMaterial.htm?id=731680 This is a free online course offered by the Saylor Foundation.'In this course, you will study basic algebraic operations and concepts, as well as the structure and use of algebra. This includes the solutions to algebraic equations, factoring algebraic expressions, working with rational expressions, and graphing of linear equations. You will apply these skills to solve real world problems (word problems). Each unit will have its own application problems, depending on the concepts you have been exposed to. This course is also intended to provide you with a strong foundation for intermediate algebra and beyond.This course will begin with a review of some math concepts formed in pre-algebra, such as order of operations and simplifying simple algebraic expressions to get your feet wet. You will then build on these concepts by learning more about functions, graphing of functions, evaluation of functions, and factorization. You will spend time on the rules of exponents and their applications in distribution of multiplication over addition/subtraction.' http://www.merlot.org/merlot/viewMaterial.htm?id=731682 This is a free online course offered by the Saylor Foundation.'Precalculus II continues the in-depth study of functions addressed in Precalculus I by adding the trigonometric functions to your function toolkit. In this course, you will cover families of trigonometric functions, as well as their inverses, properties, graphs, and applications. Additionally, you will study trigonometric equations and identities, the laws of sines and cosines, polar coordinates and graphs, parametric equations and elementary vector operations.You might be curious how the study of trigonometry, or “trig,” as it is more often referred to, came about and why it is important to your studies still. Trigonometry, from the Greek for “triangle measure,” studies the relationships between the angles of a triangle and its sides and defines the trigonometric functions used to describe those relationships. Trigonometric functions are particularly useful when describing cyclical phenomena and have applications in numerous fields, including astronomy, navigation, music theory, physics, chemistry, and—perhaps most importantly, to the mathematics student—calculus.In this course, you will begin by establishing the definitions of the basic trig functions and exploring their properties and then proceed to use the basic definitions of the functions to study the properties of their graphs, including domain and range, and to define the inverses of these functions and establish the properties of these. Through the language of transformation, you will explore the ideas of period and amplitude and learn how these graphical differences relate to algebraic changes in the function formulas. You will also learn to solve equations, prove identities using the trig functions, and study several applications of these functions.' http://www.merlot.org/merlot/viewMaterial.htm?id=731684 This is a free online course offered by the Saylor Foundation.'Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an Algebra problem, like Y = 2X + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt where t is elapsed time and X0 is the initial deposit. But with compounded interest, now things get complicated for algebra as the rate R is now itself a function of time with Y = X0+ R(t)t. Now we have a rate of change which itself is changing. Calculus “to the rescue,” as Isaac Newton introduced the world to mathematics specifically designed to handle “those things that change.” Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics. It is a wonderful, beautiful, and useful set of ideas and techniques. You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences, including physical, biological, social, economic, and engineering. However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will learn in this course are more carefully defined and have both a functional and a graphical meaning. Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use. Calculus offers a huge variety of applications and many of them will be saved for future courses you might take. This course is divided into four learning sections, or units, plus a reference section, or Appendix. The course begins with a unit that provides a review of algebra specifically designed to help and prepare for the study of calculus. The second unit discusses functions, graphs, limits, and continuity. Understanding “limits” could not be more important as that topic really begins the study of calculus. The third unit will introduce and explain derivatives. With derivatives we are now ready to handle all those “things that change” mentioned above. The fourth unit makes “visual sense” of derivatives by discussing derivatives and graphs. Finally, the fifth unit provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over.' http://www.merlot.org/merlot/viewMaterial.htm?id=731689 This is a free online course offered by the Saylor Foundation.'The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician“plays” with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler “plays” with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.'