Mastering the Fundamentals of Mathematics [TTC Video]
16 September 2016, 12:33
Course No 1014 | MP4, MPEG4, 426x320 | AAC, 96 kbps, 2 Ch | 24x30 mins | 4.45GB
Multiplying and dividing large numbers. Simplifying fractions and converting percentages. Handling square roots and exponents. These and other skills are the veritable foundation on which all of mathematics rests. To master them is to unlock the door to more advanced areas of study—such as algebra, geometry, and calculus—and to discover new levels of confidence in dealing with the math of everyday life.
Whether you're a high-school student preparing for the challenges of higher math classes, an adult who needs a refresher in math to prepare for a new career, or someone who just wants to keep his or her mind active and sharp, there's no denying that a solid grasp of arithmetic and prealgebra is essential in today's world. Knowing the fundamentals of mathematics can
- increase your chances of success in high-school and college math classes;
- prepare you for a career in a field that requires a strong foundation in math, such as economics, engineering, medicine, and the building trades;
- strengthen your everyday critical thinking skills; and
- help you handle with confidence everyday tasks such as shopping and planning a personal budget.
Yet despite how basic this kind of math may seem, the mechanics of mathematics remains a mystery to many of us because we've been taught to focus solely on our answers. But in the opinion of award-winning Professor James A. Sellers of The Pennsylvania State University, a true understanding of basic math involves more than just arriving at the right solution. It involves
- properly understanding the nature of numbers and mathematical concepts,
- paying close attention to the step-by-step processes behind different calculations, and
- thinking about what you're solving for—and why you're solving for it in a specific way.
This more well-rounded approach to the basics of mathematics is a surefire way to strengthen your current knowledge or to gain new skills for more deftly and confidently approaching and dealing with math. And it's all available to you in Professor Sellers' engaging course, Mastering the Fundamentals of Mathematics. Using the same inspirational teaching skill and experience he's brought to his other popular Great Courses in math, Professor Sellers reveals the secrets behind all the key math topics you need to know. In 24 lectures packed with helpful examples, practice problems, and guided walkthroughs, you'll finally grasp the all-important fundamentals of math in a way that truly sticks.
Explore All the Essential Areas of Basic Math
Designed for lifelong learners of all ages, Mastering the Fundamentals of Mathematics zeroes in on topics that everyone needs to know:
- Adding, subtracting, multiplying, and dividing whole numbers, fractions, negative numbers, and decimals
- Converting between fractions, decimals, and percentages
- Solving real-world problems involving ratios and proportions
- Working with whole-number exponents and square roots
With each topic, Professor Sellers shows you how to approach, understand, and solve problems of varying complexity. And, later in the course, he offers brief introductions to more advanced areas of math and prealgebra, including
- two-dimensional geometry,
- elementary number theory, and
- basic probability and statistics.
And whether he's discussing the order of operations or introducing you to methods for plotting points on a coordinate plane, Professor Sellers shows that the key to facing down more intimidating math problems is by tapping into basic concepts and calculations you've already mastered. Like an inspirational instructor who only has your success in mind, he reveals how basic math comes together—and even works together—to help you solve problems such as finding the area of a circle or breaking down a complex word problem involving statistics.Learn Tricks and Shortcuts for Solving Problems
To help you solve problems with greater ease, Mastering the Fundamentals of Mathematics is packed with tips, tricks, techniques, and shortcuts. Here's just a small sampling of what you'll find in this course.
- Reducing fractions to their lowest terms: When dealing with fractions in math, you'll often be required to express your answers in the lowest terms to make the fractions easier to understand. So how can you tell when a fraction has been reduced to its lowest term? You'll know because the only divisor or factor that the numerator (top number) and denominator (bottom number) share is 1. For example, the fraction 4/8 is not in its lowest term because both numbers share a factor of 2.
- Adding numbers with different signs: What's a less complicated way to solve an addition problem such as 7 + (- 3) without using a number line? First, figure out which number has the larger absolute value (7). Then, subtract the other absolute value from this one (7 - 3 = 4) and attach the sign of the number that had the larger absolute value (4). That's it!
- Lining up decimal points: Sometimes, performing calculations with large decimals (such as 153.46 + 5343.3) can be tricky, but the important point is knowing when to align your decimal points. In addition and subtraction problems, it's essential to line up corresponding digits in a right-justified way to get the correct answer; with multiplication and division, however, this alignment is unnecessary.
- Calculating tips in your head: Do you always find yourself unsure of how much of a tip to leave? Knowing how to work with percentages and decimals makes it easy. To calculate a 15% tip, take 10% of the bill just by moving the decimal point one place to the left (example: $12.00 would be $1.20). Then, add half of that number ($0.60) to that amount and you've got the answer ($1.80). If you want to leave a 20% tip, take 10% of the bill ($1.20) and just double it ($2.40).
An Interactive, Engaging Way to Learn Math
An added feature of Mastering the Fundamentals of Mathematics is its interactive nature. At specific points in a given lecture, Professor Sellers gives you a problem and invites you to pause the course, try your best to solve the problem, and then continue the course to check youranswer alongside his and chart your personal progress. Oftentimes, Professor Sellers roots his practice problems in everyday scenarios in which you're likely to find yourself, such as paying for groceries and tipping at restaurants.
Plus, he's crafted a free, comprehensive workbook with a complete answer key to go along with his course—one that comes filled with additional practice problems on each topic he covers in the course.
Yet even with its wealth of practice problems and exercises, what makes this course so rewarding is ultimately Professor Sellers himself. As Director of Undergraduate Mathematics at The Pennsylvania State University, he's in the unique position of knowing the specific areas math students have trouble with—and the specific ways to help them over these common hurdles. Calm and clear, this winner of the Teresa Cohen Mathematics Service Award is a constantly encouraging presence who refuses to let you give up and helps you prove to yourself that you can be successful in math.
So whether you're just setting out on your mathematical journey or whether you simply want to rediscover what you've forgotten, you'll find Mastering the Fundamentals of Mathematics to be an invaluable guide to an invaluable subject.
Mathematics, Philosophy, and the 'Real World' [TTC Video]
16 September 2016, 12:04
Course No 1440 | AVI, XviD, 640x480 | MP3, 96 kbps, 2 Ch | 36x30 mins | + PDF Guidebook | 6.27GB
Mathematics has spread its influence far beyond the realm of numbers. The concepts and methods of mathematics are crucially important to all of culture and affect the way countless people in all spheres of life look at the world. Consider these cases:
- When Leonardo da Vinci planned his mural The Last Supper in the 1490s, he employed geometric perspective to create a uniquely striking composition, centered on the head of Jesus.
- When Thomas Jefferson sat down to write the Declaration of Independence in 1776, he composed it on the model of a geometric proof, which is what gives it much of its power as a defense of liberty.
- When Albert Einstein developed his theory of general relativity in the early 20th century, he used non-Euclidean geometry to prove that the path of a ray of light, in the presence of a gravitational field, is not straight but curved.
Intriguing examples like these reflect the important dialogue between mathematics and philosophy that has flourished throughout history. Indeed, mathematics has consistently helped determine the course of Western philosophical thought. Views about human nature, religion, truth, space and time, and much more have been shaped and honed by the ideas and practices of this vital scientific field.
Award-winning Professor Judith V. Grabiner shows you how mathematics has shaped human thought in profound and exciting ways in Mathematics, Philosophy, and the "Real World," a 36-lecture series that explores mathematical concepts and practices that can be applied to a fascinating range of areas and experiences.
Believing that mathematics should be accessible to any intellectually aware individual, Professor Grabiner has designed a course that is lively and wide-ranging, with no prerequisites beyond high school math. For those with an interest in mathematics, this course is essential to understanding its invaluable impact on the history of philosophical ideas; for those with an interest in philosophy, Professor Grabiner's course reveals just how indebted the field is to the mathematical world.
Math Meets Philosophy
In a presentation that is clear, delightful, and filled with fascinating case histories, Professor Grabiner focuses on two areas of mathematics that are easily followed by the nonspecialist: probability and statistics, and geometry. These play a pivotal role in the lives of ordinary citizens today, when statistical information is everywhere, from medical data to opinion polls to newspaper graphs; and when the logical rules of a geometric proof are a good approach to making any important decision.
Mathematics, Philosophy, and the "Real World" introduces enough elementary probability and statistics so that you understand the subtleties of the all-important bell curve. Then you are immersed in key theorems of Euclid's Elements of Geometry, the 2,200-year-old work that set the standard for logical argument. Throughout the course, Professor Grabiner shows how these fundamental ideas have had an enormous impact in other fields. Notably, mathematics helped stimulate the development of Western philosophy and it has guided philosophical thought ever since, a role that you investigate through thinkers such as these:
- Plato: Flourishing in the 4th century B.C.E., Plato was inspired by geometry to argue that reality resides in a perfect world of Forms accessible only to the intellect—just like the ideal circles, triangles, and other shapes that seem to exist only in the mind.
- Descartes: Writing in the 17th century, René Descartes used geometric reasoning in a systematic search for all possible truths. In a famous exercise, he doubted everything until he arrived at an irrefutable fact: "I think, therefore I am."
- Kant: A century after Descartes, Immanuel Kant argued that metaphysics was possible by showing its kinship with mathematics. The perfection of Euclidean geometry led him to take for granted that space has to be Euclidean.
- Einstein: Working in the early 20th century with a concept of "straight lines" that was different from Euclid's, Albert Einstein showed that gravity is a geometric property of non-Euclidean space, which is an essential idea of his general theory of relativity.
Non-Euclidean Geometry Explained
The discovery of non-Euclidean geometry influenced fields beyond mathematics, laying the foundation for new scientific and philosophical theories and also inspiring works by artists such as the Cubists, the Surrealists, and their successors leading up to today.
Non-Euclidean geometry was a stunning intellectual breakthrough in the 19th century, and you study how three mathematicians, working independently, overthrew the belief that Euclid's geometry was the only possible consistent system for dealing with points, lines, surfaces, and solids. Einstein's theory of relativity was just one of the many ideas to draw on the non-Euclidean insight that parallel lines need not be the way Euclid imagined them.
Professor Grabiner prepares the ground for your exploration of non-Euclidean geometry by going carefully over several of Euclid's proofs so that you understand Euclid's theory of parallel lines at a fundamental level. You even venture into the visually rich world of art and architecture to see how Renaissance masters used Euclidean geometry to map three-dimensional space onto flat surfaces and to design buildings embodying geometrical balance and symmetry. The Euclidean picture of space became internalized to a remarkable extent during and after the Renaissance, with a far-reaching effect on the development of philosophy and science.
Change the Way You Think
Mathematics has not only changed the way specialists think about the world, it has given the rest of us an easily understandable set of concepts for analyzing and understanding our surroundings. Professor Grabiner provides a checklist of questions to ask about any statistical or probabilistic data that you may encounter. Her intriguing observations include the following:
- Statistics: Biologist and author Stephen Jay Gould, who developed abdominal cancer, was told his disease had an eight-month median survival time after diagnosis. The diagnosis sounded hopeless, but his understanding of the characteristics of the median (as opposed to the mean or mode) gave him a strategy for survival.
- Bad graphs: There are many ways to make a bad graph; some deliberately misleading, others merely badly conceived. Beware of a graph that starts at a number higher than zero, since comparisons between different data points on the graph will be exaggerated.
- Polls: The Literary Digest poll before the 1936 U.S. presidential election was the largest ever conducted and predicted a landslide win for Alf Landon over Franklin Roosevelt. Yet the result was exactly the opposite due to an unrecognized systematic bias in the polling sample.
- Probability: Intuition can lead one astray when one is judging probabilities. You investigate the case of an eyewitness to an accident who has done well on tests of identifying the type of vehicle involved. But a simple calculation shows that she is more likely wrong than not.
The Power of Mathematical Thinking
Mathematics, Philosophy, and the "Real World" focuses on mathematics and its influence on culture in the West. But for an alternative view, Professor Grabiner devotes a lecture to mathematics in classical China, where geometers discovered some of the same results as the ancient Greeks but with a very different approach. One major difference is that the Chinese didn't use indirect proof, a technique that proves a proposition true because the assumption that it is false leads to a contradiction.
In another lecture, Professor Grabiner gives time to the critics of mathematics—philosophers, scientists, poets, and writers who have argued against the misuse of mathematics. Charles Dickens speaks for many in his memorable novel Hard Times, which depicts the human misery brought by Victorian England's obsession with statistics and efficiency.
But even more memorable are the cases in which mathematics turns up where it is least expected. "We hold these truths to be self evident ..." So wrote Thomas Jefferson in the second sentence of the Declaration of Independence. He had originally started, "We hold these truths to be sacred and undeniable ... " The change to "self-evident" was probably made at the suggestion of Benjamin Franklin, a great scientist as well as a statesman, who saw the power of appealing to scientific thinking. A Euclidean proof begins with axioms (self-evident truths) and then moves through a series of logical steps to a conclusion.
With her consummate skill as a teacher, Professor Grabiner shows how Jefferson laid out America's case against Great Britain with all of the rigor he learned in Euclid's Elements, working up to a single, irrefutable conclusion: "That these United Colonies are, and of Right ought to be Free and Independent States."
There is arguably no greater demonstration of the power of mathematics to transform the real world—and it's just one of the fascinating insights you'll find in Mathematics, Philosophy, and the "Real World."
Mathematics Describing the Real World: Precalculus and Trigonometry [TTC Video]
16 September 2016, 11:52
Course No 1005 | AVI, AVC, 384x288 | MP3, 56 kbps, 2 Ch | 36x30 mins | + PDF Guidebook | 2.26GB
What's the sure road to success in calculus? The answer is simple: Precalculus. Traditionally studied after Algebra II, this mathematical field covers advanced algebra, trigonometry, exponents, logarithms, and much more. These interrelated topics are essential for solving calculus problems, and by themselves are powerful methods for describing the real world, permeating all areas of science and every branch of mathematics. Little wonder, then, that precalculus is a core course in high schools throughout the country and an important review subject in college.
Unfortunately, many students struggle in precalculus because they fail to see the links between different topics—between one approach to finding an answer and a startlingly different, often miraculously simpler, technique. As a result, they lose out on the enjoyment and fascination of mastering an amazingly useful tool box of problem-solving strategies.
And even if you're not planning to take calculus, understanding the fundamentals of precalculus can give you a versatile set of skills that can be applied to a wide range of fields—from computer science and engineering to business and health care.
Mathematics Describing the Real World: Precalculus and Trigonometry is your unrivaled introduction to this crucial subject, taught by award-winning Professor Bruce Edwards of the University of Florida. Professor Edwards is coauthor of one of the most widely used textbooks on precalculus and an expert in getting students over the trouble spots of this challenging phase of their mathematics education.
"Calculus is difficult because of the precalculus skills needed for success," Professor Edwards points out, adding, "In my many years of teaching, I have found that success in calculus is assured if students have a strong background in precalculus."
A Math Milestone Made Clear
In 36 intensively illustrated half-hour lectures, supplemented by a workbook with additional explanations and problems, Mathematics Describing the Real World takes you through all the major topics of a typical precalculus course taught in high school or college. Those who will especially benefit from Professor Edwards's lucid and engaging approach include
- high school and college students currently enrolled in precalculus who feel overwhelmed and want coaching from an inspiring teacher who knows where students stumble;
- parents of students, who may feel out of their depth with the advanced concepts taught in precalculus;
- those who have finished Algebra II and are eager to get a head start on the next milestone on the road to calculus;
- beginning calculus students who want to review and hone their skills in crucial precalculus topics;
- anyone motivated to learn precalculus on his or her own, whether as a home-schooled pupil or as an adult preparing for a new career.
The Powerful Tools of Precalculus
With precalculus, you start to see all of mathematics as a unified whole—as a group of often radically different techniques for representing data, analyzing problems, and finding solutions. And you discover that these techniques are ultimately connected in a beautiful way. Perceiving these connections helps you choose the best tool for a given problem:
- Algebraic functions: Including polynomial functions and rational functions, these equations relate the input value of a variable to a single output value, corresponding to countless everyday situations in which one event depends on another.
- Trigonometry: Originally dealing with the measurement of triangles, this subject has been vastly enriched by the concept of the trigonometric function, which models many types of cyclical processes, such as waves, orbits, and vibrations.
- Exponential and logarithmic functions: Often involving the natural base, e, these functions are built on terms with exponents and their inverse, logarithms, and describe phenomena such as population growth and the magnitude of an earthquake on the Richter scale.
- Complex numbers: Seemingly logic-defying, complex numbers are based on the square root of –1, designated by the symbol i. They are essential for solving many technical problems and are the basis for the beautiful patterns in fractal geometry.
- Vectors: Quantities like velocity have both magnitude and direction. Vectors allow the direction component to be specified in a form that allows addition, multiplication, and other operations that are crucial in fields such as physics.
- Matrices: A matrix is a rectangular array of numbers with special rules that permit two matrices to be added or multiplied. Practically any situation where data are collected in columns and rows can be treated mathematically as a matrix.
In addition, Professor Edwards devotes two lectures to conic sections, slicing a cone mathematically into circles, parabolas, ellipses, and hyperbolas. You also learn when it's useful to switch from Cartesian to polar coordinates; how infinite sequences and series lead to the concept of the limit in calculus; and two approaches to counting questions: permutations and combinations. You close with an introduction to probability and a final lecture that features an actual calculus problem, which your experience in precalculus makes ... elementary!
Believing that students learn mathematics most effectively when they see it in the context of the world around them, Professor Edwards uses scores of interesting problems that are fun, engaging, and often relevant to real life. Among the many applications of precalculus that you'll encounter are these:
- Public health: A student with a new strain of flu arrives at college. How long before every susceptible person is infected? An exponential function called the logistic growth model shows how quickly an epidemic spreads.
- Surveying: Suppose you have to measure the diagonal width of a marsh without getting wet. It's a simple matter of walking two sides of a triangle on dry land and then using trigonometry to determine the length of the third side that spans the marsh.
- Astronomy: One of the most famous cases involving the sine and cosine functions that model periodic phenomena occurred in 1967, when astronomer Jocelyn Bell detected a radio signal from space at 1.3373-second intervals. It proved to be the first pulsar ever observed.
- Acoustics: The special properties of an ellipse explain why a person standing at a given spot in the U.S. Capitol's Statuary Hall can hear a whisper from someone standing 85 feet away.
- Computer graphics: How do you make an object appear to rotate on a computer screen? Matrix algebra allows you to move each pixel in an image by a specified angle by multiplying two matrices together.
- Probability: Have you ever forgotten your four-digit ATM PIN number? What is the probability that you can guess it? A simple calculation shows that you would have to punch numbers nonstop for many hours before being assured of success.
An Adventure in Mathematical Learning
A three-time Teacher of the Year in the College of Liberal Arts and Sciences at the University of Florida, Professor Edwards has a time-tested approach to making difficult material accessible. In Mathematics Describing the Real World, he enlivens his lectures with study tips and a feature he calls "You Be the Teacher," in which he puts you in the professor's shoes by asking how you would design a particular test problem or answer one of the frequently asked questions he gets in the classroom. For example, are all exponential functions increasing? After you hear Professor Edwards's explanation, you'll know that when someone uses the term "exponentially," you should ask, "Do you mean exponential growth or decay?"—for it can go in either direction. He also gives valuable tips on using graphing calculators, pointing out their amazing capabilities—and pitfalls.