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Mastering the Fundamentals of Mathematics

Mastering the Fundamentals of Mathematics

Professor James A. Sellers Ph.D.
The Pennsylvania State University
Course No.  1014
Course No.  1014
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Course Overview

About This Course

24 lectures  |  31 minutes per lecture

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

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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.

View Less
24 Lectures
  • 1
    Addition and Subtraction
    This introductory lecture starts with Professor Sellers’ overview of the general topics and themes you’ll encounter throughout the course. Then, plunge into an engaging review of the addition and subtraction of whole numbers, complete with several helpful tips designed to help you approach these types of problems with more confidence. x
  • 2
    Multiplication
    Continue your quick review of basic mathematical operations, this time with a focus on the multiplication of whole numbers. In addition to uncovering the relationship between addition and multiplication, you’ll get plenty of opportunities to strengthen your ability to multiply two 2-digit numbers, two 3-digit numbers, and more. x
  • 3
    Long Division
    Turn now to the opposite of multiplication: division. Learn how to properly set up a long division problem, how to check your answers to make sure they’re correct, how to handle zeroes when they appear in a problem, and what to do when a long division problem ends with a remainder. x
  • 4
    Introduction to Fractions
    Mathematics is also filled with “parts” of whole numbers, or fractions. In the first of several lectures on fractions, define key terms and focus on powerful techniques for determining if fractions are equivalent, finding out which of two fractions is larger, and reducing fractions to their lowest terms. x
  • 5
    Adding and Subtracting Fractions
    Fractions with the same denominator. Fractions with different denominators. Mixed numbers. Here, learn ways to add and subtract them all (and sometimes even in the same problem) and get tips for reducing your answers to their lowest terms. Math with fractions, you’ll discover, doesn’t have to be intimidating—it can even be fun! x
  • 6
    Multiplying Fractions
    Continue having fun with fractions, this time by mastering how to multiply them and reduce your answer to its lowest term. Professor Sellers shows you how to approach and solve multiplication problems involving fractions (with both similar and different denominators), fractions and whole numbers, and fractions and mixed numbers. x
  • 7
    Dividing Fractions
    Professor Sellers walks you step-by-step through the process for speedily solving division problems involving fractions in this lecture filled with helpful practice problems. You’ll also learn how to better handle calculations involving different notations, fractions, and whole numbers, and even word problems involving the division of fractions. x
  • 8
    Adding and Subtracting Decimals
    What’s 29.42 + 84.67? Or 643 + 82.987? What about 25.7 – 10.483? Problems like these are the focus of this helpful lecture on adding and subtracting decimals. One tip for making these sorts of calculations easier: making sure your decimal points are all lined up vertically. x
  • 9
    Multiplying and Dividing Decimals
    Investigate the best ways to multiply and divide decimal numbers. You’ll get insights into when and when not to ignore the decimal point in your calculations, how to check your answer to ensure that your result has the correct number of decimal places, and how to express remainders in decimals. x
  • 10
    Fractions, Decimals, and Percents
    Take a closer look at converting between percents, decimals, and fractions—an area of basic mathematics that many people have a hard time with. After learning the techniques in this lecture and using them on numerous practice problems, you’ll be surprised at how easy this type of conversion is to master. x
  • 11
    Percent Problems
    Use the skills you developed in the last lecture to better approach and solve different kinds of percentage problems you’d most likely encounter in your everyday life. Among these everyday scenarios: calculating the tip at a restaurant and determining how much money you’re saving on a store’s discount. x
  • 12
    Ratios and Proportions
    How do ratios and proportions work? How can you figure out if a particular problem is merely just a ratio or proportion problem in disguise? What are some pitfalls to watch out for? And how can a better understanding of these subjects help save you money? Find out here. x
  • 13
    Exponents and Order of Operations
    Explore a fifth fundamental mathematical operation: exponentiation. First, take a step-by-step look at the order of operations for handling longer calculations that involve multiple tasks—complete with invaluable tips to help you handle them with ease. Then, see where exponentiation fits in this larger process. x
  • 14
    Negative and Positive Integers
    Improve your confidence in dealing with negative numbers. You’ll learn to use the number line to help visualize these numbers; discover how to rewrite subtraction problems involving negative numbers as addition problems to make them easier; examine the rules involved in multiplying and dividing with them; and much more. x
  • 15
    Introduction to Square Roots
    In this lecture, finally make sense of square roots. Professor Sellers offers examples to help you sidestep issues many students express frustration with, shows you how to simplify radical expressions involving addition and subtraction, and reveals how to find the approximate value of a square root without using a calculator. x
  • 16
    Negative and Fractional Powers
    What happens when you have to raise numbers to a fraction of a power? How about when you have to deal with negative exponents? Or negative fractional exponents? No need to worry —Professor Sellers guides you through this tricky mathematical territory, arming you with invaluable techniques for approaching these scenarios. x
  • 17
    Graphing in the Coordinate Plane
    Grab some graph paper and learn how to graph objects in the coordinate (or xy) plane. You’ll find out how to plot points, how to determine which quadrant they go in, how to sketch the graph of a line, how to determine a line’s slope, and more. x
  • 18
    Geometry—Triangles and Quadrilaterals
    Continue exploring the visual side of mathematics with this look at the basics of two-dimensional geometry. Among the topics you’ll focus on here are the various types of triangles (including scalene and obtuse triangles) and quadrilaterals (such as rectangles and squares), as well as methods for measuring angles, area, and perimeter. x
  • 19
    Geometry—Polygons and Circles
    Gain a greater appreciation for the interaction between arithmetic and geometry. First, learn how to recognize and approach large polygons, including hexagons and decagons. Then, explore the various concepts behind circles (such as radius, diameter, and the always intriguing pi), as well as methods for calculating their circumference, area, and perimeter. x
  • 20
    Number Theory—Prime Numbers and Divisors
    Shift gears and demystify number theory, which takes as its focus the study of the properties of whole numbers. Concepts that Professor Sellers discusses and teaches you how to engage with in this insightful lecture include divisors, prime numbers, prime factorizations, greatest common divisors, and factor trees. x
  • 21
    Number Theory—Divisibility Tricks
    In this second lecture on the world of number theory, take a closer look at the relationships between even and odd numbers, as well as the rules of divisibility for particular numbers. By the end, you’ll be surprised that something as intimidating as number theory could be made so accessible. x
  • 22
    Introduction to Statistics
    Get a solid introduction to statistics, one of the most useful areas of mathematics. Here, you’ll focus on the four basic “measurements” statisticians use when gleaning meaning from data: mean, media, mode, and range. Also, see these concepts at work in everyday scenarios in which statistics plays a key role. x
  • 23
    Introduction to Probability
    Learn more about probability, a cousin of statistics and another mathematical field that helps us make sense of the seemingly unexplainable nature of the world. You’ll consider basic questions and concepts from probability, drawing on the knowledge and skills of the fundamentals of mathematics you acquired in earlier lectures. x
  • 24
    Introduction to Algebra
    Professor Sellers reviews the importance of math in daily life and previews the next logical step in your studies: Algebra I (which involves variables). Whether you’re planning to take more Great Courses in mathematics or simply looking to sharpen your mind, you’ll be sent off with new levels of confidence. x

Lecture Titles

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James A. Sellers
Ph.D. James A. Sellers
The Pennsylvania State University
Dr. James A. Sellers is Professor of Mathematics and Director of Undergraduate Mathematics at The Pennsylvania State University. He earned his B.S. in Mathematics from The University of Texas at San Antonio and his Ph.D. in Mathematics from Penn State. In the past few years, Professor Sellers has received the Teresa Cohen Mathematics Service Award from the Penn State Department of Mathematics and the Mathematical Association of America Allegheny Mountain Section Mentoring Award. More than 60 of Professor Sellers's research articles on partitions and related topics have been published in a wide variety of peer-reviewed journals. In 2008, he was a visiting scholar at the Isaac Newton Institute at the University of Cambridge. Professor Sellers has enjoyed many interactions at the high school and middle school levels. He has served as an instructor of middle-school students in the TexPREP program in San Antonio, Texas. He has also worked with Saxon Publishers on revisions to a number of its high-school textbooks. As a home educator and father of five, he has spoken to various home education organizations about mathematics curricula and teaching issues.
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Reviews

Rated 4.9 out of 5 by 16 reviewers.
Rated 5 out of 5 by Dr. Sellers Grades Out At 100% Again Professor Sellers has followed up his two previous Great Courses with another absolute winner. Complementing his outstanding courses on Algebra I and Algebra II, this course reviews the fundamentals of arithmetic. He explains the basic operations of addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals. He follows with lectures on percents, ratios, and proportions and then goes into an extensive explanation of exponents and graphing in the coordinate plane. He concludes with brief introductions to Geometry, Number Theory, Statistics, and Probability. Dr. Sellers does an excellent job in presenting the topics necessary in order to succeed in higher mathematics. His logical explanations, carefully chosen examples, and practical applications progressively lead the student to the desired end. His sincere enthusiasm and pleasing personality make this a most enjoyable course. WARNING: Mathematics is not a spectator subject; you cannot just listen to an instructor, no matter how good he might be, but must actually work out many examples yourself in order to find and correct mistakes that may be unique to you. It takes practice, practice, practice along with an excellent instructor to guarantee success; therefore, I highly recommend you work out all of the problems given in the accompanying workbook in order to get the most out of this course and further increase your chances of success. July 13, 2012
Rated 5 out of 5 by I am SO glad I purchased this! I am so very thankful that I decided to purchase this. I wanted to "freshen up" my math skills, so thought a course that offered a wide range of mathematical subjects would be perfect for that. I wanted to start from beginning to end, no matter what I managed to "retain" these so many years. I learned techniques and methods that I didn't learn all throughout my education... and also learned different ways of solving equations. Prof. Sellers' teaching style, and presentation of materials - made me envious of all those students that were able to learn from him, in person. He makes MATH fun again! Math is no longer a 4-letter-word! :) I believe that even people who find themselves as math-intolerant or resistant, will thoroughly ENJOY this series! I would definitely take any/all courses available from this instructor. I have highly recommended this course to my friends and family, and will continue to do so in the future. If you, or anyone you know is struggling with math, I highly recommend taking this course - you will NOT regret it. His style and approach is personable, quirky, and not intimidating in the least. This will be a DVD that you will want to share with everyone, but will not because you don't want to let it go. :) August 7, 2014
Rated 5 out of 5 by The Heart of Middle School Math This course covers all of the topics of middle school mathematics that prepare one for subsequent success in algebra, pre-calculus and calculus. Some of the lessons are so fundamental they can be skipped. However, as the lessons progressed, I skipped fewer and fewer lessons. The course can be done in a topical way rather than a sequential way. Yet, even though I had a good understanding of the topics, I learned something from each lesson. Why did I do this course? As an avocation, I tutor middle school math students who struggle with their fundamentals. I wanted learn from a seasoned teacher as to how to present topics and concepts. I learned more than I expected--not only as how to present concepts, but I learned more about the concepts. I find that parents of middle school students struggle to help their children as most parents are many years removed from math fundamentals. Parents can use this course to re-educate themselves--thereby enabling themselves to help their children. Moreover, middle school math teachers can benefit from the course by learning from a seasoned math teacher. Dr. Sellers has produced for The Teaching Company Algebra I and Algebra II. Both are outstanding. This course is a wonderful segue to algebra. I gave the Algebra I course to my grandson when he began 7th grade. Through independent study, he was able to advance to Algebra II before he entered 8th grade. Bravo, and thank you, Dr. Sellers! July 3, 2014
Rated 5 out of 5 by Excellent base I bought this course because I started calculus , and after 3 courses, I decided I needed a good review of algebra concepts (that I had learnt 20 years ago), and this course was in a bundle with Algebra I and II, so I bought it too. It is really basic, and I did skip some parts, but as a review of the basics, you can't do better. All the concepts are there (operations, fractions, geometry, etc), and it is really nice to review how to do all those calculations by hand, without a machine or a calculator. There are also examples of how to use them in real life. I feel much more confident now to review algebra concepts and then take on calculus. A word on the professor: he is outstanding. The reviews were raving about him, and they were right! He is enthusiastic, fun, engaging, clear, and he always seem to be speaking directly to the learner (he is not reading). He has lots of examples and the visuals are absolutely helpful and always clear. This course is a must for those in need of a solid review of the basics before undertaking the math series. It is truly an exceptionally well made course. July 2, 2014
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