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Algebra I

Algebra I

Professor James A. Sellers Ph.D.
The Pennsylvania State University

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Algebra I

Algebra I

Professor James A. Sellers Ph.D.
The Pennsylvania State University
Course No.  1001
Course No.  1001
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Course Overview

About This Course

36 lectures  |  30 minutes per lecture

Algebra I is one of the most critical courses that students take in high school. Not only does it introduce them to a powerful reasoning tool with applications in many different careers, but algebra is the gateway to higher education. Students who do well in algebra are better prepared for college entrance exams and for college in general, since algebra teaches them how to solve problems and think abstractly—skills that pay off no matter what major they pursue.

Because algebra involves a new way of thinking, many students find it especially challenging. Many parents also find it to be the area where they have the most trouble helping their high-school-age children. With 36 half-hour lessons,

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Algebra I is one of the most critical courses that students take in high school. Not only does it introduce them to a powerful reasoning tool with applications in many different careers, but algebra is the gateway to higher education. Students who do well in algebra are better prepared for college entrance exams and for college in general, since algebra teaches them how to solve problems and think abstractly—skills that pay off no matter what major they pursue.

Because algebra involves a new way of thinking, many students find it especially challenging. Many parents also find it to be the area where they have the most trouble helping their high-school-age children. With 36 half-hour lessons, Algebra I is an entirely new course developed to meet both these concerns, teaching students and parents the concepts and procedures of first-year algebra in an easily accessible way. Indeed, anyone wanting to learn algebra from the beginning or needing a thorough review will find this course an ideal tutor.

Conquer the Challenges of Learning Algebra

Taught by Professor James A. Sellers, an award-winning educator at The Pennsylvania State University, Algebra I incorporates the following valuable features:

  • Drawing on extensive research, The Great Courses and Dr. Sellers have identified the biggest challenges for high school students in mastering Algebra I, which are specifically addressed in this course.
  • This course reflects the latest standards and emphases in high school and college algebra taught in the United States.
  • Algebra I includes a mini-textbook with detailed summaries of each lesson, a multitude of additional problems to supplement those presented in the on-screen lessons, guided instructions for solving the problems, and important formulas and definitions of terms.
  • Professor Sellers interacts with viewers in a one-on-one manner, carefully explaining every step in the solution to a problem and giving frequent tips, problem-solving strategies, and insights into areas where students have the most trouble.

As Director of Undergraduate Mathematics at Penn State, Professor Sellers appreciates the key role that algebra plays in preparing students for higher education. He understands what entering college students need to have mastered in terms of math preparation to launch themselves successfully on their undergraduate careers, whether they intend to take more math in college or not. Professor Sellers is alert to the math deficiencies of the typical entering high school graduate, and he has developed an effective strategy for putting students confidently on the road to college-level mathematics.

Whatever your age, it is well worth the trouble to master this subject. Algebra is indispensible for those embarking on careers in science, engineering, information technology, and higher mathematics, but it is also a fundamental reasoning tool that shows up in economics, architecture, publishing, graphic arts, public policy, manufacturing, insurance, and many other fields, as well as in a host of at-home activities such as planning a budget, altering a recipe, calculating car mileage, painting a room, planting a garden, building a patio, or comparison shopping.

And for all of its reputation as a grueling rite of passage, algebra is actually an enjoyable and fascinating subject—when taught well.

Algebra without Fear

Professor Sellers takes the fear out of learning algebra by approaching it in a friendly and reassuring spirit. Most students won't have a teacher as unhurried and as attentive to detail as Dr. Sellers, who explains everything clearly and, whenever possible, in more than one way so that the most important concepts sink in.

He starts with a review of fractions, decimals, percents, positive and negative numbers, and numbers raised to various powers, showing how to perform different operations on these values. Then he introduces variables as the building blocks of algebraic expressions, before moving on to the main ideas, terms, techniques, pitfalls, formulas, and strategies for success in tackling Algebra I. Throughout, he presents a carefully crafted series of gradually more challenging problems, building the student's confidence and mastery.

After taking this course, students will be familiar with the terminology and symbolic nature of first-year algebra and will understand how to represent various types of functions (linear, quadratic, rational, and radical) using algebraic rules, tables of data, and graphs. In the process, they will also become acquainted with the types of problems that can be solved using such functions, with a particular eye toward solving various types of equations and inequalities.

Throughout the course, Professor Sellers emphasizes the following skills:

  • Using multiple techniques to solve problems
  • Understanding when a given technique can be used
  • Knowing how to translate word problems into mathematical expressions
  • Recognizing numerical patterns
Tips for Success

Algebra is a rich and complex subject, in which seemingly insurmountable obstacles can be overcome, often with ease, if one knows how to approach them. Professor Sellers is an experienced guide in this terrain and a treasure trove of practical advice—from the simple (make sure that you master the basics of addition, subtraction, multiplication, and division) to the more demanding (memorize the algebraic formulas that you use most often). Here are some other examples of his tips for success:

  • Learn the order of operations: These are the rules you follow when performing mathematical operations. You can remember the order with this sentence: Please Excuse My Dear Aunt Sally. The first letter of each word stands for an operation. First, do all work in parentheses; then the exponents; then multiplication and division; finally, do the addition and subtraction.
  • Know your variables: It's easy to make a mistake when writing an algebraic expression if you don't understand what each variable represents. Choose letters that you can remember; for example, d for distance and t for time. If you have sloppy handwriting, avoid letters that look like numbers (b, l, o, s, and z).
  • Use graph paper: You'll be surprised at how the grid of lines encourages you to organize your thinking. The columns and rows help you keep your work neat and easy to follow.
  • Pay attention to signs: Be very careful of positive and negative signs. A misplaced plus or minus sign will give you the wrong answer.
  • Don't mix units: If you are using seconds and are given a time in minutes, make sure to convert the units so they are all the same.
  • Simplify: Straighten out the clutter in an equation by putting like terms together. Constants, such as 7, -2, 28, group together, as do terms with the same variable, such as 3x, x, -10x. Then combine the like terms. Often you'll find that the equation practically solves itself.
  • Balance the equation: When you perform an operation on one side of an equation—such as adding or subtracting a number, or multiplying or dividing the entire side by a quantity—do the exact same thing to the other side. This keeps things in balance.
  • Above all, check your work! When you have finished a problem, ask yourself, "Does this answer make sense?" Plug your solution into the original equation to see if it does. Checking your work is the number one insurance policy for accurate work—the step that separates good students from superstar students.

By developing habits such as these, you will discover that solving algebra problems becomes a pleasure and not a chore—just as in a sport in which you have mastered the rudiments and are ready to face a competitor. Algebra I gives you the inspirational instruction, repetition, and practice to excel at what for many students is the most dreaded course in high school. Open yourself to the world of opportunity that algebra offers by making the best possible start on this all-important subject.

View Less
36 Lectures
  • 1
    An Introduction to the Course
    Professor Sellers introduces the general topics and themes for the course, describing his approach and recommending a strategy for making the best use of the lessons and supplementary workbook. Warm up with some simple problems that demonstrate signed numbers and operations. x
  • 2
    Order of Operations
    The order in which you do simple operations of arithmetic can make a big difference. Learn how to solve problems that combine adding, subtracting, multiplying, and dividing, as well as raising numbers to various powers. These same concepts also apply when you need to simplify algebraic expressions, making it critical to master them now. x
  • 3
    Percents, Decimals, and Fractions
    Continue your study of math fundamentals by exploring various procedures for converting between percents, decimals, and fractions. Professor Sellers notes that it helps to see these procedures as ways of presenting the same information in different forms. x
  • 4
    Variables and Algebraic Expressions
    Advance to the next level of problem solving by using variables as the building blocks to create algebraic expressions, which are combinations of mathematical symbols that might include numbers, variables, and operation symbols. Also learn some tricks for translating the language of problems (phrases in English) into the language of math (algebraic expressions). x
  • 5
    Operations and Expressions
    Discover that by following basic rules on how to treat coefficients and exponents, you can reduce very complicated algebraic expressions to much simpler ones. You start by using the commutative property of multiplication to rearrange the terms of an expression, making combining them relatively easy. x
  • 6
    Principles of Graphing in 2 Dimensions
    Using graph paper and pencil, begin your exploration of the coordinate plane, also known as the Cartesian plane. Learn how to plot points in the four quadrants of the plane, how to choose a scale for labeling the x and y axes, and how to graph a linear equation. x
  • 7
    Solving Linear Equations, Part 1
    In this lesson, work through simple one- and two-step linear equations, learning how to isolate the variable by different operations. Professor Sellers also presents a word problem involving a two-step equation and gives tips for how to solve it. x
  • 8
    Solving Linear Equations, Part 2
    Investigating more complicated examples of linear equations, learn that linear equations fall into three categories. First, the equation might have exactly one solution. Second, it might have no solutions at all. Third, it might be an identity, which means every number is a solution. x
  • 9
    Slope of a Line
    Explore the concept of slope, which for a given straight line is its rate of change, defined as the rise over run. Learn the formula for calculating slope with coordinates only, and what it means to have a positive, negative, and undefined slope. x
  • 10
    Graphing Linear Equations, Part 1
    Use what you've learned about slope to graph linear equations in the slope-intercept form, y = mx + b, where m is the slope, and b is the y intercept. Experiment with examples in which you calculate the equation from a graph and from a table of pairs of points. x
  • 11
    Graphing Linear Equations, Part 2
    A more versatile approach to writing the equation of a line is the point-slope form, in which only two points are required, and neither needs to intercept the y axis. Work through several examples and become comfortable determining the equation using the line and the line using the equation x
  • 12
    Parallel and Perpendicular Lines
    Apply what you've discovered about equations of lines to two very special types of lines: parallel and perpendicular. Learn how to tell if lines are parallel or perpendicular from their equations alone, without having to see the lines themselves. Also try your hand at word problems that feature both types of lines. x
  • 13
    Solving Word Problems with Linear Equations
    Linear equations reflect the behavior of real-life phenomena. Practice evaluating tables of numbers to determine if they can be represented as linear equations. Conclude with an example about the yearly growth of a tree. Does it increase in size at a linear rate? x
  • 14
    Linear Equations for Real-World Data
    Investigating more real-world applications of linear equations, derive the formula for converting degrees Celsius to Fahrenheit; determine the boiling point of water in Denver, Colorado; and calculate the speed of a rising balloon and the time for an elevator to descend to the ground floor. x
  • 15
    Systems of Linear Equations, Part 1
    When two lines intersect, they form a system of linear equations. Discover two methods for finding a solution to such a system: by graphing and by substitution. Then try out a real-world example, involving a farmer who wants to plant different crops in different proportions. x
  • 16
    Systems of Linear Equations, Part 2
    Expand your tools for solving systems of linear equations by exploring the method of solving by elimination. This technique allows you to eliminate one variable by performing addition, subtraction, or multiplication on both sides of an equation, allowing a straightforward solution for the remaining variable. x
  • 17
    Linear Inequalities
    Shift gears to consider linear inequalities, which are mathematical expressions featuring a less than sign or a greater than sign instead of an equal sign. Discover that these kinds of problems have some very interesting twists, and they come up frequently in business applications. x
  • 18
    An Introduction to Quadratic Polynomials
    Transition to a more complex type of algebraic expression, which incorporates squared terms and is therefore known as quadratic. Learn how to use the FOIL method (first, outer, inner, last) to multiply linear terms to get a quadratic expression. x
  • 19
    Factoring Trinomials
    Begin to find solutions for quadratic equations, starting with the FOIL technique in reverse to find the binomial factors of a quadratic trinomial (a binomial expression consists of two terms, a trinomial of three). Professor Sellers explains the tricks of factoring such expressions, which is a process almost like solving a mystery. x
  • 20
    Quadratic Equations—Factoring
    In some circumstances, quadratic expressions are given in a special form that allows them to be factored quickly. Focus on two such forms: perfect square trinomials and differences of two squares. Learning to recognize these cases makes factoring easy. x
  • 21
    Quadratic Equations—The Quadratic Formula
    For those cases that defy simple factoring, the quadratic formula provides a powerful technique for solving quadratic equations. Discover that this formidable-looking expression is not as difficult as it appears and is well worth committing to memory. Also learn how to determine if a quadratic equation has no solutions. x
  • 22
    Quadratic Equations—Completing the Square
    After learning the definition of a function, investigate an additional approach to solving quadratic equations: completing the square. This technique is very useful when rewriting the equation of a quadratic function in such a way that the graph of the function is easily sketched. x
  • 23
    Representations of Quadratic Functions
    Drawing on your experience solving quadratic functions, analyze the parabolic shapes produced by such functions when represented on a graph. Use your algebraic skills to determine the parabola's vertex, its x and y intercepts, and whether it opens in an upward "cup" or downward in a "cap." x
  • 24
    Quadratic Equations in the Real World
    Quadratic functions often arise in real-world settings. Explore a number of problems, including calculating the maximum height of a rocket and determining how long an object dropped from a tree takes to reach the ground. Learn that in finding a solution, graphing can often help. x
  • 25
    The Pythagorean Theorem
    Because it involves terms raised to the second power, the famous Pythagorean theorem, a2 + b2 = c2, is actually a quadratic equation. Discover how techniques you have previously learned for analyzing quadratic functions can be used for solving problems involving right triangles. x
  • 26
    Polynomials of Higher Degree
    Most of the expressions you've studied in the course so far have been polynomials. Learn what characterizes a polynomial and how to recognize polynomials in both algebraic functions and in graphical form. Professor Sellers defines several terms, including the degree of an equation, the leading coefficient, and the domain. x
  • 27
    Operations and Polynomials
    Much of what you've learned about linear and quadratic expressions applies to adding, subtracting, multiplying, and dividing polynomials. Discover how the FOIL operation can be extended to multiplying large polynomials, and a version of long division works for dividing one polynomial by another. x
  • 28
    Rational Expressions, Part 1
    When one polynomial is divided by another, the result is called a rational function because it is the ratio of two polynomials. These functions play an important role in algebra. Learn how to add and subtract rational functions by first finding their common divisor. x
  • 29
    Rational Expressions, Part 2
    Continuing your exploration of rational expressions, try your hand at multiplying and dividing them. The key to solving these complicated-looking equations is to proceed one step at a time. Close the lesson with a problem that brings together all you've learned about rational functions. x
  • 30
    Graphing Rational Functions, Part 1
    Examine the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that aren't even connected on a graph. Learn to identify the intercepts and the vertical and horizontal asymptotes of these fascinating curves. x
  • 31
    Graphing Rational Functions, Part 2
    Sketch the graphs of several rational functions by first calculating the vertical and horizontal asymptotes, the x and y intercepts, and then plotting several points in the function. In the final exercise, you must simplify the expression in order to extract the needed information. x
  • 32
    Radical Expressions
    Anytime you see a root symbol—for example, the symbol for a square root—then you're dealing with what mathematicians call a radical. Learn how to simplify radical expressions and perform operations on them, such as multiplication, division, addition, and subtraction, as well as combinations of these operations. x
  • 33
    Solving Radical Equations
    Discover how to solve equations that contain radical expressions. A key step is isolating the radical term and then squaring both sides. As always, it's important to check the solution by plugging it into the equation to see if it makes sense. This is especially true with radical equations, which can sometimes yield extraneous, or invalid, solutions. x
  • 34
    Graphing Radical Functions
    In previous lessons, you moved from linear, quadratic, and rational functions to the graphs that display them. Now do the same with radical functions. For these, it's important to pay attention to the domain of the functions to ensure that negative values are not introduced beneath the root symbol. x
  • 35
    Sequences and Pattern Recognition, Part 1
    Pattern recognition is an important and fascinating mathematical skill. Investigate two types of number patterns: geometric sequences and arithmetic sequences. Learn how to analyze such patterns and work out a formula that predicts any term in the sequence x
  • 36
    Sequences and Pattern Recognition, Part 2
    Conclude the course by examining more types of number sequences, discovering how rich and enjoyable the mathematics of pattern recognition can be. As in previous lessons, employ your reasoning skills and growing command of algebra to find order—and beauty—where once all was a confusion of numbers. x

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James A. Sellers
James A. Sellers, Ph.D.
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 52 reviewers.
Rated 5 out of 5 by Remarkable Course!!! I have wanted to go back and review basic Algebra for many years, but life (you know how it is) sometimes makes that proposition difficult. Well, I finally stopped procrastinating and took a chance on Prof. Sellers. With a Ph.D. of my own and many miles on the Odometer I was amazed to find someone with his patience and clarity. I've spent my fair share of time in a classroom, but Dr. Sellers is unique. There are other Algebra courses in the Great Courses, but this is the one you want. --It's his patience and clarity that sets him apart. In fact, have re-watched some lessons just to study his pedagogy! It's really that good. Where was he when I was being bludgeoned by my Math teachers (in High School and College)?!? Oh, probably not born... :) August 15, 2015
Rated 5 out of 5 by I can not find that all the points in problem 10 lecture 10 are on the same line August 6, 2015
Rated 5 out of 5 by Just Wonderful I initially bought this course about 4 years ago or so, just before my older son was scheduled to start Algebra in school. I was a very strong Math student as a kid, but I hadn't seen this stuff in so many years. I barely remembered a thing. A fate would have it, I never got around to watching the lectures at that time, but my son actually did! He'd come home from school every day and watch a new lecture. About a month after he finished, his school came to the realization that they needed to skip him ahead a grade in Mathematics because he already had this information down cold! He's now a sophomore in High School and is taking BC Calculus, planning on taking Multivariable Calculus next year. Well, my younger son is now getting into Algebra, so I decided to sit down and go through this course myself. I get it now. Dr. Sellers is about as clear as he can be. He gives solid examples, then walks you through them at a very good pace. I never felt like I was being pushed too fast, but nor did I ever feel like things were dragging. I'm about to start Algebra 2 now & I honestly can't wait. I cannot recommend this course enough to anyone who wants to learn Algebra, teen or adult alike. March 25, 2015
Rated 5 out of 5 by ALGEBRA 1 This is by far the best Algebra 1 math course I have ever taken. This course breaks the math concepts down so it is easy to understand for non math people like me. I am an Adult Age student and wanted to "refresh" my Algebra skills, because you do actually use Algebra in everyday life. The course is taught by Professor James A. Sellers, who I wish was my math teacher back when I was in high school. The problem today is that math is taught so poorly in our schools, which is why American kids don't like math period. When I was in school as a young man I was terrible at math. These courses in Algebra have taught me to love math and embrace it. When you become an adult, you realize how important math is or you will not get a good job. These mathematical courses offered are excellent, and I would highly recommend them. I will be taking the Calculus course next. If you are afraid of math or Algebra, take this course. This course will give you an excellent foundation to go into higher math. Math is fun. March 21, 2015
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