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Algebra II

Algebra II

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

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Algebra II

Algebra II

Professor James A. Sellers Ph.D.
The Pennsylvania State University
Course No.  1002
Course No.  1002
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Course Overview

About This Course

36 lectures  |  31 minutes per lecture

Algebra II is the fork in the road. Those who succeed in this second part of the algebra sequence are well on their way to precalculus, calculus, and higher mathematics, which open the door to careers in science, engineering, medicine, economics, information technology, and many other fields. And since algebraic thinking is found in almost every sphere of modern life, a thorough grounding in this abstract discipline is essential for many nontechnical careers as well, from law to business to graphic arts.

Such benefits aside, Algebra II is a deeply rewarding subject in its own right that goes well beyond the rudiments of signed numbers, symbols, and simple equations learned in Algebra I. Indeed, the transition from Algebra I to Algebra II is like the leap from the first to the second year of a language, when you make your first steps toward genuine fluency. With the basic concepts firmly in place, you are ready to extend your skills in exciting new directions and to start to think mathematically.

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Algebra II is the fork in the road. Those who succeed in this second part of the algebra sequence are well on their way to precalculus, calculus, and higher mathematics, which open the door to careers in science, engineering, medicine, economics, information technology, and many other fields. And since algebraic thinking is found in almost every sphere of modern life, a thorough grounding in this abstract discipline is essential for many nontechnical careers as well, from law to business to graphic arts.

Such benefits aside, Algebra II is a deeply rewarding subject in its own right that goes well beyond the rudiments of signed numbers, symbols, and simple equations learned in Algebra I. Indeed, the transition from Algebra I to Algebra II is like the leap from the first to the second year of a language, when you make your first steps toward genuine fluency. With the basic concepts firmly in place, you are ready to extend your skills in exciting new directions and to start to think mathematically.

Therefore it is essential that you stay the course in your study of algebra. Among the advantages cited in Algebra II by award-winning Professor James A. Sellers of The Pennsylvania State University are these:

  • Perseverance in algebra pays off: Those who master algebra in high school are much more likely to succeed not just in college-level math courses, but in college in general.
  • Algebra is a valuable tool of reasoning: With countless daily uses that may not seem to be algebra problems, algebra comes in handy for everything from planning a party to organizing a trip to negotiating a loan.
  • Algebra is the foundation of calculus: Those skilled in algebra will find calculus more comprehensible—which makes all the difference in understanding a mathematical field that underlies physics, engineering, and so much more.

Through Professor Sellers's clear and inspiring instruction, Algebra II gives you the tools you need to thrive in a core skill of mathematics. In 36 engaging half-hour lessons, Professor Sellers walks you through hundreds of problems, showing every step in their solution and highlighting the most common missteps made by students.

A Course for Learners of All Ages

A gifted speaker and eloquent explainer of ideas, Professor Sellers shows that algebra can be an exciting intellectual adventure for any age and not nearly as difficult as many students fear. Those who will benefit from Professor Sellers's user-friendly approach include

  • high-school students currently enrolled in an Algebra II class and their parents, who seek an outstanding private tutor;
  • home-schooled students and others wishing to learn Algebra II on their own with these 18 hours of lessons and the accompanying mini-textbook;
  • college students struggling with math requirements and who need to strengthen their grasp of this fundamental subject;
  • math teachers searching for a better approach to Algebra II, guided by a professor who knows how to teach the subject;
  • summer learners who have completed Algebra I and want a head start on Algebra II;
  • anyone curious about the rigorous style of thought that underlies mathematics, the sciences, and our technological world.

Step Up to the Next Level

Taking your mathematics education to the next level, Algebra II starts by reviewing concepts from Algebra I and sharpening your problem-solving skills in linear and quadratic equations and other basic procedures. Professor Sellers begins with the simplest examples and gradually adds complexity to build confidence. As the course progresses, he introduces new topics, such as conic sections, roots and radicals, exponential and logarithmic functions, and elementary probability.

As you solve problems with Professor Sellers, you will see that the ideas behind algebra are wonderfully interconnected, that there are often several routes to a solution, and that concepts and procedures such as the following have a host of applications:

  • Functions: One of the simplest and most powerful ideas introduced in algebra is the function. Defined as a relation between two variables so that for any given input value there is exactly one output value, functions are used throughout higher mathematics.
  • Graphing: Professor Sellers notes that "algebra is much more than solving equations and manipulating algebraic expressions." By plotting an equation or a function as a graph, algebra's key properties often come sharply into focus.
  • Polynomials: By the time you meet the term "polynomial" in lecture 19, you will have dealt with dozens of these very useful expressions, including linear and quadratic equations. Professor Sellers shows how to perform complex operations on polynomials with ease.
  • Conic sections: Among algebra's countless links to the real world are conic sections, the class of curves formed by slicing a cone at different angles. These curves correspond to everything from planetary orbits to the shape of satellite TV dishes.
  • Roots and radicals: You are probably already familiar with square roots, but there are also cube roots, 4th roots, 5th roots, and so on. "Radical" comes from the Latin word for "root" and refers to symbols and operations involving roots.
  • Exponents and logarithms: Exponential growth and decay occur throughout nature and are modeled with exponential functions and their inverse, logarithmic functions. Like so many tools in algebra, the concepts are simple, their applications truly awe-inspiring.

With your growing mathematical maturity, you will learn to deploy an arsenal of formulas, theorems, and rules of thumb that provide a deeper understanding of patterns in algebra, while allowing you to analyze and solve equations more quickly than you imagined. Professor Sellers introduces these very useful techniques and more:

  • Vertical line test tells you instantly whether a graph represents a function.
  • Quadratic formula allows you to solve any quadratic equation, no matter how "messy."
  • Fundamental theorem of algebra specifies how many roots exist for a given polynomial.
  • Binomial theorem gives you the key to the coefficients for a binomial of any power.
  • Change of base formula permits you to use a calculator to evaluate logarithms that are not in base 10 or e.
  • "Pert" formula applies algebra to the real-world problem of calculating continuously compounded interest.

Professor Sellers ends the course with three entertaining lectures showing how to solve problems in combinatorics and probability, which have applications in some intriguing areas, whether you need to calculate the possible outcomes in a match of five contestants, the potential three-topping pizzas when there are eight toppings to choose from, or the probability of being dealt different hands in poker.

Dispel the Fog of Confusion!

Practically everyone who has taken algebra has spent time in "the fog," when new ideas just don't make sense. As a winner of the Teresa Cohen Mathematics Service Award from The Pennsylvania State University, Professor Sellers is unusually adept at dispelling the fog.

He does this by explaining the same concept in a variety of insightful ways, by carefully choosing problems that build on each other incrementally, and through his years of experience in addressing areas where students have the most trouble. Whenever the going gets tough, he shows you the path through to a solution and then makes doubly sure that you know the way.

This sense of ease and adventure in tackling the richly varied terrain of algebra characterizes the experience you will have with this superstar teacher and Algebra II. You will learn to solve problems that look impossible at first glance, find that you enjoy them more than you ever thought possible, and look forward to even more challenging exploits as you continue your mathematics education.

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36 Lectures
  • 1
    An Introduction to Algebra II
    Professor Sellers explains the topics covered in the course, the importance of algebra, and how you can get the most out of these lessons. You then launch into the fundamentals of algebra by reviewing the order of operations and trying your hand at several problems. x
  • 2
    Solving Linear Equations
    Explore linear equations, starting with one-step equations and then advancing to those requiring two or more steps to solve. Next, apply the distributive property to simplify certain problems, and then learn about the three categories of linear equations. x
  • 3
    Solving Equations Involving Absolute Values
    Taking your knowledge of linear equations a step further, look at examples involving absolute values, which can be thought of as a distance on a number line, always expressed as a positive value. Use your critical-thinking skills to recognize absolute value problems that have limited or no solutions. x
  • 4
    Linear Equations and Functions
    Moving into the visual realm, learn how linear equations are represented as straight lines on graphs using either the slope-intercept or point-slope forms of the function. Next, investigate parallel and perpendicular lines and how to identify them by the value of their slopes. x
  • 5
    Graphing Essentials
    Reversing the procedure from the previous lesson, start with an equation and draw the line that corresponds to it. Then test your knowledge by matching four linear equations to their graphs. Finally, learn how to rewrite an equation to move its graph up, down, left, or right—or flip it entirely. x
  • 6
    Functions—Introduction, Examples, Terminology
    Functions are crucially important not only for algebra, but for precalculus, calculus, and higher mathematics. Learn the definition of a function, the notation, and associated concepts such as domain and range. Then try out the vertical line test for determining whether a given curve is a graph of a function. x
  • 7
    Systems of 2 Linear Equations, Part 1
    Practice solving systems of two linear equations by graphing the corresponding lines and looking for the intersection point. Discover that there are three possible outcomes: no solution, infinitely many solutions, and exactly one solution. x
  • 8
    Systems of 2 Linear Equations, Part 2
    Explore two other techniques for solving systems of two linear equations. First, the method of substitution solves one of the equations and substitutes the result into the other. Second, the method of elimination adds or subtracts the equations to see if a variable can be eliminated. x
  • 9
    Systems of 3 Linear Equations
    As the number of variables increases, it becomes unwieldy to solve systems of linear equations by graphing. Learn that these problems are not as hard as they look and that systems of three linear equations often yield to the strategy of successively eliminating variables. x
  • 10
    Solving Systems of Linear Inequalities
    Make the leap into systems of linear inequalities, where the solution is a set of values on one side or another of a graphed line. An inequality is an assertion such as "less than" or "greater than," which encompasses a range of values. x
  • 11
    An Introduction to Quadratic Functions
    Begin your investigation of quadratic functions by visualizing what these functions look like when graphed. They always form a U-shaped curve called a parabola, whose location on the coordinate plane can be predicted based on the individual terms of the equation. x
  • 12
    Quadratic Equations—Factoring
    One of the most important skills related to quadratics is factoring. Review the basics of factoring, and learn to recognize a very useful special case known as the difference of two squares. Close by working on a word problem that translates into a quadratic equation. x
  • 13
    Quadratic Equations—Square Roots
    The square root approach to solving quadratic equations works not just for perfect squares, such as 3 × 3 = 9, but also for values that don't seem to involve squares at all. Probe the idea behind this technique, and also venture into the strange world of complex numbers. x
  • 14
    Completing the Square
    Turn a quadratic equation into an easily solvable form that includes a perfect square—a technique called completing the square. An important benefit of this approach is that the rewritten form gives the coordinates for the vertex of the parabola represented by the equation. x
  • 15
    Using the Quadratic Formula
    When other approaches fail, one tool can solve every quadratic equation: the quadratic formula. Practice this formula on a wide range of problems, learning how a special expression called the discriminant immediately tells how many real-number solutions the equation has. x
  • 16
    Solving Quadratic Inequalities
    Extending the exercises on inequalities from lecture 10, step into the realm of quadratic inequalities, where the boundary graph is not a straight line but a parabola. Use your skills analyzing quadratic expressions to sketch graphs quickly and solve systems of quadratic inequalities. x
  • 17
    Conic Sections—Parabolas and Hyperbolas
    Delve into the algebra of conic sections, which are the cross-sectional shapes produced by slicing a cone at different angles. In this lesson, study parabolas and hyperbolas, which differ in how many variable terms are squared in each. Also learn how to sketch a hyperbola from its equation. x
  • 18
    Conic Sections—Circles and Ellipses
    Investigate the algebraic properties of the other two conic sections: ellipses and circles. Ellipses resemble stretched circles and are defined by their major and minor axes, whose ratio determines the ellipse's eccentricity. Circles are ellipses whose eccentricity = 1, with the major and minor axes equal. x
  • 19
    An Introduction to Polynomials
    Pause to examine the nature of polynomials—a class of algebraic expressions that you've been working with since the beginning of the course. Professor Sellers introduces several useful concepts, such as the standard form of polynomials and their degree, domain, range, and leading coefficients. x
  • 20
    Graphing Polynomial Functions
    Deepen your insight into polynomial functions by graphing them to see how they differ from non-polynomials. Then learn how the general shape of the graph can be predicted from the highest exponent of the polynomial, known as its degree. Finally, explore how other terms in the function also affect the graph. x
  • 21
    Combining Polynomials
    Switch from graphs to the algebraic side of polynomial functions, learning how to combine them in many different ways, including addition, subtraction, multiplication, and even long division, which is easier than it seems. Discover which of these operations produce new polynomials and which do not. x
  • 22
    Solving Special Polynomial Equations
    Learn how to solve polynomial equations where the degree is greater than two by turning them into expressions you already know how to handle. Your "toolbox" includes techniques called the difference of two squares, the difference of two cubes, and the sum of two cubes. x
  • 23
    Rational Roots of Polynomial Equations
    Going beyond the approaches you've learned so far, discover how to solve polynomial equations by applying two powerful tools for finding rational roots: the rational roots theorem and the factor theorem. Both will prove very useful in succeeding lessons. x
  • 24
    The Fundamental Theorem of Algebra
    Explore two additional tools for identifying the roots of polynomial equations: Descartes' rule of signs, which narrows down the number of possible positive and negative real roots; and the fundamental theorem of algebra, which gives the total of all roots for a given polynomial. x
  • 25
    Roots and Radical Expressions
    Shift gears away from polynomials to focus on expressions involving roots, including square roots, cube roots, and roots of higher degrees—all known as radical expressions. Practice multiplying, dividing, adding, and subtracting a wide variety of radical expressions. x
  • 26
    Solving Equations Involving Radicals
    Drawing on your experience with roots and radicals from the previous lesson, try your hand at solving equations with these expressions. Begin by learning how to manipulate rational, or fractional, exponents. Then practice with simple equations, while being on the lookout for extraneous, or "imposter," solutions. x
  • 27
    Graphing Power, Radical, and Root Functions
    Using graph paper, experiment with curves formed by simple radical functions. First, determine the domain of the function, which tells you the general location of the graph on the coordinate plane. Then, investigate how different terms in the function alter the graph in predictable ways. x
  • 28
    An Introduction to Rational Functions
    Shift your focus to graphs of rational functions—functions that are the ratio of two polynomials. These graphs are more complicated than those from the previous lesson, but their general characteristics can be quickly determined by calculating the domain, the x- and y-intercepts, and the vertical and horizontal asymptotes. x
  • 29
    The Algebra of Rational Functions
    Combine rational functions using addition, subtraction, multiplication, division, and composition. The trick is to start each problem by putting the expressions in factored form, which makes the calculations go more smoothly. Leaving the answer in factored form also allows other operations, such as graphing, to be easily performed. x
  • 30
    Partial Fractions
    Now that you know how to add rational expressions, try the opposite procedure of splitting a more complicated rational expression into its component parts. Called partial fraction decomposition, this approach is a topic in introductory calculus and is used for solving a wide range of more advanced math problems. x
  • 31
    An Introduction to Exponential Functions
    Exponential functions are important in real-world applications involving growth and decay rates, such as compound interest and depreciation. Experiment with simple exponential functions, exploring such concepts as the base, growth factor, and decay factor, and how different values for these terms affect the graph of the function. x
  • 32
    An Introduction to Logarithmic Functions
    Plot a logarithmic function on the coordinate plane to see how it is the mirror image of a corresponding exponential function. Just like a mirror image, logarithms can be disorienting at first; but by studying their properties you will discover how they make certain calculations much simpler. x
  • 33
    Uses of Exponential and Logarithmic Functions
    Delve deeper into exponential and logarithmic functions with the goal of solving a typical financial investment problem using the "Pert" formula. To prepare, study the change of base formula for logarithms and the special function of the base called e. x
  • 34
    The Binomial Theorem
    Pascal's triangle is a famous triangular array of numbers that corresponds to the coefficients of binomials of different powers. In a lesson connecting a branch of mathematics called combinatorics with algebra, investigate the formula for each value in Pascal's triangle, the factorial function, and the binomial theorem. x
  • 35
    Permutations and Combinations
    Continue your study of the link between combinatorics and algebra by using the factorial function to solve problems in permutations and combinations. For example, what are all the permutations of the letters a, b, c? And how many combinations of four books are possible when you have six to choose from? x
  • 36
    Elementary Probability
    After a short introduction to probability, celebrate your completion of the course with a deck of cards. Can you use the principles of probability, permutations, and combinations to calculate the probability of being dealt different hands? As with the rest of algebra, once you know the rules, it's simplicity itself! x

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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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Rated 4.4 out of 5 by 19 reviewers.
Rated 5 out of 5 by The Most Important Math Course You Will Ever Take Let me begin by stating I'm a retired mathematics instructor who had a very successful and rewarding career of 39 years and I can categorically state, 'Algebra II is the most important math course you will ever take if pursuing a career in mathematics or science.' I encountered numerous situations where a frustrated student understood the concept I was presenting in Precalculus or Calculus but could not perform the necessary algebraic steps in order to solve the problem. I spent countless hours reviewing and reteaching topics from Algebra in order to get students up to speed. Professor Sellers does an excellent job in presenting the topics necessary in order to succeed in higher mathematics. His logical explanations and carefully chosen examples progressively lead the student to the desired end. Dr. Sellers' 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 increase your chances of success in this critically-important subject. December 15, 2011
Rated 5 out of 5 by Excellent Course! Outstanding Professor! With Algebra I to set the foundation, this is an excellent course! It builds on that first course, adds complexity to the problems, and then teaches brand new concepts in Dr. Sellers outstanding way. He is never boring to listen to, makes learning fun and enjoyable while teaching these sometimes complex ideas. He is so good at explaining concepts, that I can go right to the workbook and complete the problems successfully, even though I have usually struggled with math in the past. An Excellent Course, an Outstanding Professor! March 26, 2015
Rated 1 out of 5 by II ? What's " II" about it? I don't necessarily have a problem with the content or presentation, what I have a problem with is misrepresentation. After watching Algebra I , I bought" II" thinking it was going to keep going where " I" left off. But to my dismay, the content is the same. He even explains it as if you have never seen that material before. I don't understand why they made this course and why no one that has reviewed this course has mentioned this. Might as well view Algebra I all over again if you need a review and save the money. November 18, 2014
Rated 2 out of 5 by TTC should be embarrassed. I was going to give this course as a gift to a friend's daughter that could use some help with Algebra but, I will be looking at a different product as I do not find this one adequate. Specifically, this course does not encourage the viewer to think mathematically and logically. It promotes memorization and the mechanical application of rules without much thought. It fails to teach the viewer that a complex problem is solved by breaking a problem into simpler parts. I understand the statement above is harsh and for that reason it should be supported with facts. I cannot pick apart the entire course in a review. Additionally, after lecture 12, I had seen enough. Here are a couple of example that led me to my unfavorable conclusion: In the second lecture which covers Solving Linear Equations, Mr. Sellers solves the equation 5x + 4 = -11 by first getting rid of the 4 (a good choice) but, he justifies that choice by stating that the reason to get rid of the 4 first is, in Mr. Sellers own words, "believe it or not, the answer to that question is wrapped up in order of operations". That assertion is dismal, order of operations has absolutely nothing to do with choosing to deal with the sum first. Dealing with the sum first is done because it is the easiest way to simplify the equation, the multiplier 5, can then be dealt with in the simpler resulting equation. As anyone with a minimum knowledge of mathematics and, algebra in particular knows, the order in which the operations are performed will not alter the result (at least not when dealing with the basic operators.) The equation could be solved by dealing with the multiplier first and the sum afterwards. Doing it that way is just more work, therefore choosing to deal with the sum first is strictly based on solving the problem the easiest and simplest way. That is what Mr. Sellers should have made clear to the viewer, seek to simplify. That is just one example, there are quite a few instances where the justification given by Mr. Sellers for the steps chosen is very questionable and arguably incorrect as in the case above. Mr. Sellers harps on the viewer that the answers obtained should be verified. Mr. Sellers should take his own advice. In lecture 12, Quadratic Equations - Factoring, Mr. Sellers obtains an obviously incorrect result to a trivial problem and, completely fails to see that the answer is wrong. He should have verified his answer. Specifically, at the end of lecture 12, Mr. Sellers poses the problem of a rectangular garden that is 10 units long by 8 units wide giving an area of 80 units and, asks the viewer to calculate the increment "X" necessary on all sides (4 of them of course) to increase the area to 88. The resolution presented of this trivial problem cannot be characterized as anything but appalling. The reasoning is incorrect, the equations are incorrect and the result is, of course, incorrect. His solution states that, the length and the width on all sides must be increased by 2. This is obviously wrong. If 10 x 8 = 80, then 14 * 12 is obviously WAY more than 88 (11 * 8 would yield 88). Since 11 * 8 yields the desired 88 with an increase of just 1 unit on one side, a quick and dirty approximation to the result is given by X = 1/4 (since the increment occurs on the four sides of the rectangle). Given the quadratic nature of the problem, X is then known to be less than 1/4. The value of 2 yields 14 * 12, just 14 * 10 would make it obvious that an increment of 2 on each side is incorrect. The correct answer is X = 0.217 (approx). The correct equation (not the one he presented) to solve the problem is (10 + 2x) * (8 + 2x) = 88 I cannot give any examples past lecture 12 as, at that point, I stopped watching. I will be looking for an Algebra product that promotes and develops a feel for Algebra in the viewer. TTC should rework this course. Also, the moving shades in the background are distracting. The blue background is nice but movement only serves to disrupt the attention of the viewer. Consider a static background. July 8, 2014
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