Overview

Course No. 1001

Learn or review all the concepts of first-year algebra—including variables, order of operations, and functions—with these lectures taught by a Professor of Mathematics. Algebra I is one of the most critical courses that students take in high school and the gateway to higher education. Students who do well in algebra are better prepared for college entrance exams and for college in general. But it can be challenging for some students (and their parents). With 36 half-hour lessons, this course was developed to teach 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.

Professor James Sellers, an award-winning educator at The Pennsylvania State University, takes the fear out of learning algebra by approaching it with 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.

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. 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. Algebra is loved, not feared, by students of Dr. Sellers!

Course Structure

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.

Duration: 33 min

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.

Duration: 30 min

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.

Duration: 30 min

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).

Duration: 30 min

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.

Duration: 31 min

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.

Duration: 28 min

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.

Duration: 30 min

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.

Duration: 29 min

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.

Duration: 28 min

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.

Duration: 31 min

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.

Duration: 30 min

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.

Duration: 31 min

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?

Duration: 31 min

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.

Duration: 30 min

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.

Duration: 30 min

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.

Duration: 32 min

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.

Duration: 31 min

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.

Duration: 31 min

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.

Duration: 31 min

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.

Duration: 32 min

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.

Duration: 30 min

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.

Duration: 31 min

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.”

Duration: 29 min

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.

Duration: 32 min

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.

Duration: 31 min

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.

Duration: 31 min

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.

Duration: 30 min

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.

Duration: 30 min

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.

Duration: 32 min

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.

Duration: 31 min

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.

Duration: 32 min

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.

Duration: 32 min

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.

Duration: 32 min

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.

Duration: 32 min

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.

Duration: 32 min

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.

Duration: 33 min