Algebra I

Professor Sellers introduces the general topics and themes; describing his approach; a strategy for making the best use of the lessons and supplementary workbook; warming up with simple problems that demonstrate signed numbers and operations.

The order in which one does simple operations of arithmetic can make a big difference; how to solve problems that combine adding, subtracting, multiplying and dividing; raising numbers to various powers; simplifying algebraic expressions.

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.

Using variables as building blocks to create algebraic expressions, which are combinations of mathematical symbols that might include numbers, variables and operation symbols; tricks for translating the language of problems into the language of math.

By following basic rules on how to treat coefficients and exponents, one can simplify complicated algebraic expressions; using the commutative property of multiplication to rearrange the terms of an expression, making combining them relatively easy.

Exploring the coordinate plane, also known as the Cartesian plane, using graph paper and a pencil; how to plot points in the four quadrants of the plane; how to choose a scale for labeling the x and y axes; how to graph a linear equation.

Working 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 on how to solve it.

Investigating more complicated examples of linear equations; learning 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.

Exploring the concept of a slope, which for a given straight line is its rate of change, defined as the rise over run; the formula for calculating a slope with coordinates only; what it means to have a positive, negative and undefined slope.

Graphing linear equations in the slope-intercept form, y = mx + b, where m is the slope and b is the y intercept; experimenting with examples in which one calculates the equation from a graph and from a table of pairs of points.

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; several examples; determining the equation using the line, and vice versa.

Applying equations of lines to parallel and perpendicular lines; how to tell if lines are parallel or perpendicular from their equations alone, without having to see the lines themselves; word problems that feature both types of lines.

Linear equations reflect the behavior of real-life phenomena; practicing evaluating tables of numbers to determine if they can be represented as linear equations; concluding with an example about the yearly growth of a tree.

Investigating more real-world applications of linear equations; deriving the formula for converting degrees Celsius to Fahrenheit; determining the boiling point of water in Denver, Colorado; calculating the speed and time of various objects.

When two lines intersect, they form a system of linear equations; discovering two methods for finding a solution to such a system, which are by graphing and by substitution; a real-world example involving a farmer who wants to plant different crops.

Expanding the tools for solving systems of linear equations by exploring the method of solving by elimination; this technique allows people to eliminate one variable by performing addition, subtraction or multiplication on both sides of an equation.

Considering linear inequalities, which are mathematical expressions featuring a less-than sign or a greater-than sign instead of an equal sign; these kinds of problems have interesting twists and come up frequently in business applications.

Transitioning to a more complex type of algebraic expression, which incorporates squared terms and is therefore known as quadratic; how to use the FOIL method to multiply linear terms to get a quadratic expression.

Finding solutions for quadratic equations; starting with the FOIL technique in reverse to find the binomial factors of a quadratic trinomial; Professor Sellers explains the tricks of factoring such expressions, which is almost like solving a mystery.

In some circumstances, quadratic expressions are given in a special form that allows them to be factored quickly; focusing on two such forms, which are perfect square trinomials and the differences of two squares; learning to recognize these cases.

For cases that defy simple factoring, the quadratic formula provides a technique for solving quadratic equations; discovering that this expression is not as difficult as it appears to be; how to determine if a quadratic equation has no solutions.

Investigating an additional approach to solving quadratic equations, which is 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.

Analyzing the parabolic shapes produced by quadratic functions when represented on a graph; using 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".

Quadratic functions often arise in real-world settings; exploring a number of problems, which include calculating the maximum height of a rocket; learning that graphing can often help in finding a solution.

Because it involves terms raised to the second power, the famous Pythagorean theorem is actually a quadratic equation; how techniques for analyzing quadratic functions can be used to solve problems involving right triangles.

What characterizes a polynomial; 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.

The role of linear and quadratic expressions in adding, subtracting, multiplying and dividing polynomials; how the FOIL operation can be used to multiply large polynomials; a version of long division works for dividing one polynomial by another.

When one polynomial is divided by another, the result is called a rational function because it is the ratio of two polynomials; rational functions play an important role in algebra; how to add and subtract rational functions.

Multiplying and dividing rational expressions; the key to solving these complicated-looking equations is to proceed one step at a time; closing the lesson with a problem that brings together all one has learned about rational functions.

Examining the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that are not even connected on a graph; learning to identify the intercepts and the vertical and horizontal asymptotes of these curves.

Sketching the graphs of several rational functions by calculating the vertical and horizontal asymptotes, the x and y intercepts, and then plotting several points in the function; simplifying the expression to extract the needed information.

What mathematicians call a radical; how to simplify radical expressions; how to perform operations on them, such as multiplication, division, addition and subtraction, as well as combinations of these operations.

How to solve equations that contain radical expressions; isolating the radical term and squaring both sides; it is important to check the solution by plugging it into the equation to see if it makes sense; how this is true for radical equations.

Moving from radical functions to the graphs that display them; it is important to pay attention to the domain of the functions to ensure that negative values are not introduced beneath the root symbol.

Pattern recognition is an important mathematical skill; investigating two types of number patterns, which are geometric sequences and arithmetic sequences; how to analyze such patterns and work out a formula that predicts any term in the sequence.

Concluding by examining more types of number sequences; discovering how rich and enjoyable the mathematics of pattern recognition can be; employing reasoning skills and algebra to find order and beauty in a sea of confusing numbers.