Edu Technology and Algebra

Monday, April 23, 2012

Algebra: Real Numbers and Their Operations (Study Guide)

Chapter 1: Real Numbers and Their Operations

Extra video examples to accompany Elementary Algebra - the free to read online open textbook published by Flat World Knowledge.

--> Click on a problem below to see it worked out on YouTube. <--

Real Numbers and the Number Line:

Adding and Subtracting Integers:

Multiplying and Dividing Integers:

Fractions:

Review of Decimals and Percents:

Exponents and Square Roots:

Order of Operations:

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Thursday, April 12, 2012

Algebra: Solving Quadratic Equations by Factoring

Previously we have learned how to solve linear equations, ax + b = 0, now we will outline a technique that can be used to solve factorable quadratic equations of the form ax^2 + bx + c = 0. Unlike linear equations, which usually have one solution, quadratic equations can have up to two solutions. We begin with the zero product property:

This property is the key to solving quadratic equations by factoring. Basically, if the quadratic equation is equal to zero you factor it, then set each factor equal to zero, and solve.

Tip: You can always check your answers by substituting them into the original equation to see if a true statement results.

Solve.

Note: When one solution is obtained, as in the previous problem, we say that the solution is a double root. This technique requires the zero product property, so you first make sure that the quadratic equation is set equal to zero before factoring. Do this by adding all terms to one side of the equation. The check is usually optional.

A common mistake is to set each factor to 56 here - which leads to incorrect results. You must have zero on one side of the equation for this technique to work.

Solve.

Clearing fractions from our equations can be done by multiplying both sides of our equation by the LCM of the denominators. After doing this, the equation becomes a bit easier to solve.

Solve.

Finding equations given the solutions requires you to work the entire process in reverse. Given the solutions you can find factors that can be multiplied to obtain a quadratic equation.

Find a quadratic equation with the given solution set.

You can find much more in the free to read online open Elementary Algebra textbook available on the Flat World Knowledge website:

[ Free Algebra Textbook ]

UPDATE: Here are some corresponding exercises. Enjoy.

[ Exercises PDF ] [ Download Source DOCX ]

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Friday, March 9, 2012

Make It Your Own

Here is a link to a blog post that I wrote for Flat World Knowledge:

http://blog.flatworldknowledge.com/2012/03/08/3-steps-to-a-perfect-open-textbook/

Unlike most open source materials, the FWK customization tools allow you to easily customize their books.

Friday, March 2, 2012

Algebra: How to Solve Linear Systems

In our Algebra course, we are currently learning how to solve systems of two linear equations with two variables. This topic is covered in Chapter 4 of our open Elementary Algebra textbook published by Flat World Knowledge. To find the simultaneous solution of such a system, or point of intersection, we can use one of three methods.

1. The Graphing Method: Rewrite the two equations in point-slope form ( y = mx + b), graph them on the same set of coordinates and then determine the point of intersection. A video example follows:

2. The Substitution Method: This completely algebraic method, which requires that we first isolate one of the variables. We then substitute the result into the other equation. Once we solve for one of the variables, back substitute to find the value of the other variable. Remember that the answer is an ordered pair (x, y).

3. The Elimination Method: Usually this is the method of choice. The idea is to multiply one or both of the equations by appropriate numbers so that one of the variables will eliminate if the equations are added together. This is sometimes called the "addition method." Always back substitute to find the value for the other variable and present the solution as an ordered pair.

Each method has its strengths and weaknesses, but whichever method you choose the answer will be the same. Most of the time, a system will produce one solution, a single point of intersection. However, sometimes the equations are actually equivalent and in that case there are infinitely many simultaneous solutions. This describes a dependent system and the algebraic methods will lead to a true statement like 5 = 5 or 0 = 0. Click here for a video example:

It is also important to note that there is not always an answer to some linear systems. Sometimes the lines are parallel and do not intersect. This describes an inconsistent system and the algebraic methods will lead to a false statement like 0 = 4. Click here for a video example.

These take time and practice to master. Try some yourself and you will soon see that solving systems of two linear equations is actually fun. Also, be sure to look at the many applications found in section 4.4. Hope this helps.

UPDATE: More Solving Linear Systems videos. Visually searchable, click on a problem and view it worked out on youTube.

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Thursday, March 1, 2012

Beginning Algebra Course Worksheets

Just posted my full set of course worksheets for Elementary Algebra. We cover chapters 1 - 7 of the open textbook found here. I have included the docx source files as well as pdf versions. Please feel free to modify them to suit your course needs.

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Tuesday, February 28, 2012

Algebra: How To Graph a Line

We have learned three methods for graphing lines in Chapter 3 of Elementary Algebra:

It is important that we know how to quickly sketch the graph of a line using slope-intercept form: y = mx + b. However, the equation of the line may not be given in this form and you do not always get the x-intercept so easily. The following video outlines the steps for graphing a typical line.