Elimination vs. Substitution

When solving systems of equations, we want to find the simplest method possible. Depending on the equations being solved for, different methods help. Here are two general rules to help determine the best possible system of solving.

1. If in either equation, one variable has already been solved for, use the substitution method.

-This is because when one variable is already solved for, it is much simpler to plug in the value of x or y and from there, solve the equation.

For example:

y = 3 x - 5

y - 3 = 3 (x - 7)

In this case, Y has been solved for. Simply plug in or "substitute" the value of y (3x-5) in for the Y of the second equation.

2. If the X values or Y values of both equations cancel out, or if the values can be multiplied simply so they do cancel out, use elimination.

-This is because if the X's or Y's simply cancel out, the equations will be much easier to solve for because only one variable is left to deal with.

For example:

5x + 4y = 20

3x - 4y = 60

Notice the Y variables cancel out because one is a positive 4 while the other is a negative.

Also:

3x + y = 10

-9x + 3y = 30

By observing carefully, you can see that one of the equations can be multiplied so that one of the variable will fall out. Which equation is it? What can it be multiplied by?

Multiple everything in equation one by 3. By doing this, the coefficient to X becomes positive 9, which is directly lined up with a negative 9 causing them to cancel out.

Using these two rules, solving systems of equations will turn out to be much simpler.

Effect of Outliers on Mean and Median

Outliers, are numbers that stray from the "most popular" area of data. This could be a number that is very low, very very high, or negative, as long as it does not fit in with the general data area. These numbers affect the mean because mean is the average of all numbers on the graph no matter where they lie. An extremely large number in a group of single digit data provides an inaccurate understanding of the mean of the data. This can also be a technique in advertising to make a certain product look more popular, less expensive, etc. Outliers have a fairly dramatic affect on the mean of a set of data but do not have much affect on median. They will not change the median of a data set any more or less than another number would. For more information, keep up with what we're doing in class on Mr. Fisch's Algebra Blog.

Conference Reflective Post

To start off, I am for the most part pleased with this class. It has met most of my expectations with the exception of the difficulty level. I am open minded knowing Algebra 1 concepts will increase in difficulty, but so far with 2-3 question assessments, and simple review topics, I find the class not pushing me to my full potential. The topics we discuss in class are going well for me right now. Even if the concept is a little bit new, it takes little time to catch on. One challenge is that I do not have as many friends in that class as other classes, but am working on that. One thing you could do as a teacher that would benefit me as well as the rest of the class is check/grade the homework you assign, especially videos and self check questions. I have noticed not a ton of people do as much of the homework you give as you might want them too, and I think this might tempt them to do it. As a student, I can help myself by completing the class notes fully, and to the best of my ability as well as asking questions when I'm stumped. My parents are helpful to me when I need it but haven't needed to help me with much work at all. I don't believe there is anything else to say about this class so this concludes my Conference Reflective Post.

SLOPE

Intercepts and slopes are two building blocks for many things we will learn this year. It is crucial to understand slope in the context of an equation referred to as y=mx+b. Can you identify which part of this equation is the slope? (m) is the slope of the equation, but what on earth is a slope? A slope is a mathematical word used for the steepness of a line. Many have learned it as rise over run, or as I learned it, delta Y over delta X. The slope can be used to get from any point on a line to the next. By starting at a point on aline, the slope can be used to rise up, and run over to the next point. There are a few different kinds of slope as well. A positive slope is a line that ascends when looking left to right. A negative slope descends looking left to right, and no slope moves neither up nor down looking left to right. Hope this leaves you with a better understanding of the critical concept of slope.

CRAZY EQUATIONS!!!

3(x-5) = -7x + 12

If you are like me, you panic when this equation is thrown in front of you; however, HAVE NO FEAR! There is a simple way to solve this crazy equation. Let's do it step-by step. Each picture below illustrates a step in the process of solving this equation with a variable on each side.

Here is our original equation. I learned to set it up like a table, to keep my steps organized.

To begin, the 3 must be distributed to everything in the parentheses. 3 multiplied by X is 3x, and 3 multiplied by -5 is -15. Copy down the rest of the equation.
This cannot be solved until all X's are on the same side of the equals sign. I could either subtract 3x from the left, or add 7x to the right. I like working with whole numbers, so I chose to add seven. Copy down your new equation.
Now that all X's are on the same side of the equal sign, we need to isolate it. This happens by performing the opposite of subtraction, which is multiplication. Add 15 to both sides of the equation remembering that anything that happens to one side of an equation, must happen to the other.
Good work. We are close to finished, but we haven't isolated our variable. 10 is being multiplied by X here, so we will perform the opposite and divide by 10 on both sides of the equation.

CONGRATULATIONS MASTER OF CRAZY EQUATIONS!!!

Direct and Inverse Variation

Direct and Inverse Variation are two very important concepts to know in Algebra 1, as well as future math classes. We'll start with Direct variation. In direct variation, an equation formula such as y=kx will be used. This means, that there is a constant number that can be multiplied by an X value to get a Y value. As X increases, so does Y. When you divide the increase in Y by the increase in X, you get your constant, or K. Take a look at the below picture:

As, for Inverse Variation, it's a little different. We use the equation y=k/x to solve. You can always divide the constant by the X value to get a Y value. When the X value increases, the Y value decreases. Take a look at this example:

HAPPY LEARNING!

Clubs and Sports

As for the clubs and sports I want to participate in, I know I am going to join the AHS swim team during the winter season. this fall I am trying lacrosse for the first time through South Surburban to see if I like it. The clubs I have attended so far are Student council, Drama Club, and Fellowship of Christian Athletes (FCA). I have enjoyed all of these and hope to continue coming regularly.

Reflective Blog Post #1

Looking back at my first week in High School, I feel like I am ready to take on Arapahoe and all that it has to offer. I am excited not only about the teachers I have this semester, but the copious ways to get involved outside of academics. I do not have many, if any concerns yet about my High School year as a Freshman. I am ready to take on challenges and learn all that I have to learn this year. My goal relating to algebra is to become proficient quickly in topics I have not yet learned. As for AHS, my goal is to participate in at least one club regularly, and one sport. Outside of AHS, I hope to become a more organized person and a better typer.