D.E.V

When started this project, I realized I didn't have the technology or patience to do a video or anything fancy like that. So I chose to do an extended blog post.

The first unit I chose to make an example for is the Long Division Unit. 
The reasoning behind long division is to reduce the degree or the power of the variables.  When you reduce the power of the given equation, you can then factor and find x-intercepts to graph the equation.  The way you do this is to find an x to be able to divide out of the given equation.  The equation is put underneath a radical with the x value when equal to zero on the outside.  So, if the given x in this situation is -6, that means it will be (x+6) on the outside of the radical.

Once you get the equation set up, the way you start it is divide the first value by the first variable on the outside of the radical (shown in red). You then distribute that to both parts of the given x value (also shown in red).  After that, you pull down the next part of the equation on the top (shown by arrows).  You continue that process over and over until you are left with zero.

With what you are left with up top allows you to Long Divide again to be able to factor and create a graph.

Now, in doing this process, it can also help you find a remainder left over.

The next way to factor is to group two parts of the equations; factor by grouping.  In doing this, you can isolate the two sides and find common factors within them.  But the thing with factoring by grouping is when you simplify both sides, they need to have the same thing left over when their common factor is taken out.  An example is shown below.

As you can see, the squared x was taken out to create an (x+5) in the first section, and a thirty six taken out to create an (x+5) on the other side.  When you take the (x+5) out, you are left with (x^2 +36) which, when factored out, simplifies to (x+6)(x-6).  The x's you are left with are -5, -6, and 6.

The last example comes from Graphing Large Degree Polynomials.
First, you must understand the breakdown of the Polynomial you are given.  The a value is the leading coefficient in front of all of your x values.  ex) -x(x-5)
The next part is deciding what happens when the graph gets to the x value. Below is a chart to demonstrate what the correct thing to do is.

So, when given a polynomial, you first have to count to see what the degree is.  For example, if you are given -x(x-4)(x+7)(x-19), the degree is 4.  You count each one in parenthesis and the negative x out in front.  Now, if what is in parenthesis has a power, for say 2, you count that one as two toward the degree, instead of one.  When you graph a polynomial, you can use the graph to find the domain, the place where the y-values are positive or zero.  Below is an example.

As you can see, the x value out in front is negative.  So you would go look up at the chart from earlier and eliminate two ways to make the graph.  You then count up the x value to find the degree, which you see is 9; an odd number.  When looking at your x values, you see that (x+3) is to the power of two.  So you know that at -3, the graph will bounce.  And also with (x-16), it is raised to the power of 3, so it will curve through.  

After creating the graph, one can find the domain.  The y-values are positive negative infinity to negative three, from negative three to negative one, from zero to two, and from ten to sixteen.  

Reflection:

I chose the concepts I did to create my problem set because they were the concepts I had the best level of understanding and grasp of.  In these problems, I learned how to go the opposite way; going from what I wanted the answers to be, to the problem they solve.  I have learned so much more than what I put in my D.E.V.  To me, this was sort of educationally valuable in the face that it is very hard for me to put my thoughts into words sometimes, especially in math.  But this project was time consuming and problem solving which is what you want your students to do.  I would continue this project in future years.