In this semester, we have learned many things in Ms. Burtons pre calc. Math 10, but here are five things that really stood out to me.

1. Exponent laws.

The exponent laws stood out to me, because they where one of the first things we learned, and I always saw them reoccurring and tying into each chapter in different ways. If I didn’t know what the exponent laws where, I would probably not be very successful in this class.

(Product law, Quotent law, Product of quotent law, Power of a product law, Power of a power law and Negative exponents)

2. Trigonometry – SOH CAH TOA

Out of all the chapters we completed in this class the trigonometry chapter was probably my second favorite, due to the fun of measuring and plugging in different equations to find out a hidden length or angle. I also believe that trigonometry can be useful in real life, which may be another reason I enjoyed it so much.

(SINE ratio, COSINE ratio, TANGENT ratio)

3. Prime factorization and GCF and LCM

Another early chapter 1 subject(s), these where very helpful, and being taught and refreshed with these will more than likely be useful in my future, as well as them not being to challenging to use and having factor trees be very fun at times, as well as finding the GCF or LCM.

4. Multiplying fractions

I think that fractions where my biggest challenge in the math category, but I think that they where the biggest improvement I made during this class because when I came into this class, fractions where very scary to me and intimidating because I have always have had trouble with them in the past, but then now, I feel like I have diminished that feeling quite a bit. Not only with Multiplying fractions but also with fractions in general.

5. Polynomials

I wanted to put polynomials in my top five list because they where my favorited chapter, and I believe it was the chapter that I was most invested in. Not too easy and not too challenging, I believe that this chapter was the best and most educational for me personally.

For this week’s blog post we will be taking a look at the 4 different ways of solving systems.

Our first way to solve is using graphs

The first step to solving a system with a graph is to get two slopes and of course have it on a graph

ex

The next thing to do is just look at our graph and just see where the two slopes cross through each other, and mark down the point where it goes through the points.

The next method is to use inspection

If we have two equations that are in simple form

ex:

x + y = 9

x – y = 1

we can just look at the equation, and inspect on what two numbers add to seven, and subtract to 1

in this case, the answer would be (5,4)

The next way to solve an equation is inspection, which mainly focuses on isolating a variable.

The first step is to have an x y equation

ex: 6x + y = 8  and 2x + y = 4

(note that this example is made to be an easy format to solve, and most equations will not be formatted this simple)

The first step, is to isolate a variable, and since we are given two equations where y does not have a coefficient, so we will isolate.

y= -6x + 8 and y= -2x + 4

since we have both variables isolated, and the statement is y=, we can an equation into y

for example y= -6x + 8 and y= -2x + 4 will now be turned into -6x + 8 = -2x + 4

Now what we need to do is to add like variables together, so we will move the x variables to the same side, as well as the constants to each side.

ex -6x + 2x = -8 + 4

now to add each variable together

-4x = -4

now to isolate x, we will divide -4 by -4

this will give us x=1

after this, plug x into the original equation, add like variables, and isolate y to get the final answer.

The last way to solve a system is elimination.

8x + y = 5

4x + 2y =10

first step is to make a zero pair, by multiplying a equation to make two numbers = 0

we will multiply 4x + 2y =10 by -2, because it will make a 0 pair with 8x + y = 5’s 8x

our equation now will be

8x + y = 5

-8x + -4y = -20

Now just add each like variable together

-3y = -15

now to isolate -3y just divide each side by -3 giving us

y = 5

now we would just go back to the original equation and plug 5 into y and solve for x to get our system point.

 

I chose this topic for my blog post, because I needed to review on the topic and display my understanding to myself.

This week’s Blog post will be about how to solve a system using elimination.

What is elimination?

Elimination is simply a way to solve a system, where the key is a word called zero pairs, which means two numbers which make each other ultimately equal zero. We usually get a zero pair by using multiplication to mold a number into a desired zero pair, but at times we are given equations that are pre-set with a zero pair.

So our first step for using elimination to solve a system, is to of course have our 2 systems preferably in standard form.

Ex:

2x + 3y = 18

4x + 4y =20

The next step is to make a zero pair, and to do this we need to use multiplication to make two numbers a zero pair. In this case, we could multiply the first equation and the second equation to make 3y and 4y a zero pair, but that would be a lot of multiplication and unnecessary work. Instead, we can easily just multiply the first system by -2 to make a zero pair with 4x in the second system.

-2 plugged into 2x+3y=18

=

-4x -6y = -36

4x + 4y = 20

As you can see, -4x and 4x make a zero pair, which is the ultimate goal in using elimination. So after those two numbers eliminate each other, we just add up both equations with their like variabes.

So, we will add

-6y + 4y and -36+20

this will give us

-2y = -16

Now we use division to isolate y, so we will divde both sides by 2

-2y/-2 = -16/-2

this will give us

y=8

Now that we found our base answer, we need to plug in 8 into one of our previous systems

We will use

2x + 3(8) = 18

So the first step is to plug in 8 into y

2x + 3(8) = 18

2x + 24 = 18

after this we want to rearrange the system to add like terms.

2x = 18 -24

2x = -6

Now just like before isolate and divide

2x/2=-6/2

x=-3

This gives us our solved system of (-3, 8) where our two slopes will cross each other.

I chose elimination for this weeks blog post because it is my favorited way to solve a system, and I believed that I could make a well demonstrated blog post on the topic.

Direct Characterization Quotations
Quote #1:
– “One of these days, thought Winston with sudden deep conviction, Syme will be vaporized. He is too intelligent. He sees too clearly and speaks too plainly. The Party does not like such people. One day he will disappear. It is written in his face.” (68)

Quote #2:
– “Syme was a philologist, a specialist in Newspeak. Indeed, he was one of the enormous team of experts now engaged in compiling the Eleventh Edition of the Newspeak Dictionary. He was a tiny creature, smaller than Winston, with dark hair and large, protuberant eyes, at once mournful and derisive, which seemed to search your face closely while he was speaking to you.” (62)

Indirect Characterization Quotations

Quote #3:
– “‘Don’t you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it.” (67)

Quote #4:
– “‘I wanted to ask you whether you’d got any razor blades,’ he said. ‘Not one!’ said Winston with a sort of guilty haste. ‘I’ve Free eBooks at Planet eBook.com 63 tried all over the place.” (62-63)

Quote #5:
– “‘It was a good hanging,’ said Syme reminiscently. ‘I think it spoils it when they tie their feet together. I like to see them kicking. And above all, at the end, the tongue sticking right out, and blue—a quite bright blue. That’s the detail that appeals to me.’” (64)
Quote #6:
– “One of the notices carried a printed list of the members of the Chess Committee, of whom Syme had been one. It looked almost exactly as it had looked before—nothing had been crossed out—but it was one name shorter.” (186)

For this weeks blog post, here is my wonky initials Desmos activity.

In George Orwell’s novel “1984” we are presented with many different instances where we can dissect pieces of characterization in the novel. Characterization is when the author tells us either directly or indirectly who or what the character is, and what they are about. Direct characterization is when the author tells the reader straight-up information about the character, for example, “The 6’1 Caucasian male with red hair had braces.”. In the novel, we can find many well written examples where our protagonist Winston Smith is described with  direct characterization, “His hair was very fair, his face naturally sanguine, his skin roughened by coarse soap and blunt razor blades and the cold of the winter that had just ended.” (Orwell 4). This is direct characterization because the illustrator used words directly telling us about what he looks like, for example now we know he has fair hair and roughened skin. Our second type of characterization is indirect, where the author will use a abbreviation named S.T.E.A.L (speech, thoughts, effects on others, actions and looks) to help the reader have a better understanding of who the character is. A well illustrated example from the novel is, “At last they were face to face, and it seemed that his only impulse was to run away. His heart bounded violently. He would have been incapable of speaking.” (Orwell 198). This is indirect characterization, because Winston’s actions and thoughts tell us that he is very unsure and anxious or even scared, which is very valuable information when trying to figure out more about a certain character. When reading, watching or even speaking of a story, it is always important to be able to include direct and indirect characterization to add meaning and interest to a character, and it is just as important to know the difference.

In the novel “1984” by George Orwell, there is many different instances where mood and setting are both illustrated in different manners. The two types of setting is physical and emotional, where physical setting is when the actual facts, such as location, time, date, and weather are stated to help ignite the mood in a story. Emotional setting is when the literature is setting a whole atmosphere which may provoke different emotions in the reader. An example of well illustrated setting in the novel, “The weather was baking hot. In the labyrinthine Ministry the windowless, air-conditioned rooms kept their normal
temperature, but outside the pavements scorched one’s feet
and the stench of the Tubes at the rush hours was a horror.” (Orwell 186). This quotation gives us a direct idea of what is going on in the story because it provides concrete facts like the temperature and how the pavement was “scorching” as well as the Tubes being a horrible stench, which sets the mood to being uncomfortable and unenjoyable. Here is an example of emotional setting in the novel, ” As soon as they arrived they would sprinkle everything with pepper bought on the black market, tear off their clothes, and make love with sweating
bodies, then fall asleep and wake to find that the bugs had
rallied and were massing for the counter-attack.” (Orwell 189). This is a emotional mood setting example because it does not state what the environment is like, but we can take the fact that they are making love with a sense of urgency that the mood may be something of a romantic and frantic but comfortable mood as we see routine like behavior from both Winston and Julia. Although it may be challenging to uncover at times, being able to tell what kind of mood is being set and used is very useful and a fun skill to have.

For this weeks blog post, I will be using the general form equation, to get my x and y intercepts.

To get our x and y intercepts, we must first understand what general form is using an example.

12x -y +6 = 0

In general form, both x and y are on the same side of the equation, with x never being a negative, or a fraction if we are using general form.  General form is regarded to be the most useless out of the three equations of slope, because it does not tell us anything about our slope, unless we convert it into either point slope, or y intercept.

Our next step, using the same example, will be to pull out an old trick and do the x or y = 0.

So if our equation is 12x -y +6 = 0

12x -y +6 = 0 if x = 0 will give us something like this.

y int = (0,6)

Because if we make x 0 we still have two variables on one side of the equation. but if we take the y and put it on the other side of the equation, it gives us the answer, 6=y

To find the x intercept we will do something  like this.

12x -y +6 = 0

y=0

6=-12x

Here, we need to use some division.

6/-12 = -0.5

So this makes our x intercept.

x=-0.5

Here is a photo of our general form equation on a graph for those visual learners.

I chose this topic for my week 15 blog post because I believed that I understood it well, and was able to make it into a informative and understanding post.

 

The slope of a line is defined as the steepness, or rise over run. Finding what the slope of a line is, is quite simple.

Our first step for finding our slope, will be to have a graph with at least 2 coordinates.

Here we have quite a large slope, but the next simple thing to do is just to determine our points on the graph. As we can see our points are 2,2 and 30, 7.

what we can do now is just count how much it takes to rise and run.

In this case, it would take us 28 units to get to 30x from 2x so 28 would be our run. To get to 7y from 2y it takes us 5 units, so our rise over run slope will be 5/28.

For our next example, I will be using a different kind of slope: (-2,4) (5,4)

One may notice that something is different with this slope, because there is a point in the second quadrant, and this makes our slope a different value. That value will be negative.

All we do for this slope now is just count rise and run again. We can see that we have to rise 2 and run 7, but we are going towards the negative side of the x axis, so it will be -7. So our slope will be 2/-7.

A simple way to remember this rule is by using the “Mr. Slope face”

as you can see from the face the slope rising to the right, is positive and to the left is negative, and straight down is a undefined line, and this is a line where the x coordinate never changes, and there is no y intercept, and a line that is completely horizontal is 0 because there is no x intercept.

I chose this concept for my weekly blog post because I found it relatively easy to explain and I understood the concept fairly well, so I thought of making my blog post on it.