Chapter 8 Vocabulary

013114

Here's our prezi on the vocabulary for chapter 8.

http://prezi.com/poqksnaiqwz_/?utm_campaign=share&utm_medium=copy

Lesson 8.1

013114

Today we learned two different methods of elimination using matrices, the Gaussian Elimination and the Gauss Jordan Elimination.  These two methods are somewhat similar, but personally, I think the Gauss Jordan Elimination is way easier and faster.

The Gaussian Elimination deals with inputting a given system of equations in matrices, and then creating your diagonal of 1's, which will give you a new set of equations.  Eventually you will have a given solution to one of your terms, and then you just plug and chug and you'll be able to solve for your other variables.

The Gauss Jordan Elimination is a little different, because it deals with finding your constants within your matrices, just by making all the numbers outside of your diagonal 1's zeroes.  It gets a little tricky to do this, but once you do, everything is really simple and your answers come right on the page.

Here are some examples:

#3

In addition, referring to the error analysis, the problem was incorrect due to the fact that the student did not correctly zero out the first zero, because he or she did not ally the (-1) to the whole row. In conclusion, the answer should actually be (7,-3).

The Unsolvable Math Problem

013114

As I was looking for things to post today, I came across an article about a college student who had ACCIDENTALLY solved a math problem that mathematicians since Einstein have been trying to solve.
The student had been studying very late the night before a test and ended up sleeping in and attending the class late.  When he arrived, he quickly began the three problems that had been posted on the board. He was having a hard time on the last one, and it almost seemed impossible, but he suddenly found a method that worked and was able to finish just before time was called.  Later that night, the student received a call from his professor whom seemed frantic.  The student thought that he had failed the whole test, but in reality, the professor told the student that the third problem was an example of an "impossible equation" and the student was able to solve it!  The student was quite amazed with himself, especially since he wasn't even supposed to do the problem in the first place!
So I guess you could say that you never what's going to happen when you just do all the problems on the board, maybe you'll find a solution to something that hasn't been solved in centuries!

The name of this man was George Dantzig.  During the time of his revelation, he was a student completing his Doctorate at UC Berkeley in 1946.

Math Soup for the Body and Soul

012814

Not really, but I always thought the title of those books were funny, which I thought would be a good title for a post about math jokes.

How do you make seven an even number? 

You take the s out!

Why should the number 288 never be mentioned?

It's two gross.

Why did I divide sin by tan?

Just cos.

What do you call friends who love math?

Algebros.

Even though these were a little corny, I still found them pretty funny, but also kind of sad that people could even think of Math jokes. This just shows how everybody has their own preference, and that some people actually enjoy math and wouldn't mind using it as a subject of a joke, or of a talk, or of anything really.

Chapter 7 Review

012814

Chapter 7 was filled with lots of different concepts including old and new. From elimination am substitution, to linear programming. Another thing we learned that was somewhat new but also old was breakeven. There's a possibility this was done in Algebra 2, but all it involves is substitution, which was definitely from algebra. Break even is used to find the point at which a business' cost and revenue are the same. In order to find this, you must use two equations.

Total revenue= (price per unit)(# of units sold)

Total cost = (cost per unit)(# of units sold) + initial cost

Here's an example:

Sorry this i late, but I'm battling illnesses.

Dyscalculia

011614

Dyscalculia means difficulty in learning arithmetic, such as difficulty in understanding numbers, and learning math facts!  This is a disability similar to dyslexia.  But, it is less common and only exists between 3 and 6% of the population, in addition a quarter of children with dyscalculia have ADHD.

Dyscalculia comes from Greek and Latin which literally means "counting badly".  This disorder might seem quite silly and an excuse to have a bad Math grade, but it is a real disorder with many problems that come with it.  For instance, one with dyscalculia can have difficult reading analog clocks, difficulty stating which of two numbers is larger, problems with differentiating between left and right, as well as an inability to concentrate on mentally intensive tasks.

Though people with Dyscalculia have a more difficult time with math, there are treatments to help remediate it.  For instance, forms of educational therapy as well as direct stimulation have been proved to demonstrate selective improvement in results.

Personally, I think that Dyscalculia is a disorder that many people are unaware of, and that there's a possibility that I could have it too! Especially when it comes to Math Analysis...

Lesson 7.5

011614

Systems of inequalities are the next step after solving a system of equations, but we'll still be using everything eventually. In order to solve a system of inequality, a graph will be needed in order to find the solution that will satisfy all the inequalities.

 

Here are the steps:

Replace the inequality sign with a equal sign, and sketch the graph of the resulting equation.

 >dashed lines (greater than or less than) 

 >solid lines (greater than or equal to or less than or equal to)

Test one point in each regions formed by the graph in step 1. If the point satisfies the inequality, shade the entire region to denote that every point in the region satisfies the inequality.

Solution of a system of inequalities in x and y is a point (x,y) that satisfies each inequality in the system

For a system of inequalities it is helpful to find the vertices of the solution region.

Here's an example: 

Just always remember to check which vertices satisfy the inequality that includes x and y!

Lesson 7.4

011414

We finally left easy peasy Algebra, and now we have to do these wild things called partial fractions. To be honest, they're much scarier at the beginning, but once you do a ton of practice, it's super easy. Especially if you follow the steps!

These steps are:

Multiply by the lowest common denominator

Distribute

Collect terms with the same variable

Factor out the variable

Equate coefficients

Solve the system of equations 

Write as a partial fraction

Here's an example:

It's quit simple, you just always have to remember your keep your terms organized and you'll get it easily.

I feel like Math Analysis is going to get harder, so we should all buckle up and get ready!

Lesson 7.3

010914

Today, we used elimination, but with three variables, adding another step.  Another thing we learned was a non-square system.  Here's an example of a system of equations with three equations.  

Educreations video by Joe and Axie

Underwater Math

010914

All week I've been thinking about how on earth I was going to think of a topic for our creative math post.  Some came up, but they were truly boring and didn't actually apply to my personal life really.  Until today, during swim practice, I realized that I use math ALL THE TIME, so do many other sports.  Obviously it's not just the simple adding of goals, touchdowns, and points in team sports, but also math related in time, especially in swimming and track.

For instance, today Coach Simon and Zimbrano taught us about a "Russian Time", which was something completely new to us all.  Basically, it's a time we received on a 50 yard swim, plus an average amount of strokes we took per lap.  The goal at the end of the set, was to reduce your Russian Time, but keep the average stroke count the same.  

Another thing I realized was how swimming is solely based on time standards, which really correlate to math in practice.  For instance, in order to reach a certain time during our meets, we practice by going a certain average time in practice, making sure we hit that pace every single time.  Eventually, our goals get faster and we have to calculate it all again.  In addition, when we're practicing.  We use math all the time, because every swimmer calculates their own personal time using the clock on the scoreboard.  So basically, we all become professionals at the 60 second clock system, being able to understand how much we need to subtract compared to the time we leave.  We also figure out our averages by dividing bigger times, finding the average we need for shorter distances.

Even though the math we use isn't as complicated as parabolas, trig functions, or systems of equations, it's still math and it's all mental math.  Which is a lot harder than you think, especially when your body is tired AND you still have to use your mental juices.  All in all, math is used all the time, even in your daily routines, such as swim practice.  But sometimes, too much math is just too much and you can't handle it!

Lesson 7.2

010814

Now it's time to add another way to solve a system of equations, ELIMINATION. In order to eliminate a variable in an equation, you should follow these steps.

Obtain coefficients that differ only in sign.

Add equations to eliminate a variable.

Back substitute to solve for second equation.

Check your solution.

Here's an example. 

The process of elimination can also be applied to real life situations, such as one dealing with the speed of an airplane with a headwind compared to the speed of an airplane with a tailwind.

Here's an example.

And that's how you use elimination to solve a system of equations. So far second semester is good! Hopefully we can keep it up! 

Lesson 7.1

010614

It's the first day of second semester and Ms. Van Spronsen is already giving us a hard time. Today we learned TWO things, substitution AND breakeven. Which was way too much for one day. 

Anyways, substitution is really easy peasy and quite obvious. All you have to do is isolate a variable on one equation then substitute it in the other. Then you solve and then back substitute.

And that's how you substitute! Now, onto breaking even.

There are two equations used to find your break even, which is total cost = total revenue

In order to find your total cost, you multiply your cost per unit and numbers of units sold, then add the initial cost.

In order to find total revenue, you multiply the price per unit and the number of units sold.

And that's how you find the break even. Basically you just need to make sure you're plugging everything in its correct place and just go back to using substitution for the missing variables!

Well that's Day 1 of Second Semester and blogging!