Example Problems of Matrix Problems

Below is a series of pictures of example problems. The first couple are basic system of equations problems, while the bottom page consists of matrix problems. These were completed by my team that includes Jacky V, Mancy, Alec, and Fifi.

8.1 Vocab Prezi

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

Here is a link to our Prezi. Credz to my girl Jacky V, who put in a lot of work and finished putting together the prezi when I was extremely frustrated and hated technology too much to complete it.

The Prezi includes:
1. ALL 23 Vocab terms for 8.1
2. The definition of every term.
3. A picture depicting each vocab term in action.

Trig Identities Review

We started reviewing for trigonometric functions today, because they will be on our final. Apparently, knowing trig will also be very helpful for calculus next year. I attached a video from khan academy that gives some helpful examples. It took me forever to find stuff on Khan Academy because that website is very advanced and my brain is very underdeveloped when it comes to comprehending technology. The lower link to PurpleMath is also very helpful; it's a comprehensive list of all the trig identities we need to know. They are also grouped together conveniently so it's easier to memorize. Be sure to memorize the first 11 fundaental identities. Those are the ones we need to know first!

http://www.purplemath.com/modules/idents.htm

NBA All-Star Game

Below is a link to one of many polls done by ESPN.com. After the NBA all-star game rosters were announced, a poll was released asking which fully healthy starting lineup would be better. The East has Kyrie Irving, Dwyane Wade, Lebron James, Paul George, and Carmelo Anthony. The West has Kobe Bryant, Stephen Curry, Kevin Durant, Kevin Love, and Blake Griffin. Although some of these players (particularly Lebron James, Paul George, Kevin Durant, and Blake Griffin) are very good defensively, these players were certainly chosen for their offensive capabilities to add to the entertainment value of the game. Because of this, I tried to see if the poll was correct by offensive standards by adding the points per game of each player.

East:

Irving 21.7

Wade 18.9

James 26.2

George 23.3

Anthony 26.1

Total: 116.1

West:

Bryant 13.8

Curry 23.5

Durant 31.0

Griffin 22.6

Love 25.0

Total: 115.9

51% of fans in the poll chose the West, but according to this logic, the East has a slight scoring advantage. But it's so close, that maybe the fans just went with Durant, who is the highest with 31 points per game. These scoring numbers are so close that I'm sure this all-star game will be very entertaining!


http://espn.go.com/sportsnation/poll/conversation/_/id/4157579

Review Post: Break Even Problems

        The goal of break even problems is to find when total revenue is equal to total cost. Total revenue equals price per unit times the number of units. Total cost equals cose per unit times number of units plus any initial cost. Since you want revenue to equal cost, set price per unit times # of units equal to cost per unit times # of units plus initial cost.

        Companies might use this equation when they first start off to try to find out how many units they must create/sell to stop losing money. This equation assumes that every unit created will be sold, as the # of units is the same is in most cases the same in both the revenue and cost equations. In essence, this equation is finding how many units must be sold at an increased selling price (more than the cost) to balance out the initial cost, as businesses always need money to start up and need to make sure they can survive until they sell enough units to break even, and eventually start making money.

Remember, the final equation is:

(Price per unit)(# of units) = (cost per unit)(# of units) + initial cost.

The Math Behind Field Goals

Some people say that field goals, extra points, punts, and kickoffs are useless and unexciting parts of football. Some may argue that the NFL would be better off without it, as these plays (particularly kickoffs) are the most dangerous in football and allow teams to gain an advantage from some scrawny kicker that isn't even really part of the team. But in my opinion, kicking plays can be some of the most exciting parts of the games, like a clutch field goal that gives a kicker's team the lead. And in terms of math, kicking is one of the most interesting parts of the game.

On Sunday, December 8th, Denver's kicker Matt Prater broke the NFL record with a 64-yard field goal. The optimal angle for kicking a field goal is approximately 40 degrees, but it is difficult to get that exact angle when 11 defenders are running after you. According to the link below, for a kicker to make a 70-yard field goal, he would have to kick at an initial velocity of 35 m/s, which is about 78.3 miles per hour! Fortunately for Prater, his home turf of Denver is nearly exactly one mile above sea level. Because of elevation, air is a lot less dense in Denver, so there isn't as much air resistance. According to the model below, because of a loss of air resistance, a kicker whose normal range would be 57 yards is about 61 in Denver (at 1609.3 m above sea level):

Prater can allegedly make field goals of 70+ yards in practice, so his range would be impressive even at a sea level stadium. But with the fact that he plays at least 8 games a year this far above sea level, Prater may break his own record by kicking unbelievable field goals. His amazing ability goes to show that kicking really is an important and exciting part of football.

http://www.wired.com/wiredscience/2011/12/are-field-goals-easier-in-denver/

Linear Programming

This lesson is very similar to the previous one about solving and graphing systems of inequalities. However, this section is more focused on word problems that apply the shaded graphs to everyday life. The problems often ask for the maximum amount or minimum amount of something, so you have to find the vertices of the shaded region, as the best answer always lies at one of these. Here are the steps:

1. Read the problem carefully.
2. Write the constraints or inequalities.
3. Graph the inequalities. Find the feasible region.
4. Find the vertices of the feasible region.
5. Write a function to find the minimum or maximum value. (This format: z = ax + by).
6. Plug the vertices into the function.
7. Find the maximum or minimum.

Many of these problems involve maximizing profit or minimizing costs for a company. Because of this, the variables x and y often correspond to certain commodities, and the a and b in the above equation (in step 5) are usually either the cost of each commodity or the profit gained from selling each commodity.

Fraction Decomposition

On last Friday and Monday, we learned about Fraction Decomposition. This is one of the most complicated math concepts we've learned this year, so let's get right into it:

This is a simple example, but some of these problems can get very complex. Since there are so many steps, remember to work at a reasonable pace, because a very simple mistake can be devastating. You don't want to have to redo this whole thing if you get a funky answer!

Ray Allen's Unlikely Shot

I watched a really interesting "Sport Science" video about Ray Allen's game-winning shot at the end of game 6 of the NBA Finals last season (the link is below). Without hitting this ridiculously unlikely shot, the Heat would have lost game 6 and the NBA Finals all together, changing the course of history. There are a lot of interesting physics explained in the video that made the shot significantly harder than the average 3-pointer. For instance, because the shot was rushed, Allen's release was at 40 degrees, way flatter than the optimal 48 degrees. This reduces the area of the basket in which the ball could successfully pass through the hoop, because it is not coming down from as high and is more likely to hit one of the edges of the rim. In addition, his unusually far distance from the 3-point line made it so that if he shot it even 1 degree more to the left or the right, the ball would have hit the rim and missed. It's really interesting to see how such minute differences in form can drastically change the chances of a shot's success.

https://www.youtube.com/watch?v=wEyXm6-MMVQ

Elimination

Elimination Strategies

If substitution fails, another method to solve systems of equations is elimination.

The first step is to multiply one or both of the equations in such a way that the resulting coefficients on one variable have the same absolute value.

• One foolproof way to accomplish is to multiply one equation by the coefficient of the desired variable in the other equation, and do the same with the other equation.   
• Ex: 5x + 3y = 20 ; 3x + 10y = 30  -----> multiply 1st equation by 3, and the 2nd equation by 5 to eliminate x.

There are also certain shortcuts that allow you to only multiply one of the equations, such as if 3 and 6 are the coefficients of x, you could just double the 1st equation, or cut the 2nd in half.

• Ex: 2x + 3y = 7 ; 4x + 5y = 22 -------> simply multiple the 1st equation by 2.

Next, you must add/subtract the equations to eliminate the variable for which you made both variables have the same absolute value so that the variable is eliminated. Then, solve to find the other variable

• Ex: 2x + 3y = 8         

           +   -2x + 3y = 4
           =             6y = 12
                           y = 2

Then, substitute your solution (in this case, 2) into one of the original equations to find the value of the variable you eliminated.

• 2x + 3(2) = 8  ----> 2x = 2 -----> x=1

Finally, check your solutions by substituting both variables in the original equations.

First Day in Mathland 2014

Ms. V likes to refer to today as "the day we go back to kindergarten." Today, we learned a short lesson about solving equations with two variables. In these problems, you will always have at least as many equations as you have variables. First, you have to rearrange one equation so that one variable is isolated (e. g. x=y-3), then substitute that same variable in the other equation with the other side of your rearranged equation so that the equation contains only one variable. Then, solve for the variable in the equation you just created. Finally, use the value(s) of that variable to find the value of the other variable. I don't think I was learning this level of math in kindergarten, but it was nice to take a step back for the first day after break.

Hi Miss V and friends1

Blogging for math is so much fun, but it seems like it's gonna be a lot of work...