Mathematical Intuition

Math is a crucial topic to any child who wants a good educationa and future. Then how can a child improve in mathematics? Well, through the use of psychology and experiments, some people have found out that there is a link between kids who practiced intuition with math. This means that kids that are exposed to mathematics and practice to think unconciously about numbers and ammounts by actions such as guessing ammounts, are more likely to do better in math at school. This was experimented on by psychologists, and this link can clearly be seen through some of their designed experiments. Learn more about this in this article: http://psychcentral.com/news/2014/01/30/intuitive-training-improves-math-scores/65193.html

Matrices

The matrix is a way of changing equations with variables into a form where nunbers are listed in boxes that can be manipulated for the values of variables. A matrix is called by the number of rows times the number of columns. for example,  a matrix with 2 rows and 5 columns will be called (5x2). The three ways that matrix rows can be manipulated are:

1) interchange rows

2) add a row to another row

3) multiply or divide a row by a constant

The main goal of the Gaussian way is to get zeros below a diagonal of ones that start from the top left corner and end in the bottom right corner. Then, you create equations from the numbers you have and use back substitution into each equation to gain the values of each variable. However, in the Gaussian methods, zeroes are wanted everywhere except in the aforementioned diagonal. With the Gaussian, there is no need to find or use equations and substitution. all you need to do is state the value of each variable from interpreting the matrix.

The example shown online was wrong because he/she forgot to add the value of the last row in both of the steps. The line does not divide where the addition of subtraction can occur. Below are examples my group did:

Trigonometric Identities Review

We had learned about the trigonometric identities beforehand, but we never actually memorized them all! In a lesson this week, we learned that we need to memorize all forty eight! We have started slow with practicing the 11 fundamental ones on a worksheet, but we will gradually have to review more. The identities will help us all in many ways in Calculus if we choose to take it next year. The identities help us manipulate equations concerning angles to the simplicity or complexity that we desire. It will be invaluable for students in calculus next year because they will be able to do problems way faster and easier. Below, I have a picture of most of the identities:

Math problems

It has been discovered in Seattle that problems in mathematics had started even before kindergarten! They had learned that students who are entering kindergarten lacked the necessary skills for success in mathematics in the school. It was done by an assessment of their abilities in the many parts of their academics, including math. This means that kids did not receive enough exposure to math before school starts. This probably explains why students in Seattle and other states may lack math skills later on in higher levels of education including middle school and high school. With further studies, we may learn why math maybe lacking the states compared to other nations and a solution can be provided.

http://blogs.seattletimes.com/educationlab/2014/01/22/math-struggles-start-even-before-kindergarten-state-says/

Non-Square System

A topic that most people in my class are shaky one is the non-square sytem. In non-square systems, there are more variables to solve for than there are equations that are given. For example, there might be three variables including x, y, and z, while there is only two equations. This means that we have to solve the variables in terms of Z or another common variable, but in this class, we will use Z. Therefore, in this scenario, x and y will both equal some form of Z (for example, y=5z and x=z+7). Then you will put the answer in terms of "a" instead of z. In this step, you will just replace z with "a" in the answer. 

Below, is an example Miss V did for review for a non-square system: in this scenario, there was two equations with 3 variables present.

The Birthday Problem

The birthday problem is an interesting mathematical phenomenon/equation. It shows the probability of two or more people sharing the same people in a room full of n amount of people. This equation is actually more interesting than people think! One would assume that there needs to be more than 365 people (or at least i would), but it is far less. In fact, there only needs to be 100 people for there to be a 99.99997 percent of two people sharing the same birthday! This is extremely interesting to me for it is not something i would try to figure out at all before finding out there is an actual equation for it! It is cool that someone actually took alot of time to figure this out.

www.youtube.com/watch?v=qRLVcr-pk5I

Partial Fraction Decomposition

Partial fraction decomposition is basically th breakdown of a complex partial fraction into fractions that add together with simpler denominators and numerators. As a student, i have done the opposite before in a lower math class, which is combining two fractions or more into one with a common and complex denominator. Partial fraction decomposition is way harder and takes more time due to many more steps. However, it is very straightforward with clear cut steps unlike some other things we did with trigonometry. 

The steps to Partial Fraction Decomposition include:

1) multiply

2) distribute

3) collect like terms

4) factor

5) equate

6) solve

 

Below, i have examples:

As shown in the examples, there are also many ways the partial fraction decomposition can be aet up according to the denominator of the initial fraction. 3 different ways are shown below

Linear Programming

Today in Mathland, we actually learned something quite usefull. Ms.Vanspronsen even stated that her friend did this for a living. Linear programming is done to find things such as maximum and minimum profit from a system of inequalitites. There are seven steps in all which may take up quite some time. Linear programming is used in businesses' such as wheat farms to find out what they should produce and how much of each quantity. For example, one company can find out how much pool tables and air hockey tables they should make with the amount of time they have. This same example is shown below:

In-Class Educreation video on Solving 3 equations with 3 unknowns

http://www.educreations.com/lesson/view/group-4-period-7-math-analysis/15535608/?s=QIW0vA&ref=app

This is on number 4. I did this with Astyn.

Through this post on reddit.com, I have learned some interesting ways knowing some more math can benefit us in our everyday lives. For example, One would normally not assume that a 14 inch pizza is that much larger than a 10 inch pizza. Normally, these 14 inch pizzas aren't much more expensive than the 10 inch ones. However, it is twice the size, and can be confirmed with a little bit of math! This is one of the many math life "hacks" one can observe from the commenters on this post. However, don't believe just believe everything you read, but confirm it yourself by doing some math. Some of these facts might be useful in your life later on.

http://us.reddit.com/r/math/comments/mn4bj/a_14_pizza_has_twice_as_much_pizza_as_a_10_pizza/

Elimination

Today, we learned about the Method of Elimination in Mathland! The process of elimination is, as previously mentioned in the last blog, for solving the variables among two problems/equations. However, it is sometimes easier to use substitution over elimination. There are also three possible solutions for the process of elimination. There can be one unique solution, which means that that is one point of intersection between the two lines. There can be no solutions or inconsistency, which means that they are parallel lines. Finally, there can also be infinite solutions which means that they are the same lines.There are four steps to the method of elimination which includes:
 

Method of Elimination:

1) obtain coefficients that differ only in sign

2) add equations to eliminate a variable

3) back substitute to solve for second equation

4) check your solution!

The picture above shows how to solve a problem with two equations by using the method of elimination to figure out the speed of the wind and plane from given information such as distance and time going and coming back from mathland.

Substitution

On January 7th, 2014 in Mathland, we went back to middle school and learned about a concept of basic algebra called substitution. We use substitution among two problems/equations to solve out the variables in them. In reality, we can also use elimination to do the same exact thing, but there are some cases where it will be easier for one or the other. There are four total steps in substitution which includes:

1) Isolate one variable in one equation
2) Substitute the variable found in step one the OTHER equation
3) Solve
4) Back substitution to find remaining variables

We had also learned about breaking even, losing money, or making profits. When finding out which category your business will bring you, you can also use substitution to solve your equations. To find out whether you break even, you follow three steps which includes:

1) total cost = cost per unit * initial cost
2) total revenue = price per unit * number of units
3) You break even when: total cost = total revenue

The picture above shows how to figure out the variables from two equations using substitution. As seen, two solutions were found.