Pascal's Triangle

One can use the Binomial Theorem or the Pascal Triangle to expand and equation such as (2x+3y)^4 without just multiplying it all out manually! However, as Miss V quietly states, Pascal's Triangle is the one that a person should use. With the Pascal's triangle method, you find the row according to the power of your equation. Then, you fill in the number after each constant starting with the highest power of the first term and descending and the lowest number of the second term and ascending after each term in your expansion. With the use of the Pascal's Triangle, one can also find the term that Combinations are trying to find by looking at the first number before the C for the row number and the number after the C for the term. However, you must subtract 1 to the term number to get the correct answer.
Here is a Pascal Triangle: 

Mathematical Induction

In Mathland, we learned about how we can prove a statement using the platform of Mathematical induction. Mathematical induction is very simple, but is long and has clear steps and things you must write every time for it to hold true! For example, without the statement: "assume the statement is true for n" at the start of step 2, it would not hold true! With mathematical induction, you must memorize the format in order to do it. Mainly, there are two steps. The first step is prove the statement is true at the starting point while step 2 is to assume the statement is true for n and then prove that the statement is true for n+1.
Here is a picture of the format you must follow, the orange parts hold three for all mathematical induction proofs

Well Ordering Principle

Basically, the Well Ordering Principle or Well-ordering theorem is a term in Math that within a set of positive integers, there are is a least element, which is self-explanatory for it is the number that is the least in value. this principle turns out to be logically equivalent to the mathematical induction, the fifth axiom of Peano, which is quite surprising. If mathematical induction is true for all natural numbers, then so is the well-ordering principle. Basically, the well-ordering principle can be proved with the mathematical induction. Ultimately, both imply each other to work because they are based on the same type of logic.
Here is an the proof of well-ordering principle according to mathematical induction: the http://www.math.wustl.edu/~chi/310notesIV.pdf

0!

Most factorials are easy to understand and to find the solution to, except the factorial of a certain number. This number is 0. It is weird that every other number above zero is basically that number multiplied by all numbers below it towards one. However, 0 x 0 should be zero for even if it is multiplied by one, it is still zero. However, through this article, one can basically understand why mathematicians had defined 0! As 1. One huge reason is due to permutations. A set with no values should be arranged in only one way. Also, the formulas for permutations and combinations need the value of zero factorial to be one so that it is consistent with the equations that are derived. It is basically made up so that math equations stay true. One can learn more from this article: http://statistics.about.com/od/ProbHelpandTutorials/a/Why-Does-Zero-Factorial-Equal-One.htm

Arithmetic Sequences

We learned about arithemetic sequences. Arithmetic sequences are sequences of numbers where there is a common difference between each number such that a number is is being added to each number to get the next number in the sequence. There are both explicit and sum formulas. Explicit formulas are used to find a single number in a sequence when the first number and the change or common difference between each number is known. The sum formulas are used to find the sum of the whole arithmetic sequence when the first and last number and the amount of numbers are known. The equations and examples will be in a picture below. 

Sequences and Summation

In Mathland, we learned about sequences and summation. A sequence can be infinite and or finite and that appellation pretty much sums up how they are. Numbers with an exclamation mark can also be known as factorials. A factorial is basically the number that is being factorized times all the integers lower than it to zero. For example: 4!=4x3x2x1. However, the factorial of zero is one due to some important reasons. Summation notation is a way one can sum up sequences when the upper limit, explicit formula, and lower limit are known. There are also many properties of summation notations. Below are summations notations and their properties.

Determinants

This was my group's educreation video on how to use determinants:
http://www.educreations.com/lesson/view/p-635-36/16782864/?s=RdR6UY&ref=app

Website

This was my group's website:
https://sites.google.com/a/maranathastudents.org/mathlandgroup5/

Equation of a Line with Determinants

This was my group's voicethread: https://voicethread.com/?#q.b5437871.i27642718

Cryptography

Today in Mathland, we learned how to do some basic cryptography with matrices and their inverses. Cryptography is basically the way of using a code to hide a message. This was done many times in history to bring certain messages securely to another individual without people knowing what was written. Cryptography can be extremely complex, but we will first try it with simple matrices. The way this was done is that the original matrix is the one that encrypts the numbers which corresponds to letters of the alphabet. For example, "A" would be 1, while "C" would be 3. This is done by grouping the numbers we would like to encrypt into a form that can multiply with the encryption matrix to create a code. This code then can be decrypted by multiplying it to the inverse of the original encryption matrix to once again obtain the numbers that correspond to letters. Below, there is the example of the cryptography we tried in class where we decrpyted a code and encrypted the message with a new code:

Inverses of Matrices

Today in Mathland, we learned about how to make inverses of matrices. An in inverse of a matrix or (A^-1) when multiplied to the matrix it is an inverse of or (A) becomes an identity matrix of the same dimensions. They can be multiplied in any order. The steps to do this includes:

1) link the matrix you want to make an inverse of with an identity matrix of the same dimensions

2) use matrix rules to manipulate the matrix so that the left side becomes the identity matrix while the right becomes an inverse matrix.

3) then you can multiply the inverse with its inverse to see if it makes an identity matrix. 

If the left side of the bottom row becomes all zeroes, it is a singular matrix with no inverse matrix. Below, i have examples of both normal and matrices with no solution: (the first one is singular and has no solution while the second one is normal and has a solution)

Matrices Solutions

Hello guys, today I will be explaining when a matrix has solutions and when they have infinite solutions! First of all, a matrix that has no solution will have a row with zeroes in it from the left until it reaches a non-zero number on the far right(the number that is separated by a line. This makes sense because if we put that row in an equation form, 0=a number that is not zero and that is not possible! On the other hand, a matrix that has infinite solutions is one where there are less rows than columns. This means that there will be less equations than variables! To solve theses kinds of problems, we will have to solve in terms of a variable which makes the possibilities infinite! By the way, The row echelon form of a matrix is when it is set up a specific way where the main diagonal in the matrix will only consist of 1's and the numbers surrounding that will all be 0's. Good luck everyone!