Here’s the first post. Hopefully everything is understandable, and if you have any questions, feel free to ask! Everything here should be a review, but ask if something doesn’t make sense to you. I wanted this to get out of the way we could get to the more interesting bits, so tell me if I rushed something too much. Thanks for reading!
Topic 1: Arithmetic review
Topic 1
Studying the structure of an equation (a review of arithmetic)
Now, all of you should have a good understanding of arithmetic before attempting to understand algebra. If you don’t, go finish that first. Arithmetic is the core of all math, and it is of utmost importance that you understand it before continuing on to more complicated subjects.
The equals sign
So first off, what is an equation? Well let’s look at the etymology of the word equation. Just by dissecting the word, you can see that it is similar to equal + tion. For two things to be equal, they are essentially equivalent. So basically what an equation is mathematical statement that two values or equal, equivalent, and the same. The mathematical symbol that denotes equality is the equals sign, =.
So, the most simple equation is:
0 = 0
Zero is equal to zero. Or:
7 = 7
Seven is equal to seven.
This is important to understand. Unlike arithmetic, where the equals sign is usually used to say “what’s the answer?”, for example:
2 + 2 =
In algebra, the equals sign is commonly used to denote that two things are equal, and it is up to you to figure out how. That doesn’t mean that the equals sign in arithmetic is somehow a different symbol than the one in algebra. They are very much the same. Algebra just unveils the complexity of this seemingly simple symbol.
Arithmetic operators
If you have indeed listened to my advice, you should be familiar with these fundamental operators:

Plus (+)  this is the symbol for addition. It takes two numbers and combines their values. Eg. 4 + 2 = 6. What exactly is this saying? If you take 4 of anything, and add 2 more of that something, you will end up with 6 of that something. For instance, if you have 4 sticks, which we can represent as , and 2 more sticks (), in total you will have 4 + 2 sticks, which is  or 6 sticks.

Minus ()  this is the symbol for subtraction. It takes two numbers and tells you the difference between them. For example. Eg. 4  2 = 2. This means that if you have four of anything, and you take away 2 of that something, you’ll be left with two of that something. Let’s use our stick example. Our four sticks look like this: , now if we take away two sticks (), we’ll be left with  or 2.

Times (×)  this is the symbol for multiplication. It takes two numbers and gives you what you would have if you had some number of times the other value. This is essentially repeated addition. If you have 4 sticks, and someone else has 3 times that many sticks, that means they have 4 × 3 sticks, or 4 + 4 + 4 sticks, which is 12.

Divide (divided by) (÷)  this is the symbol for division. If you have two values, it tells you how many times one value can fit into another. If you have 6 sticks and you give separate them equally among two of your friends, each friend will get 6 ÷ 2 = 3 sticks.
There are a few more points we must go over about these operators. First off, if you haven’t noticed, addition and subtraction or what we call “inverse” or “opposite” operations. If you add 2 to something and then subtract 2, you’ll end up with the same value you started with. (This happens no matter which operation you use first). For example: 10 + 6  6 = 16  6 = 10.
The same is true with multiplication and division. If you, for example, multiply something by 2 and then divide that by 2, you will end up with what you began with. (This too, is not affected by the operation you decide to use first). For example: 10 × 6 ÷ 6 = 60 ÷ 6 = 10
Another important note is that addition and multiplication are commutative. This means that if you add 2 and 7, that is the same as adding 7 and 2. This is pretty intuitive if you think about what exactly addition is doing. Think about it with the stick example I was using earlier. 2 + 7 can be written as  +  =  = 9. 7 + 2 =  +  =  = 9
Now, let’s think about why multiplication would be commutative. Let’s say you have 7 × 6. This can be written like 7 + 7 + 7 + 7 + 7 + 7. So let’s say we’re dealing with little pebbles, it can be written as:
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. + . + . + . + . + . = . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
6 × 7 can be written as:
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. + . + . + . + . + . + . = . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
Which you may notice is just 7 × 6 flipped on its side! So there are the same number of pebbles!
Subtraction and division are not commutative, though. For instance: 9  6 = 3, but 6  9 = 3. 4 ÷ 2 = 2, but 2 ÷ 4 = 0.5 (I will discuss fractions, decimal points, and negative numbers soon.)
There are two more operators you should be aware of:

Power  this can be written as something like 6^{3} or 4^{2}. What this means is that you take the big number on the bottom and multiply it by itself the number of times the number on the top says. For example: 5^{3} = 5 × 5 × 5 = 25 × 5 = 125. Sometimes this will be called “squared” or “cubed”. If you have some number squared, it is multiplied by itself twice (ie. 5 squared is 5^{2} = 5 × 5 = 25). If a number is cubed, it is multiplied by itself three times (ie. 4 cubed is 4^{3} = 4 × 4 × 4 = 16 × 4 = 64)

Root (√) this can be written as something like ^{2}√9 or ^{3}√64 (if no number is given, like √25, then it is the “square” root, the number is assumed to be 2). What this means is you’re looking for a number that multiplied by itself the number of times the small number says, you’ll get the number inside the √. For example:
^{3}√64 = 4 because 4^{3} = 64. √25 = 5 because 5^{2} = 25.
Both powers and roots are what we call “exponents”.
The order of operations
The order of operations is the universally accepted order in which operations are performed. It goes:
 Parentheses (sometimes called brackets)
 Exponents
 Multiplication and division (multiplication does not always come before division. They are performed from left to right.)
 Addition and subtraction (again, they have the same priority, and are performed from left to right.)
A common acronym for this is PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction. BUT REMEMBER: multiplication and division have the same priority and are evaluated from left to right, and so is addition and subtraction.
For instance:
6 + 2 × 4 ÷ (9  7) = 6 + 2 × 4 ÷ 2 = 6 + 8 ÷ 2 = 6 + 4 = 10
Now, here’s a trick one:
4 ÷ 2(3  1)
Is it 4 ( 2 × 2 = 4 )
Or 1 ( 4 ÷ (2 × 2) = 4 ÷ 4 = 1)?
The answer is actually 1. I’ll explain why this is later when we discuss variables and parameters.
Fractions and decimals
Fractions are really just another notation for division. They can be written like: ^{2}/_{9} which is really just 2 ÷ 9. The upper part of the fraction is called the “numerator” and the lower half is the “denominator”. If you have a fraction like: ^{4}/_{16}, you can divide (or multiply) both halves of the fraction. So ^{4}/_{16} = ^{4 ÷ 4}/_{16 ÷ 4}.
Decimals are just numbers written in terms of powers of 10. So like: 0.2 = 2/10 = 1/5, 0.34 = 34/100 = 17/50 and so on.
Negative numbers
Negative numbers are numbers that come before zero on the number line. They are written with a minus sign before them, like 1. 1’s distance from 0 is 1, but it comes before 0. Subtraction can be written as the addition of negative numbers. For instance: 4  2 = 4 + (2). Notice the parentheses around 2. These aren’t strictly necessary, but I suggest you use them.