 In life, we sometimes don't know the actual identity of a person, and so we use pronouns, him, her, they, it. In mathematics, we sometimes don't know the actual value of a number, and so we use variables, x, y, p, q, and so on. Now, just as there's an etiquette to using pronouns, there's also an etiquette to using variables. Always identify what the variable represents, and always use different variables for different quantities. It's also helpful to use variables evocative of what they represent, and avoid variables that look like numbers or each other. So let's say we want to name variables that represent length, width, and volume, and so we could use x to represent our length. Actually, since length starts with the letter L, we might want to use L as our variable. But since L looks like 1, it might be better to use the capital version so that we can distinguish between our variable and the number. Remember, if it's not written down, it didn't happen. So if L represents length, I've got to write that down. We also want a variable to represent width, so let's think of a good variable named for width, how about w? And in a fit of inspired creativity, we'll use v to represent volume. Or we might want to name variables that represent two numbers, and so we can use x to represent the first number. We could use n, but I'm narrating this video, so I get to pick which variable we use. Now we do need a variable to represent the second number, and here's the important thing. Unless you're willing to guarantee the two numbers are the same, which is to say unless you're willing to put $20 on the table and say that the two numbers are the same, we should use a different variable to represent our second number. And so our second number, how about y? Now that we have variables, we can talk about an expression. An expression is any valid sequence of variables, numbers, and operators. So if I write down 5 plus times, well that's nonsense. This doesn't make any sense at all because 5 is a number, but plus and times are operators, and we need to figure out what we're plus-ing, what we're adding, or what we're multiplying. And so this is nonsense, and this is not an expression. On the other hand, something like 5 plus 7, well these are numbers and an operator, plus, and it's a valid sequence. 5 plus 7 is a valid expression. And we can introduce something like x plus 3y, so these are numbers, variables, and operators, and this is a valid expression because the plus has the things that we're adding. So a very important and very useful idea is the following. The type of expression is determined by the last operation performed. Now when your expression involves variables, it's helpful to ignore the operations once you've done them. And one way we might represent that is to cover things up as we evaluate them, and so at the end of this process we might have a sum. That's something plus something. Where the green and the blue represent some expression that we've evaluated, or maybe we have a difference, something minus something, again where the green and the blue represent some expression that we've evaluated, or we could have a product, something times something, or a quotient, something over something. And again if the idea is in every case the college squares overlap something that we've already evaluated. So let's try to identify the type of expression 3x plus 5 times y plus 8. Now you might say how can you perform these operations when you don't know what x or y are, and that's where this idea of ignoring the operations once you've done them becomes useful. So if we look at this, the order of operations requires us to first of all find y plus 8, since it's in parentheses. Now we can't actually find y plus 8 because we don't know what it is, but we do have to take care of this first, so let's pretend we've done that and so y plus 8 is not there anymore. And maybe it will represent that by covering this up with a blue tab. The order of operations then tells us the next thing we have to do is multiply 3 and x, since this is a multiplication. And again we'll represent the fact that we've done that multiplication by covering it up with a green tab. Now we have another multiplication, 5 times blue tab, and since we've done that we'll cover that up, how about with a red tab. And now the last thing I have to do is I have to add green tab and red tab. Since the last operation performed is an addition, this is a sum. You could also call it an addition, there isn't any important difference between the two. Well how about something like this? So remember that the numerator and denominator of a fraction have implied parentheses, so in our expression we have to take care of the things inside the parentheses first. So let's take a look at that numerator. The first thing we have to do is find 3x because it's a multiplication. So 3x is blue tab. Now since we're still inside our set of parentheses we need blue tab plus 5, so maybe that'll be green tab. Next we have to subtract 8 minus y because our denominator is in parentheses and so when we do that we'll get something, we'll call it red tab. And finally we need to divide numerator, green tab, by denominator, blue tab. And since the last thing we did was a division, this is a quotient or a division. One important idea to keep in mind how you speak influences how you think. This is not a fraction. The type of expression is determined by the operation that you perform. Fraction is not an operation. Now typically expressions don't fall out of the sky and hit you on the head. If they do you're walking in the wrong neighborhood. What we really want to do is to be able to write an expression given some basic information. So for example maybe when we want to write an expression representing the product of 3 numbers. And here's why knowing terms like sum, product, difference, quotient and so on is useful. You're told that this is a product of 3 numbers. Well that's a multiplication and so since this is a product we're multiplying 3 numbers. And our product of 3 numbers, well that's really first number times second number times third number. Now what we could do is let's set up our variables. X is our first number. Y is our second number. Z is our third. But our first number we're calling that X so instead of first number we'll write X. And we'll use parentheses to keep everything from running together. Instead of second number we'll write Y. And instead of third number we'll write Z. And there's our expression for the product of 3 numbers. And we have a few more parentheses than we really need so while we could leave the answer in this form let's get rid of some of these access parentheses that don't actually do anything because there's nothing inside of them to group.