 In a perfect world, we'd always get mathematical problems in mathematical forms. Equations would fall out of the sky. We don't live in that world, and it's often necessary to translate a problem phrased in a natural language into a mathematical statement. To understand writing equations, it's important to keep in mind that the most important mathematical symbol is the equals symbol. The important thing to remember about this equals means replaceable. When you write A equals B, you mean that everywhere you see A, you can replace it with B, and everywhere you see B, you could replace it with A. If you keep this in mind, translating a statement in English or any other natural language like Mandarin or Klingon into a mathematical statement becomes much easier. For example, let Z represent the number of students in a class express five more than the number of students. A useful strategy is to ask yourself, what would I do if I had the actual number? Remember paper is cheap, so let's write stuff down. If Z represents the number of students in a class, then we could write down Z equals the number of students. So suppose you knew the actual number of students. If there are twelve students in the class, then five more than the number of students would be seventeen. Now an important component of mathematics and life is asking yourself, how did I get that? And you probably got the number seventeen by adding twelve plus five. And here's an important idea. It helps not to do the arithmetic. So we want to write down five more than the number of students as twelve plus five. So remember, equals means replaceable. If Z is equal to the number of students, then anytime we see number of students, we can replace it with Z. Well how about here? So we'll replace the number of students with Z. But equals means replaceable. Twelve equals Z means that anytime we see twelve, we can replace it with Z. Well how about here? And equals means replaceable. So our expression for five more than the number of students is going to be Z plus five. And we'll do one more thing. We don't actually know the number of students, so we'll hide this line twelve equals Z. Or again, let Z represent the number of students in a class. Let's express six less than the number of students. So again, a useful strategy is to ask yourself, what would I do if I had the actual number? And again, it helps not to do the arithmetic. So suppose you had ten students. Six less would be four, which you got by subtracting ten minus six. Equals means replaceable. So anytime we see number of students, we can replace it with Z. Equals means replaceable. So anytime we see ten, we can replace it with Z. And so our expression for six less than the number of students is going to be Z minus six. And again, since we don't actually know the number of students is ten, let's cover that information up. Or let's talk about half of the students. So suppose there were five students, then half would be, well, suppose there were six students. What's important to understand here is the number of students we actually begin with doesn't matter because we're going to get rid of that number anyway. So let's use numbers that are easy for us to work with. So half the number of students would be three. And we probably got that by dividing six by two. And again, it helps not to do the arithmetic. So half the number of students will record as six divided by two. Equals means replaceable. So the number of students can be replaced with Z. Equals means replaceable. So six can be replaced by Z. And we'll hide the number of students. So our expression for half the number of students is going to be Z divided by two. One very important caution is to pay careful attention to grammatical grouping symbols, especially commas. Now we sometimes joke about misplaced commas, but in fact they are a serious problem. Commas matter. For example, let's consider two very similar statements. Three times the number of students, comma, and four, versus three times the number of students and four, no comma. What's important to understand here is that the comma in the first sentence means that the phrase preceding the comma is one thing. On the other hand, in the second phrase, the lack of the comma means the number of students and four is one thing. To see why this is different, suppose there are ten students. Then three times the number of students and four, well, that's going to be three times ten plus four, or 34 students. On the other hand, three times the number of students and four, since number of students and four is one thing, we should consider those as being inside a set of parentheses. And three times that amount is going to be three times parentheses ten plus four, that's 42 students. This shows up in our algebraic expressions. So again, let's see represent the number of students in a class. Let's express twice the number of students, comma, and eight, and compare this to twice the number of students and eight, no commas. So again, we'll let Z be the number of students, and we'll pick some number of students, about eight. Actually, eight is a bad choice. As a general rule, you should avoid using any number that's actually part of the problem. And since eight is part of the problem, as is two, we don't want to choose either one of those. So let's pick, oh, I don't know, about ten. Now because of the comma, we want to group this twice the number of students. So we find twice the number of students. Well, that'll be twenty, and we got that by multiplying two by ten. To get and eight, we'll want to add eight to the number. So twice the number of students and eight will be two times ten plus eight. Equals means replaceable, so every time we see the number of students, we'll replace it with Z. Equals means replaceable, so every time we see ten, we'll replace it with Z. And since we have an algebraic expression, we don't need the explicit time symbol. And that gives us our algebraic expression for twice the number of students, comma, and eight. What if we didn't have a comma? Again, we'll take ten students. This time, since there's no comma, the number of students and eight is one number. So we'll find the number of students and eight, well, that'll be ten plus eight. And we want twice that number, so that's going to be two times parenthesis ten plus eight. Equals means replaceable, so every time we see the number of students, we'll replace it with Z. Equals means replaceable, so every time we see ten, we'll replace it with Z. And this gives us our algebraic expression for twice the number of students and eight. And for comparison, here's what we wrote for twice the number of students, comma, and eight.