Basic Math and Pre-Algebra
PART 2. Into the Unknown
It’s nice to take a world tour, even in the world of numbers, but sometimes it can be a scary trip. If you don’t know where you’re going, or you think you do but find yourself lost, you can feel unsettled. Whether you’re in a foreign country or in the middle of solving a problem, realizing that you’re lost is a frightening feeling. The best way to prevent that situation is good preparation, so that’s what we aim to do.
In this part, you’ll prepare to deal with those situations in which one or more of the numbers in your problem is unknown. You’ll look at ways to find those missing numbers, or if that’s not possible, at least to organize the possibilities. Just as having a good map and a good plan will keep you from getting lost, having a clear picture and knowing what steps to take will help you through a problem. You’re about to draw the map that will let you find your way into algebra.
CHAPTER 8. Variables and Expressions
One of the markers of the shift from arithmetic to algebra is the introduction of variables. In arithmetic, you know the numbers and you know what operations need to happen. In algebra, you step into the world of the unknown. You know some, but not all, of the numbers, and you know the result of the operations. A lot of your work is going backward to try to fill in the unknown pieces. Until you figure out what they are, those unknown numbers are represented by variables.
In this chapter, you’ll learn what variables are and how to write phrases and sentences that use them. When you start using variables, it’s important to consider what kind of number the variable might be standing in for. Those numbers are called the domain of the variable. This is also a time to look at some of the arithmetic you can do with variables and learn where you need to be especially careful and why.
A variable is any symbol that stands for a number. In algebra, the symbol is usually a letter, but that doesn’t have to be the case. You might have looked at questions that said 3 + ? = 5 and you understood that you were supposed to find the number that replaced the ? to make the statement true. In that case, ? is a variable. You might have seen puzzles that said ∆ + ◊ = 9. There are many different right answers for that one, but the ∆ and the ◊ are acting as variables.
A variable is a letter or symbol that takes the place of a number.
A variable is used to take the place of a number because the value of the number is unknown, or because the number that goes in that place may be changing. Perhaps a pattern is being represented in which different values are possible.
For example, imagine you are taking a test with 20 questions. Each question is worth 5 points, so if you get all the questions correct, you earn 100 points. But what if you don’t get them all right? Your teacher might use a rule or pattern that says your score is 5 points times the number of questions you answer correctly. Using the variables 5 for your score and n for the number of questions you answer correctly, you can write the rule as S = 5 x n. That rule will apply to everyone who takes the test, but each person may get a different number of questions correct. The variable n can have many different values. It varies. That’s where the name variable comes from.
Suppose on a different test you got your paper back with a grade of 87 on it. You want to know how many points each question was worth. You can find out by counting the number of questions you answered correctly and using the variable p to stand for the point value of one question. If you answered 29 questions correctly, you can say 29 x p = 87. Here you’re using a variable because you don’t know the number of points. The variable stands for a number that is unknown.
The Language of Variables
When you start to use variables in your phrases and sentences, it’s a lot like learning a new language. You need to learn your vocabulary, understand the grammar of the language, including the idioms, and practice, practice, practice.
Let’s start with parts of speech: nouns, verbs, and such. In algebra, numbers and variables are your nouns, and symbols like =, <, and ≥ serve as verbs. Operation signs, like + and —, act as conjunctions, the way “and” and “or” would in English.
To translate algebra into English, you can usually just read the symbols and make a few shifts, like saying “and” for + and “is” for =. The sentence x + 3 = 8 is the algebra language equivalent of “some number and three combine to make eight.” The x, the 3, and the 8 are nouns. The x and the 3 are connected by the conjunction +, and the verb, the action of the sentence, is the =. The = says “is,” or “makes.”
The difference between a phrase and a sentence is whether or not you have a verb. Sentences have verbs, phrases do not. x + 9 is a phrase. It has no equal sign or inequality sign. It sounds like “a number increased by 9.” On the other hand, x + 9 > 21 is a sentence. The > is the verb “is greater than.” It says “some number increased by 9 is greater than 21.”
When parentheses appear, it’s often easier to read if you refer to “the quantity.” The sentence 5(x + 7) = 45 can be read as “five times the quantity x plus 7 is 45.” The words “the quantity” tell your listener that parentheses are grouping some symbols into one quantity.
The larger your vocabulary, the better able you are to say what you really mean, so let’s look at some common language. That sentence 5(x + 7) = 45 could also be read as “five times the sum of x and 7 is 45.” A sum is the result of an addition. If you change the plus sign to a minus sign, 5(x - 7) = 45 can be read as “five times the difference of x and 7 is 45.” A difference is the result of a subtraction. Whenever you talk about a difference, you read the numbers in the order they appear. The difference of x and 7 is x - 7. The difference of 7 and x is 7 - x.
Idioms in Algebra
Idioms are phrases used in a language that just can’t translate word for word. If I tell you to give me break, I’m not asking you to hurt me. I’m telling you to stop teasing me or trying to fool me. “Give me a break” is an idiom.
Algebra has idioms, too. One in particular has to do with subtraction. The phrase “6 less than some number” sounds as though it would begin with a 6 if you wrote it in symbols, but it actually translates to x - 6. To see why, put a number in place of the words “some number.” If I say “6 less than 25,” you think of 25 - 6, or 19. You take 6 away from the number. So 9 less than some number is t - 9, and 83 less than some number is y - 83.
Don't confuse “6 less than some number” with “6 is less than some number.” The word is signals a verb. “6 less than some number” translates to x - 6, but “6 is less than some number” would be 6 < x.
If you want to read the phrase 7(x + 5), you could say “seven times the quantity x and 5,” or you could say “the product of 7 and the quantity x plus 5.” A product is the result of multiplication. When you want to say something about division, like x/5 = 11, you can say “a number divided by 5 is 11,” or you can say “the quotient of x and 5 is 11.” A quotient is the result of division. The order in which the numbers are named tells you which is the dividend and which is the divisor. The quotient of x and 5 is x ÷ 5, but the quotient of 5 and x is 5 ÷ x.
Write each sentence using variables.
1. The quotient of some number x and twelve is greater than four.
2. The difference of a number t and nineteen is twenty-two.
3. The sum of a number n and the quantity n increased by three is one hundred fifty-four.
4. The product of a number y and the quantity five less than y is eighty-four.
5. The quotient of a number p and the quantity p increased by one is two.