Variables and Expressions



VariablesSometimes, math problems can be challenging, especially if they are word problems and represent events that happen in real-life. In algebra, we are able to turn real-life problems into mathematical phrases or expressions and sentences that help us solve problems. We usually use variables when we write mathematical expressions and sentences. A variable is a symbol that can stand for any unknown number or object. There are common variables that are used, like the letters n or x, but we can use any letter of the alphabet. Sometimes we might even see a box () used as a variable. When we use a variable in math, we say “n represents (a number of something).” This number might be the number of apples we want to buy at the store, or the cost of a movie ticket, or the length of the side of a rectangle. Example:Mrs. Smith went to the store to buy some apples to bake pies. She spent $5.00. The apples were $2.00 a pound. Mrs. Smith needed lots of apples for her pie. How many pounds of apples did she buy? 1428750317500 To solve this problem, we can start by saying, “Let p (or any other letter) represent the number of pounds.” If p represents the number of pounds, then $2.00 x p stands for the total cost of the apples. “$2.00 x p” is called a variable expression.Here are some other examples of variable expressions: 5x n – 3 t + 8 A variable expression is like a word phrase. A word phrase like, “in the house” does not have a verb. To make it into a complete sentence, we would have to add a subject and a verb: “My dog is asleep in the house.” A variable expression does not have an equal sign. For example n – 3 is a variable expression. Notice that it does not contain an equal sign. To make it a complete sentence, we would have to add an equal sign with a value on the other side of the equals sign, such as: n – 3 = 4.PRACTICE!Which variable expression below represents “seven less than a number?” Explain.n + 7n ÷ 7n – 77 x nWrite the variable expression for “three more than a number.”Write the variable expression for “twice a number.”Sandra ate 4 fewer M&Ms than Billy. If m represents the number of M&Ms Billy ate, what variable expression could be written to find out how many M&Ms Sandra ate?Open SentencesA sentence is open when it is not known if the sentence is true or false. For instance, n is a counting number. If n = 5, the open sentence would be true. If n = 0, the open sentence would be false. An open sentence is not true or false until we replace the variable with a number.582930062230000In mathematics, an open sentence contains both a variable and an equal sign. A simple sentence like, “The number of crayons is 15.” would be written as c = 15, where c is a variable that stands for the number of crayons.594360012573000514350035433000The sentence, “A full bag of candy and two extra equal 24 pieces of candy,” would be written as b + 2 = 24. The variable b represents the number of candies in one full bag. The 2 represents the two extra.“Three full bags of candy equal 60 pieces of candy” would be written as 3b = 60.Think! What does the 3b represent?We have compared a variable expression to a written phrase. We can also compare an open sentence in math to a complete sentence in writing. A complete sentence in writing must have a subject and verb. An open sentence in math must have a variable as a subject and an equal sign as a verb. Example:365760026225500Joseph has 18 fish in his fish tank. If 5 are goldfish, how many are angelfish? Let f be the number of angelfish. The open sentence would written as: f + 5 = 18 Variable Expressions and Number OperationsBelow are examples of variable expressions with the different number operations. Notice that we can write expressions involving multiplication and division in two different ways. AdditionSubtractionMultiplicationDivisiona + 5x – 36y or 6 x yd ÷ 8 or We may think that using the variable x can be confusing because the multiplication sign (x) is very similar to the letter “x”. Because it can be confusing, we might see multiplication indicated by the letter x, parentheses ( ), with a dot, or with nothing at all. 3 x n or 3(n) or 3 · n or 3n all mean the same thing: three times a number (n).PRACTICE!1. Which of the following represents the open sentence, “Eight movie tickets cost $72.”? Let t be the number of tickets.8 + t = $728t = $72 = $72t – 8 = $722. Write the open sentence for the following problem: Twenty-four children are signed up for soccer. There will be 3 teams. How many children will be on each team? Let c stand for the number of children. ................
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