Variables, Expressions, and Open Sentences
Variables
Sometimes, 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?
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
d
4
Variables, Expressions, and Open Sentences
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!
1. Which variable expression below represents “seven less than a number?”
Explain.

n+7

n÷7

n–7

7xn
2. Write the variable expression for “three more than a number.”
3. Write the variable expression for “twice a number.”
4. 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?
Variables, Expressions, and Open Sentences
Open Sentences
A 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.
In 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.
The 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:
Joseph 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
Variables, Expressions, and Open Sentences
Variable Expressions and Number Operations
Below are examples of variable expressions with the different number operations. Notice
that we can write expressions involving multiplication and division in two different ways.
Addition
Subtraction
Multiplication
a+5
x–3
6y or 6 x y
Division
d ÷ 8 or
d
8
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 = $72

8t = $72

t
= $72
8

t – 8 = $72
2. 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.