Algebraic expressions are the phrases used in algebra to combine one or more variables (represented by letters), constants, and the operational (+ - x / ) symbols. Algebraic expressions, however, don't have an equals (=) sign.

When working in algebra, you will need to change words and phrases into some form of mathematical language. For instance, think about the word sum. What comes to your mind? Usually, when we hear the word sum, we think of addition or the total of adding numbers.

When you have gone grocery shopping, you get a receipt with the sum of your grocery bill. The prices have been added together to give you the sum. In algebra, when you hear "the sum of 35 and n" we know it refers to addition and we think 35 + n. Let's try a few phrases and turn them into algebraic expressions for addition.

### Testing Knowledge of Mathematical Phrasing for Addition

Use the following questions and answers to help your student learn the correct way to formulate Algebraic expressions based on mathematical phrasing:

- Question: Write seven plus n as an Algebraic expression.
- Answer: 7 + n
- Question: What Algebraic expression is used to mean "add seven and n."
- Answer: 7 + n
- Question: What expression is used to mean "a number increased by eight."
- Answer: n + 8 or 8 + n
- Question: Write an expression for "the sum of a number and 22."
- Answer: n + 22 or 22 + n

As you can tell, all of the questions above deal with Algebraic expressions that deal with the addition of numbers — remember to think "addition" when you hear or read the words add, plus, increase or sum, as the resulting Algebraic expression will require the addition sign (+).

### Understanding Algebraic Expressions with Subtraction

Unlike with addition expressions, when we hear words that refer to subtraction, the order of numbers cannot be changed. Remember 4+7 and 7+4 will result in the same answer but 4-7 and 7-4 in subtraction do not have the same results. Let's try a few phrases and turn them into algebraic expressions for subtraction:

- Question: Write seven less n as an Algebraic expression.
- Answer: 7 - n
- Question: What expression can be used to represent "eight minus n?"
- Answer: 8 - n
- Question: Write "a number decreased by 11" as an Algebraic expression.
- Answer: n - 11 (You can't change the order.)
- Question: How can you express the expression "two times the difference between n and five?"
- Answer: 2 (n-5)

Remember to think subtraction when you hear or read the following: minus, less, decrease, diminished by or difference. Subtraction tends to cause students greater difficulty than addition, so it's important to be sure to refer these terms of subtraction to ensure students understand.

### Other Forms of Algebraic Expressions

Multiplication, division, exponentials, and parentheticals are all part of the ways in which Algebraic expressions function, all of which follow an order of operations when presented together. This order then defines the manner in which students solve the equation to get variables to one side of the equals sign and only real numbers on the other side.

Like with addition and subtraction, each of these other forms of value manipulation come with their own terms that help identify which type of operation their Algebraic expression is performing — words like times and multiplied by trigger multiplication while words like over, divided by, and split into equal groups denote division expressions.

Once students learn these four basic forms of Algebraic expressions, they can then begin to form expressions that contain exponentials (a number multiplied by itself a designated number of times) and parentheticals (Algebraic phrases which must be solved before performing the next function in the phrase). An example of an exponential expression with parentheticals would be 2x^{2} + 2(x-2).