Lesson 5: Order of Operations

New Concept

The Order of Operations is the universal grammar of math, ensuring one correct answer for any expression. You'll master simplifying expressions with multiple operations and learn to use your calculator effectively.

What’s next

You'll start with interactive examples on basic operations, then move on to practice cards involving parentheses and algebraic expressions.

Property

Associative Law for Addition.
If $a$, $b$, and $c$ are any numbers, then

Associative Law for Multiplication.
If $a$, $b$, and $c$ are any numbers, then

Examples

• For addition: $(4+7)+3 = 11+3 = 14$ is the same as $4+(7+3) = 4+10=14$.

Property

The associative laws do not hold for subtraction or division. So, if there are no parentheses in the expression, how do we know which operations to perform first?

1. In a string of additions and subtractions, we perform the operations in order from left to right.
2. Similarly, we perform multiplications and divisions in order from left to right.

Examples

• A subtraction string is calculated from left to right: $30 - 10 - 5$ becomes $(30-10)-5 = 20-5=15$.

• A division string is also calculated from left to right: $48 \div 8 \div 2$ becomes $(48 \div 8) \div 2 = 6 \div 2 = 3$.

Property

Always perform multiplications and divisions before additions and subtractions.

In longer expressions, it can be helpful to group the expression into its terms before beginning. Terms are expressions separated by addition or subtraction symbols. We simplify each term before combining them.

Examples

• To simplify $10 + 3 \cdot 5$, you multiply first: $10 + 15 = 25$.

Property

Perform any operations inside parentheses first.

Like parentheses, a fraction bar is a grouping device. Expressions that appear above or below a fraction bar should be simplified first.

Examples

• Using parentheses, $3 \cdot (9-4)$ becomes $3 \cdot 5 = 15$.

Property

1. First, perform any operations that appear inside parentheses, or above or below a fraction bar.

2. Next, perform all multiplications and divisions in order from left to right.

3. Finally, perform all additions and subtractions in order from left to right.