What Is the Order of Operations?

Why Order Matters

Look at this problem: 3 + 4 × 2. If you work left to right, you get 3 + 4 = 7, then 7 × 2 = 14. But if you do the multiplication first, you get 4 × 2 = 8, then 3 + 8 = 11. Two different answers from the same expression — that's a problem. Mathematics needs every person to get the same answer from the same expression, so we need a set of agreed-upon rules about which operations come first. That's the order of operations.

PEMDAS — The Six Steps

The standard order is remembered by the acronym PEMDAS, which many students learn as "Please Excuse My Dear Aunt Sally":

P — Parentheses first. Always do whatever is inside parentheses (or brackets) before anything else. They're like VIP passes that move operations to the front of the line.

E — Exponents next. Powers and square roots come second (like 3² = 9 or √16 = 4).

M and D — Multiplication and Division are equals — do them left to right, whichever comes first. They don't rank one above the other.

A and S — Addition and Subtraction are also equals — do them left to right, whichever comes first.

Walking Through an Example

Let's solve: 2 + 3 × (8 − 2)² ÷ 6

Parentheses: 8 − 2 = 6. Now we have: 2 + 3 × 6² ÷ 6.

Exponents: 6² = 36. Now we have: 2 + 3 × 36 ÷ 6.

Multiplication and Division (left to right): 3 × 36 = 108, then 108 ÷ 6 = 18. Now we have: 2 + 18.

Addition: 2 + 18 = 20.

Without the order of operations, you might get wildly different answers depending on where you start. With it, there's exactly one correct answer.

The Most Common Mistake

The biggest trap is thinking M comes before D and A comes before S. They don't. Multiplication and division have the same priority — you simply go left to right. Same for addition and subtraction. In the expression 12 ÷ 4 × 3, you do the division first (12 ÷ 4 = 3) then multiply (3 × 3 = 9), because the division is to the left. If you multiplied first, you'd get 12 ÷ 12 = 1, which is wrong.

Why These Rules Exist

The order of operations isn't arbitrary — it's designed so that mathematical expressions can be written efficiently and read the same way by everyone, everywhere in the world. It's a universal convention, like driving on a specific side of the road. Without it, every formula in science, engineering, and finance would need excessive parentheses to be clear, and communication between mathematicians would break down.

Why the Order of Operations Exists

The order of operations isn't an arbitrary set of rules — it exists so that every person reading a mathematical expression gets the same answer. Without it, the expression 3 + 4 × 2 could mean 14 (if you add first) or 11 (if you multiply first). Mathematicians agreed on a standard order centuries ago so that formulas, equations, and calculations would be unambiguous everywhere in the world. It's like grammar for math: without agreed-upon rules, communication breaks down.

This concept becomes critical in algebra, science, programming, and finance. Every spreadsheet formula, every line of computer code, and every physics equation depends on the order of operations being applied consistently. A child who masters it now will never be tripped up by a formula later.

Where Kids Get Stuck

The most common mistake is always working left to right without considering operation priority. Children solve 5 + 3 × 2 as 16 instead of 11 because they add 5 + 3 first. The fix is practice with visual highlighting — circling the multiplication and division first, then going back for addition and subtraction.

Another tricky area is nested parentheses. In expressions like 2 × (3 + (4 − 1)), students must work from the innermost parentheses outward. Thinking of it as "unwrapping layers" — solve the deepest layer first — helps make the process logical.

Students also confuse left-to-right order within the same priority level. In the expression 12 ÷ 3 × 2, multiplication and division have equal priority, so you work left to right: 12 ÷ 3 = 4, then 4 × 2 = 8. Doing the multiplication first would give 12 ÷ 6 = 2, which is wrong. The same applies to addition and subtraction at their level.

Beyond PEMDAS: Understanding the Levels

PEMDAS (or BODMAS in some countries) is helpful as a memory aid, but it can mislead if students think multiplication always comes before division. It's better to think of the order as having four levels, not six steps:

• Level 1: Parentheses (or brackets) — always first
• Level 2: Exponents (or powers/indices)
• Level 3: Multiplication and Division — equal priority, left to right
• Level 4: Addition and Subtraction — equal priority, left to right

Understanding these as levels rather than a strict sequence prevents the "multiplication before division" error that catches so many students.

Try This at Home

• Calculator challenge — Type an expression into a calculator and predict the answer first. Does the calculator follow order of operations? (Scientific calculators do; basic ones may not.)
• Write your own puzzle — Ask your child to create an expression where the answer changes depending on whether you follow the order of operations or just go left to right.
• Parentheses power — Give the expression 2 + 3 × 4 and ask: "Where would you put parentheses to make the answer 20? What about 14?"
• Real-world expressions — "If you buy 3 notebooks at $2 each and 1 pen for $1.50, write an expression for the total cost: 3 × 2 + 1.50."

For help with overall math confidence, see: Signs Your Child Is Struggling with Math.

🔣 Practice Order of Operations

Last reviewed: May 2026