Why Do We Need Rules for Math?
Imagine you and a friend are solving the same math problem. You get 10, but your friend gets 14. Who is right? Without a set of rules, people could get different answers to the same problem!
That is exactly the problem that mathematicians faced hundreds of years ago. As math grew more complicated, people needed to agree on which step to do first. Over time, they created a special set of rules called the order of operations.
So here is the big question: when a math problem has more than one operation (like adding AND multiplying), which do you do first? That is exactly what this lesson will teach you. Let's find out!
The Core Rules: PEMDAS
The order of operations is a set of rules that tells you which step to do first in a math expression. An expression is a math phrase with numbers and operations (like 3 + 4 × 2). We use the memory helper PEMDAS to remember the order.
P — Parentheses
E — Exponents
M & D — Multiply & Divide
A & S — Add & Subtract
See the Order in Action
Let's look at a picture that shows the order of operations as a staircase. You always start at the top step and work your way down!
Notice how multiplication and division share the same step. That means neither one is "more important." You just go from left to right. The same is true for addition and subtraction — they also share a step.
How the Rules Work Step by Step
Let's see how PEMDAS works with real numbers. We will solve the expression 3 + 4 × 2 two ways to prove that the order matters.
Now let's try one with parentheses: (5 + 1) × 3.
Breaking Down a Harder Problem
Some ISEE problems mix several operations together. Let's trace through a bigger expression and watch each step carefully. We will solve: 2 × (3 + 5) − 4 ÷ 2.
See how each step simplifies the expression a little more? By the end, you are left with just one number. That is your answer! On the ISEE, circle or underline each step as you go — it helps you stay organized.
Worked Example: Full PEMDAS Problem
Let's walk through a problem just like one you might see on the ISEE. Follow along step by step!
Common Mistakes and How to Avoid Them
Many students make the same mistakes when using order of operations. The good news? Once you know about them, you can avoid them! Let's look at the most common ones.
| Mistake | What Really Happens | How to Fix It |
|---|---|---|
| Going left to right without checking for × or ÷ first | 3 + 4 × 2 is NOT 14. The answer is 11. | Scan the whole expression for × and ÷ before you start adding. |
| Thinking M always comes before D | 12 ÷ 3 × 2 = 8, not 2. Go left to right. | Remember: M and D are partners. Do whichever comes first from left to right. |
| Forgetting to do what is inside parentheses first | (2 + 3) × 4 = 20, not 14. | Circle any parentheses before you start. Solve them first every time. |
| Doing only one × or ÷ and skipping others | In 2 × 3 + 8 ÷ 4, do BOTH × and ÷ before adding. | Underline ALL × and ÷ parts. Solve them all before moving to + and −. |
From Simple to More Advanced
Right now, you are learning order of operations with basic numbers. As you grow in math, these same rules will help you with bigger problems. Here is a peek at what comes next!
| What You Know Now | What Comes Later |
|---|---|
| Parentheses: (3 + 5) | Brackets and braces: [2 × (3 + 5)] |
| Whole numbers: 2 × 4 + 1 | Fractions and decimals: 0.5 × 4 + 1 |
| Missing number: 3 + __ = 10 | Variables: 3 + x = 10 |
| Simple expressions | Long expressions with many steps |
The great news is that PEMDAS stays the same forever! Once you master it now, you will use the same rules in middle school, high school, and even college. You've got this!
Practice Problems
Time to practice! Try each problem on your own before looking at the answer. Remember: always answer every question on the ISEE — there is no penalty for guessing.
Order of Operations — Quick Review
The order of operations is a set of rules that tells you which math step to do first. Use PEMDAS to remember: Parentheses first, then Exponents, then Multiplication and Division from left to right, and finally Addition and Subtraction from left to right. Remember: "Please Excuse My Dear Aunt Sally."
Key reminders for the ISEE: Multiplication and division are equal partners — just go left to right. The same is true for addition and subtraction. Always do what is inside parentheses before anything else. There is no penalty for wrong answers on the ISEE, so always answer every question. You've got this!