Why Do We Need Rules for Math?
Imagine you and a friend both try to solve 3 + 4 × 2. You add first and get 14. Your friend multiplies first and gets 11. You used the same numbers and the same symbols, but you got different answers! That's a big problem, especially when math is used in science, engineering, and banking.
Mathematicians realized centuries ago that everyone needs to follow the same set of rules. Without a shared agreement, math becomes confusing and unreliable. The order of operations is that shared agreement. It tells you exactly which calculations to do first, second, and third.
Today, every calculator and computer on the planet follows these same rules. When you master the order of operations, you're speaking the universal language of math. Let's learn exactly how it works.
The Core Rules: PEMDAS
The order of operations is a set of rules that tells you the exact sequence to follow when an expression has more than one operation. The memory trick PEMDAS helps you remember the order. Each letter stands for a step.
P — Parentheses
E — Exponents
MD — Multiply & Divide
AS — Add & Subtract
See the Steps in Action
The diagram below shows the PEMDAS staircase. You start at the top and work your way down. Each step must be completed before you move to the next one. Notice that multiplication and division share the same step, and addition and subtraction share the same step.
Notice how Multiply and Divide share one step, and Add and Subtract share another. When two operations are on the same step, you just read left to right. This is the key detail that many students miss, and it's exactly what the ISEE likes to test.
The Rules Written Out
Let's write the order of operations in a clear, step-by-step way. When you see a long expression on the ISEE, follow these rules one at a time. Think of it as a checklist.
Tracing Through a Full Expression
Let's trace through a medium-difficulty expression step by step. The diagram below shows how the expression 5 + 3 × (8 − 2)² ÷ 6 gets simplified one piece at a time. Each colored bar shows which operation we do at that step.
Take a close look at Steps 3 and 4. We multiplied before we divided, but only because multiplication came first from left to right. If the division had been on the left, we would have divided first. This left-to-right rule is the most important detail to remember on test day.
Worked Example: Full Solution
Let's work through a problem just like one you might see on the ISEE. We'll solve it slowly, showing every step and explaining our reasoning.
Common Mistakes & How to Avoid Them
The ISEE test-makers design wrong answer choices based on the most common student errors. If you know what mistakes people usually make, you can avoid falling into those traps. Here's a comparison of correct approaches versus common mistakes.
| Mistake | What Students Do Wrong | Correct Approach |
|---|---|---|
| Left-to-right ignore | Always multiply before dividing (e.g., in 12 ÷ 4 × 3, they do 4 × 3 first) | Do ÷ and × left to right: 12 ÷ 4 = 3, then 3 × 3 = 9 |
| Adding before multiplying | In 2 + 5 × 3, they add 2 + 5 = 7, then multiply: 7 × 3 = 21 | Multiply first: 5 × 3 = 15, then add: 2 + 15 = 17 |
| Exponent error | Think 3² means 3 × 2 = 6 | 3² means 3 × 3 = 9. The exponent tells you how many times to multiply the base. |
| Forgetting parentheses | In (4 + 6)², they do 4 + 6² = 4 + 36 = 40 | Parentheses first: (4 + 6) = 10, then 10² = 100 |
Order of Operations and Algebra
The order of operations isn't just for arithmetic. It's the foundation for everything you'll do in algebra and beyond. When you start working with variables (letters that stand for numbers), PEMDAS still applies in exactly the same way.
| What You Know Now | What's Coming Next |
|---|---|
| Evaluate 3 + 4 × 2 = 11 | Evaluate 3 + 4x when x = 2 → 3 + 4(2) = 11 |
| Simplify (5 + 1)² = 36 | Simplify (a + 1)² when a = 5 → same PEMDAS rules |
| Follow left-to-right for × and ÷ | Solve multi-step equations using reverse order of operations |
| Use parentheses to group numbers | Use the distributive property: 3(x + 2) = 3x + 6 |
On the ISEE, you may see problems that combine the order of operations with substituting a number into a variable expression. Don't worry — if you follow PEMDAS step by step, those problems work exactly the same way. You'll just replace the variable with the given number first, then follow the rules.
Practice Problems
Try these five problems. They get harder as you go. Remember to follow PEMDAS and work left to right for multiplication/division and addition/subtraction. There is no penalty for guessing on the ISEE, so always pick an answer!
Order of Operations: Your Quick Review
The order of operations is a universal set of rules for evaluating math expressions. Use the memory aid PEMDAS: first solve inside Parentheses, then calculate Exponents, then handle Multiplication and Division from left to right, and finally Addition and Subtraction from left to right.
The biggest trap on the ISEE is the left-to-right rule: multiplication does NOT always come before division, and addition does NOT always come before subtraction. When operations are at the same level, go left to right. Write out each step on scratch paper, check your work against PEMDAS, and you'll avoid the trap answers that trip up other students. You've got this!