Order of Operations Guide: PEMDAS, Common Mistakes, and Practice Tips

Overview

The order of operations is the agreed sequence for simplifying a math expression. Its job is simple: it makes sure everyone reading the same problem gets the same answer. Without that shared order, a problem like 8 + 2 × 5 could produce different results depending on where a person starts.

The most common memory aid is PEMDAS:

• P = Parentheses
• E = Exponents
• M = Multiplication
• D = Division
• A = Addition
• S = Subtraction

That mnemonic is useful, but students often misunderstand it. The biggest correction to remember is this: multiplication and division are at the same level, and addition and subtraction are at the same level. That means you do them from left to right, not by choosing multiplication before division or addition before subtraction just because of the letters.

Here is the core rule set in plain English:

1. Simplify inside grouping symbols first, such as parentheses or brackets.
2. Evaluate exponents next.
3. Work through multiplication and division from left to right.
4. Work through addition and subtraction from left to right.

Try a basic example:

3 + 4 × 2

Do multiplication first: 4 × 2 = 8

Then add: 3 + 8 = 11

If you added first, you would get 7 × 2 = 14, which is not correct because it ignores the standard order.

Now try one with parentheses:

(3 + 4) × 2

Do the parentheses first: 3 + 4 = 7

Then multiply: 7 × 2 = 14

These two examples look similar, but the grouping changes the answer. That is exactly why the order of operations matters.

It also helps to know what kind of problem you are solving. This article focuses on expressions, not equations. An expression has no equals sign and is simplified. An equation has an equals sign and is solved. The order of operations still matters in both cases, but the goal is slightly different.

If you are using this for math homework help, a good habit is to rewrite each line cleanly after each step. That makes it easier to catch mistakes before they multiply. For a broader checking process, see How to Check Your Homework Answers Without Copying or Guessing.

How to solve mixed operations step by step

Use this repeatable method whenever a problem contains more than one operation:

1. Scan the whole expression before touching any numbers.
2. Mark any parentheses or grouping symbols.
3. Check for exponents.
4. Move left to right through multiplication and division.
5. Move left to right through addition and subtraction.
6. Rewrite the expression after every step.

Example:

18 ÷ 3 × 2 + 5

There are no parentheses or exponents. Start with multiplication and division from left to right:

18 ÷ 3 = 6

Now the expression becomes:

6 × 2 + 5

Continue left to right:

6 × 2 = 12

Now:

12 + 5 = 17

Final answer: 17

Notice that you do not multiply 3 × 2 first. The left-to-right rule matters.

One more example with exponents

2 + 3² × 4

Evaluate the exponent first:

3² = 9

Now the expression is:

2 + 9 × 4

Multiply:

9 × 4 = 36

Add:

2 + 36 = 38

Final answer: 38

Maintenance cycle

This topic is worth revisiting because order of operations mistakes tend to return when students get rushed, distracted, or move into harder math. A short maintenance cycle helps keep the rule fresh without turning it into a huge review session.

A practical review schedule looks like this:

• Weekly: do 3 to 5 mixed-operation problems.
• Before a test: review one page of worked examples and solve a few problems without notes.
• When starting a new math unit: check whether the new topic still depends on careful simplification.
• When homework errors repeat: slow down and return to line-by-line rewriting.

The goal is not memorizing a chant. The goal is building a reliable habit. A student may remember “PEMDAS” perfectly and still get problems wrong if they skip the left-to-right rule or combine steps mentally.

Here is a simple 10-minute refresh routine you can return to:

1. Minute 1: write the rule in your own words.
2. Minutes 2 to 6: solve 3 sample expressions.
3. Minutes 7 to 8: compare your steps, not just your answers.
4. Minutes 9 to 10: note one mistake pattern to watch next time.

For example, your note might say:

• I keep adding before multiplying.
• I forget exponents.
• I do division after multiplication even when it appears first.
• I try to do too much in one line.

That kind of short reflection makes review more useful than repeating random practice.

If you want to make the review stick, pair it with a study routine rather than waiting until the night before a quiz. A short timed session can help you focus without overloading. Related reading: Pomodoro for Studying: Best Session Lengths for Different Subjects.

A quick self-check method

After solving any expression, ask these questions:

• Did I simplify all parentheses first?
• Did I evaluate exponents before multiplying or dividing?
• Did I treat multiplication and division as equal steps?
• Did I work left to right?
• Did I treat addition and subtraction as equal steps?
• Did I rewrite the expression after each operation?

If the answer to any one of those is no, redo the problem slowly. In many cases, the answer changes immediately.

Signals that require updates

You should revisit your understanding of PEMDAS whenever your work starts showing signs of confusion. Most students do not need a full re-teach every time. They need a quick refresh at the moment mistakes begin to repeat.

Common signals include:

• You get different answers than your teacher, calculator, or answer key.
• You understand word problems but miss points on the arithmetic steps.
• You rush through expressions and skip rewriting lines.
• You are moving into algebra, functions, or formulas with more symbols.
• You remember the PEMDAS letters but still feel unsure about division and subtraction order.

Search intent can shift too. Sometimes students look for “PEMDAS explained” because they want a basic definition. Other times they need “math homework help order of operations” because they are already stuck in a worksheet problem. If your need changes, your review should change too. That means not just reading the rule again, but practicing the exact type of expression that caused trouble.

For example:

• If decimals are causing errors, practice mixed operations with decimals.
• If negative numbers are causing errors, practice sign rules alongside order of operations.
• If word problems are causing errors, translate the words into an expression first, then apply the order.

Students often discover that the real problem is not PEMDAS itself but the transition from words to symbols. If that is happening, this can help: Homework Help for Math Word Problems: A Step-by-Step Solving Framework.

Expressions that deserve extra attention

Some problem types create more confusion than others. Revisit the topic if you are seeing these:

• Multiple operations in a row: 24 ÷ 6 × 2
• Nested parentheses: 3(2 + (4 − 1))
• Exponents with surrounding operations: 5 + 2³ × 3
• Negative values: 8 − 3 × (−2)
• Fractions or decimals: 1.5 + 3 ÷ 0.5

When expressions get more crowded, neat writing matters even more. Do one operation per line if necessary. That is not a sign of weakness. It is a way to stay accurate.

Common issues

The fastest way to improve is to know what usually goes wrong. Here are the most common PEMDAS mistakes and how to fix them.

1. Thinking multiplication always comes before division

This is one of the biggest mistakes. In PEMDAS, multiplication and division are equal in priority. Work from left to right.

Example:

16 ÷ 4 × 2

Correct order:

16 ÷ 4 = 4, then 4 × 2 = 8

Incorrect shortcut:

4 × 2 = 8, then 16 ÷ 8 = 2

The mistake comes from treating the letters as a strict ranking instead of paired levels.

2. Thinking addition always comes before subtraction or vice versa

The same rule applies here. Addition and subtraction share a level. Go left to right.

Example:

10 − 3 + 2

Correct:

10 − 3 = 7, then 7 + 2 = 9

If you add first, you get the wrong answer.

3. Ignoring parentheses

Parentheses change the structure of the problem. You cannot skip over them just because another operation feels easier.

Example:

6 × (2 + 3)

Do the parentheses first: 2 + 3 = 5

Then multiply: 6 × 5 = 30

4. Skipping exponents

Exponents must be handled before multiplication, division, addition, or subtraction.

Example:

2 × 3²

Correct: 3² = 9, then 2 × 9 = 18

Not: 2 × 3 = 6, then 6²

5. Doing too many steps in your head

Mental math can be helpful, but in mixed operations it often hides mistakes. If your teacher asks for work shown, that is a clue that the process matters.

Fix: rewrite the expression after every operation.

Instead of jumping from:

12 − 2 × 3 + 4

straight to an answer, write:

12 − 6 + 4

6 + 4

10

6. Confusing a negative sign with subtraction

In later problems, signs can create extra confusion. For example, in 5 + (−3), the negative sign is part of the number. It is not the same as choosing the subtraction step first. Slow down and identify the number clearly before applying the order of operations.

7. Misreading the expression

Some errors begin before solving. Students copy the problem incorrectly, miss a symbol, or forget parentheses. A clean copy can prevent a wrong answer before the math even starts.

This is why checking matters. Review the original expression and compare each line. If you need a structured way to do that, return to How to Check Your Homework Answers Without Copying or Guessing.

Practice tips that actually help

• Circle grouping symbols before you start.
• Underline exponents so you do not forget them.
• Draw light arrows left to right for multiplication and division chains.
• Use one line per step on homework and test review.
• Practice short sets often instead of one long cramming session.

For memorization, some students also benefit from turning the rule and mistake patterns into a quick review deck. If that sounds useful, see Flashcards vs Notes vs Practice Questions: Which Study Method Works Best?.

When to revisit

Come back to this order of operations guide whenever mixed-operation problems start appearing again, especially in homework, placement review, algebra warm-ups, test prep, or self-study. The best time to revisit is before errors pile up, not after several assignments go wrong.

Use this practical checklist:

• Revisit after a school break if you feel rusty.
• Revisit before quizzes on expressions, integers, or algebra basics.
• Revisit when working with formulas in science or finance problems.
• Revisit if answer-key checking shows repeated arithmetic mistakes.
• Revisit when helping someone else and you want to explain the rule clearly.

Here is a short action plan you can use today:

1. Write down the four real steps: grouping symbols, exponents, multiplication/division left to right, addition/subtraction left to right.
2. Solve three problems slowly and show every line.
3. Check whether your mistake, if any, came from order, arithmetic, or copying.
4. Save one worked example in your notes so you can review it later.
5. Repeat the process next week in under 10 minutes.

If you are building stronger homework habits, the bigger lesson is this: good math work is organized work. Clear steps reduce confusion, improve accuracy, and make checking possible. That is why the order of operations is not just a one-time lesson. It is a tool you return to whenever expressions get crowded and time pressure rises.

Keep this guide bookmarked as a quick reference. When the next worksheet, placement test, or review packet includes mixed operations, you will not need to guess where to start. You will have a repeatable method, a list of common PEMDAS mistakes to avoid, and a simple routine for refreshing the skill whenever you need it.