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Teacher Guide: Order of Operations (PEMDAS)

Learn the correct order to solve math problems with multiple operations using PEMDAS.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Basic Operations. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Students will correctly apply the order of operations (PEMDAS) to evaluate expressions with multiple operations
  • Students will explain why the order of operations is necessary for consistent mathematical results
  • Students will identify and correct common errors in order of operations
  • Students will solve expressions containing parentheses, exponents, and all four basic operations
Prerequisites
  • • Understanding of addition, subtraction, multiplication, and division
  • • Familiarity with basic exponents (squares)
  • • Ability to recognize mathematical symbols
Discussion Starters
  • 1. Why do you think mathematicians needed to agree on a specific order for operations?
  • 2. What would happen if everyone solved math problems in different orders?
  • 3. Can you think of a real-life situation where the order of steps matters?
  • 4. How does your calculator know which operation to do first?
  • 5. What would happen if we didn't have parentheses?
Common Misconceptions

Multiplication always comes before division

Remediation: Show examples like 12 ÷ 3 × 2 and demonstrate working left to right. Have students practice with expressions where division comes first.

PEMDAS means do operations in that exact letter order

Remediation: Emphasize that M/D are equal priority and A/S are equal priority. Use visual groupings: P → E → (M and D) → (A and S)

A number written right before parentheses, like 2(3 + 1), is not a multiplication

Remediation: Show that 2(3 + 1) means 2 × (3 + 1): solve the parentheses first (4), then multiply (2 × 4 = 8). Have students rewrite such expressions with the × sign before solving.

Differentiation Ideas

For Struggling Students:

  • • Provide a PEMDAS checklist to reference
  • • Use color-coding for different operation types
  • • Start with expressions that only have 2 operations
  • • Use manipulatives to visualize grouping

For On-Level Students:

  • • Practice with standard 3-4 operation expressions
  • • Include expressions with parentheses and exponents
  • • Challenge with word problems requiring PEMDAS

For Advanced Students:

  • • Include nested parentheses
  • • Introduce expressions with variables
  • • Have students create expressions that equal specific values
  • • Explore order of operations in programming
Standards Alignment
  • 5.OA.A.1 (CCSS.MATH.CONTENT.5.OA.A.1)

    Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols

  • 6.EE.A.2.C (CCSS.MATH.CONTENT.6.EE.A.2.C)

    Evaluate expressions at specific values of their variables, including expressions that arise from formulas

Lesson Resources
  • visualWhiteboard or Projector

    For demonstrating worked examples step by step

  • visualPEMDAS Anchor Chart

    Visual reference showing the order of operations

  • worksheetPractice Worksheets

    Graduated difficulty problems for student practice

  • activityCalculator

    For checking answers and exploring patterns

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The order of operations is a set of rules that tells us which calculations to perform first when solving a math expression with multiple operations.
The acronym PEMDAS helps us remember the order:
  • P = Parentheses (do these first)
  • E = Exponents (powers and roots)
  • M/D = Multiplication and Division (left to right)
  • A/S = Addition and Subtraction (left to right)
> Important: Multiplication and Division have the same priority - work left to right. The same applies to Addition and Subtraction.

Worked Examples

Solve: 3+4×23 + 4 \times 2
1

Look for Parentheses

No parentheses → Move to next step
2

Look for Exponents

No exponents → Move to next step
3

Do Multiplication first

4×2=84 \times 2 = 8 → 3+83 + 8
4

Do Addition

3+83 + 8 → 1111

Common Mistakes

Adding before multiplying

Why it's wrong: Students often work left to right, doing 3+4=73 + 4 = 7 before the multiplication in 3+4×23 + 4 \times 2

Correct: Always do multiplication and division before addition and subtraction. The answer is 3+8=113 + 8 = 11, not 7×2=147 \times 2 = 14

Thinking M comes before D

Why it's wrong: PEMDAS lists M before D, making students think multiplication always comes first

Correct: Multiplication and Division have EQUAL priority. Work left to right. In 8÷2×48 \div 2 \times 4, do 8÷2=48 \div 2 = 4 first, then 4×4=164 \times 4 = 16

Forgetting to work inside parentheses completely

Why it's wrong: Students may only do part of what is inside parentheses

Correct: Solve EVERYTHING inside parentheses before moving on. In (3+2×4)(3 + 2 \times 4), you must do 2×4=82 \times 4 = 8 first, then 3+8=113 + 8 = 11

Ignoring exponents

Why it's wrong: Students sometimes treat exponents like regular multiplication

Correct: Exponents come before multiplication. In 2×322 \times 3^2, first calculate 32=93^2 = 9, then 2×9=182 \times 9 = 18

Why It Matters

Without a standard order of operations, the same math expression could give different answers depending on how you solve it. For example, 3+4×23 + 4 \times 2 could equal 1414 or 1111 depending on whether you add or multiply first.
The order of operations ensures everyone gets the same answer. This is essential in:
  • Science - calculating formulas correctly
  • Programming - computers follow strict operation orders
  • Finance - calculating interest, taxes, and budgets
  • Engineering - designing safe structures and systems

Real World Applications

Shopping Discounts

When shopping, you often need to calculate final prices after discounts and additional costs. The order in which you apply these calculations matters!

Example:

A 50 dollar shirt is 20% off, then you add 5 dollar shipping. Calculate: 50×0.8+5=40+5=4550 \times 0.8 + 5 = 40 + 5 = 45 dollars

1Try It Yourself

You want to buy a video game that costs 60 dollars. The store has a 25% off sale, and there is a 4 dollar tax added at the end.

What is the final price you will pay?

Step 1: Write the mathematical expression

Write the mathematical expression using the numbers from the problem:

Cooking & Recipes

When scaling recipes or combining ingredients, the order of operations ensures you get the right amounts.

Example:

A recipe needs (2+1)×3=3×3=9(2 + 1) \times 3 = 3 \times 3 = 9 cups total (2 cups flour + 1 cup sugar, tripled)

2Try It Yourself

You are making cookies. The recipe calls for 2 cups of flour plus 1 cup of sugar, and you want to make 4 batches.

How many total cups of dry ingredients do you need?

Step 1: Write the mathematical expression

Write the expression using parentheses to group what goes in one batch:

Video Game Scoring

Video games use mathematical formulas to calculate damage, scores, and rewards. Understanding order of operations helps you predict outcomes.

Example:

Damage = base × multiplier + bonus: 10×2+5=2510 \times 2 + 5 = 25 damage

3Try It Yourself

In a game, your base attack is 15, you have a 3× critical hit multiplier, and a +10 bonus from your weapon.

How much total damage do you deal?

Step 1: Write the mathematical expression

Write the damage formula (base × multiplier + bonus):

Key Takeaways

  • 1PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
  • 2Always solve inside Parentheses first, starting with the innermost
  • 3Exponents (powers) come after parentheses
  • 4Multiplication and Division have equal priority - work left to right
  • 5Addition and Subtraction have equal priority - work left to right
  • 6When in doubt, use parentheses to make your intention clear
  • 7BODMAS and PEMDAS are the same thing - different regions use different names

Frequently Asked Questions

What does PEMDAS stand for?

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It's the order in which you should perform operations in a math expression.

Why does multiplication come before addition?

This is a convention mathematicians agreed upon to ensure everyone gets the same answer. Without this rule, 3 + 4 × 2 could equal either 11 or 14 depending on the order.

What is the difference between PEMDAS and BODMAS?

They are the same thing! BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is used in UK/Australia, while PEMDAS is used in USA. Both give the same results.

Do I always multiply before dividing?

No! Multiplication and division have equal priority. You work from left to right. In 12 ÷ 3 × 2, first divide (12 ÷ 3 = 4), then multiply (4 × 2 = 8).

What if there are no parentheses?

If there are no parentheses, skip that step and move to exponents. If no exponents, move to multiplication/division, and so on.

How do I handle nested parentheses?

Start with the innermost parentheses and work your way out. For example, in 2 × (3 + (4 - 1)), first solve (4 - 1) = 3, then (3 + 3) = 6, then 2 × 6 = 12.

Glossary

Order of Operations
A set of rules that determines which calculations are performed first in an expression
PEMDAS
An acronym to remember the order: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
BODMAS
British version of PEMDAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction
Expression
A mathematical phrase containing numbers and operations, like 3 + 4 × 2
Parentheses
Symbols ( ) used to group parts of an expression that should be calculated first
Exponent
A number that shows how many times to multiply a base by itself, written as a superscript

Formula Card

PEMDAS Order

P→E→M/D→A/S\text{P} \rightarrow \text{E} \rightarrow \text{M/D} \rightarrow \text{A/S}

The order to follow: Parentheses, Exponents, then Multiplication/Division (left to right), then Addition/Subtraction (left to right)

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