Learn on PengiBig Ideas Math, Course 1Chapter 1: Numerical Expressions and Factors

Lesson 3: Order of Operations

In this Grade 6 lesson from Big Ideas Math Course 1, students learn to evaluate numerical expressions using the order of operations, following the sequence of parentheses, exponents, multiplication or division left to right, and addition or subtraction left to right. The lesson covers key vocabulary including numerical expression and evaluate, and students practice applying these rules to expressions with whole-number exponents as required by standard 6.EE.1. Activities also explore how inserting parentheses changes the value of an expression and why a standard order of operations is necessary for consistent results.

Section 1

Order of Operations

Property

Step 1. Parentheses and Other Grouping Symbols
Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
Step 2. Exponents
Simplify all expressions with exponents.
Step 3. Multiplication and Division
Perform all multiplication and division in order from left to right. These operations have equal priority.
Step 4. Addition and Subtraction
Perform all addition and subtraction in order from left to right. These operations have equal priority.

Examples

To simplify $30 - 5 \cdot 4$ , we perform multiplication first: $30 - 20 = 10$ .
In the expression $(3+2)^2 \div 5$ , we start with parentheses $(5)^2 \div 5$ , then the exponent $25 \div 5$ , and finally division to get $5$ .
For $4 + 2[10 - 3(2)]$ , we work inside the innermost parentheses first: $4 + 2[10 - 6]$ , then inside the brackets $4 + 2[4]$ , then multiply $4 + 8$ , and finally add to get $12$ .

Explanation

The order of operations (PEMDAS) is a set of rules everyone follows to solve math problems. This ensures that every expression has only one correct answer, preventing confusion and making sure our calculations are consistent.

Section 2

Understanding Numerical Expressions

Property

The term expression means a “phrase that makes sense” made up of numbers, letters, and operations.
A numeric expression is made up of numbers and arithmetic operations.
Two numeric expressions are equivalent if they compute the same number.

Examples

Evaluate $10 + 4 \cdot 2$ . Multiplication comes first: $4 \cdot 2 = 8$ . Then add: $10 + 8 = 18$ .
Evaluate $(10 + 4) \cdot 2$ . Parentheses come first: $10 + 4 = 14$ . Then multiply: $14 \cdot 2 = 28$ .
Show that $3 \cdot 6 + 3 \cdot 2$ is equivalent to $3 \cdot (6 + 2)$ . The first expression is $18 + 6 = 24$ . The second is $3 \cdot 8 = 24$ . Since they both equal 24, they are equivalent.

Explanation

Think of a numeric expression as a recipe. The numbers are your ingredients and the operations $(+, -, \cdot, \div)$ are your cooking steps. Parentheses tell you which steps to do first to get the right delicious result!

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Order of Operations

Property

Examples

To simplify $30 - 5 \cdot 4$ , we perform multiplication first: $30 - 20 = 10$ .
In the expression $(3+2)^2 \div 5$ , we start with parentheses $(5)^2 \div 5$ , then the exponent $25 \div 5$ , and finally division to get $5$ .
For $4 + 2[10 - 3(2)]$ , we work inside the innermost parentheses first: $4 + 2[10 - 6]$ , then inside the brackets $4 + 2[4]$ , then multiply $4 + 8$ , and finally add to get $12$ .

Explanation

Section 2

Understanding Numerical Expressions

Property

Examples

Evaluate $10 + 4 \cdot 2$ . Multiplication comes first: $4 \cdot 2 = 8$ . Then add: $10 + 8 = 18$ .
Evaluate $(10 + 4) \cdot 2$ . Parentheses come first: $10 + 4 = 14$ . Then multiply: $14 \cdot 2 = 28$ .
Show that $3 \cdot 6 + 3 \cdot 2$ is equivalent to $3 \cdot (6 + 2)$ . The first expression is $18 + 6 = 24$ . The second is $3 \cdot 8 = 24$ . Since they both equal 24, they are equivalent.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Order of Operations

Property

Examples

To simplify $30 - 5 \cdot 4$ , we perform multiplication first: $30 - 20 = 10$ .
In the expression $(3+2)^2 \div 5$ , we start with parentheses $(5)^2 \div 5$ , then the exponent $25 \div 5$ , and finally division to get $5$ .
For $4 + 2[10 - 3(2)]$ , we work inside the innermost parentheses first: $4 + 2[10 - 6]$ , then inside the brackets $4 + 2[4]$ , then multiply $4 + 8$ , and finally add to get $12$ .

Explanation

Section 2

Understanding Numerical Expressions

Property

Examples

Evaluate $10 + 4 \cdot 2$ . Multiplication comes first: $4 \cdot 2 = 8$ . Then add: $10 + 8 = 18$ .
Evaluate $(10 + 4) \cdot 2$ . Parentheses come first: $10 + 4 = 14$ . Then multiply: $14 \cdot 2 = 28$ .
Show that $3 \cdot 6 + 3 \cdot 2$ is equivalent to $3 \cdot (6 + 2)$ . The first expression is $18 + 6 = 24$ . The second is $3 \cdot 8 = 24$ . Since they both equal 24, they are equivalent.