Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 1: Follow the Rules

Lesson 3: When Does Order Matter?

Grade 4 students explore the commutative and associative properties of addition and multiplication in this lesson from AoPS Introduction to Algebra. Students learn why order matters for subtraction and division but not for addition and multiplication, using variables to express general rules like a + b = b + a. The lesson also shows how applying these properties together can simplify complex calculations involving multiple numbers.

Section 1

Commutative properties

Property

Commutative Property of Addition: If $a$ and $b$ are real numbers, then

a + b = b + a

Commutative Property of Multiplication: If $a$ and $b$ are real numbers, then

a \cdot b = b \cdot a

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same. Subtraction and division are not commutative.

Section 2

Why Subtraction and Division Are Not Commutative

Property

Subtraction and division are not commutative operations because changing the order of the numbers changes the result:

For subtraction: $a - b \neq b - a$ (when $a \neq b$ )

Section 3

Addition and Multiplication Rules

Property

Associative Law for Addition.
If $a$ , $b$ , and $c$ are any numbers, then

(a + b) + c = a + (b + c)

Associative Law for Multiplication.
If $a$ , $b$ , and $c$ are any numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

Examples

For addition: $(4+7)+3 = 11+3 = 14$ is the same as $4+(7+3) = 4+10=14$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Commutative properties

Property

Commutative Property of Addition: If $a$ and $b$ are real numbers, then

a + b = b + a

Commutative Property of Multiplication: If $a$ and $b$ are real numbers, then

a \cdot b = b \cdot a

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same. Subtraction and division are not commutative.

Section 2

Why Subtraction and Division Are Not Commutative

Property

Subtraction and division are not commutative operations because changing the order of the numbers changes the result:

For subtraction: $a - b \neq b - a$ (when $a \neq b$ )

Section 3

Addition and Multiplication Rules

Property

Associative Law for Addition.
If $a$ , $b$ , and $c$ are any numbers, then

(a + b) + c = a + (b + c)

Associative Law for Multiplication.
If $a$ , $b$ , and $c$ are any numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

Examples

For addition: $(4+7)+3 = 11+3 = 14$ is the same as $4+(7+3) = 4+10=14$ .

Overview

Lesson overview

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

Overview

Section 1

Commutative properties

Property

Commutative Property of Addition: If $a$ and $b$ are real numbers, then

a + b = b + a

Commutative Property of Multiplication: If $a$ and $b$ are real numbers, then

a \cdot b = b \cdot a

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same. Subtraction and division are not commutative.

Section 2

Why Subtraction and Division Are Not Commutative

Property

Subtraction and division are not commutative operations because changing the order of the numbers changes the result:

For subtraction: $a - b \neq b - a$ (when $a \neq b$ )

Section 3

Addition and Multiplication Rules

Property

Associative Law for Addition.
If $a$ , $b$ , and $c$ are any numbers, then

(a + b) + c = a + (b + c)

Associative Law for Multiplication.
If $a$ , $b$ , and $c$ are any numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

Examples

For addition: $(4+7)+3 = 11+3 = 14$ is the same as $4+(7+3) = 4+10=14$ .