Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 1: Properties of Arithmetic

Lesson 2: Addition

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students learn the commutative property (a + b = b + a), the associative property ((a + b) + c = a + (b + c)), and the any-order principle of addition. The lesson also introduces variables as a way to express general arithmetic rules. These foundational properties of addition are drawn from Chapter 1 of the AMC 8 curriculum and prepare students to simplify complex calculations efficiently.

Section 1

Commutative Property of Addition

Property

Changing the order of the addends $a$ and $b$ does not change their sum.

a + b = b + a

Examples

The sum of $9 + 7$ is $16$ , which is the same as the sum of $7 + 9$ , also $16$ .
A recipe calls for 2 cups of flour and 1 cup of sugar. The total volume is $2+1=3$ cups, which is the same as adding the sugar first: $1+2=3$ cups.
Calculating $50 + 112$ gives $162$ , and reversing the order to $112 + 50$ also gives $162$ .

Explanation

This property means you can swap the numbers in an addition problem, and the answer will be the same. It's like putting on your shoes; whether you put on the left or right one first, you end up with both on.

Section 2

Identity Property of Addition

Property

The sum of any number $a$ and 0 is the number. Zero is called the additive identity.

a + 0 = a

0 + a = a

Examples

For the number 25, adding zero gives the same number back: $25 + 0 = 25$ .
If you start with zero and add a number, the result is that number: $0 + 150 = 150$ .
Maria has 8 books and receives 0 new books for her birthday. She still has a total of 8 books, demonstrating that $8 + 0 = 8$ .

Explanation

Think of zero as the 'do nothing' number in addition. Adding zero to any number doesn't change it at all—the number keeps its original identity. It's like getting zero extra toys; you still have the same number of toys.

Section 3

Associative Property of Addition

Property

Associative Property of Addition: If $a$ , $b$ , $c$ are real numbers, then $(a+b)+c=a+(b+c)$

When adding three or more numbers, changing the grouping gives the same result.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Commutative Property of Addition

Property

Changing the order of the addends $a$ and $b$ does not change their sum.

a + b = b + a

Examples

The sum of $9 + 7$ is $16$ , which is the same as the sum of $7 + 9$ , also $16$ .
A recipe calls for 2 cups of flour and 1 cup of sugar. The total volume is $2+1=3$ cups, which is the same as adding the sugar first: $1+2=3$ cups.
Calculating $50 + 112$ gives $162$ , and reversing the order to $112 + 50$ also gives $162$ .

Explanation

Section 2

Identity Property of Addition

Property

The sum of any number $a$ and 0 is the number. Zero is called the additive identity.

a + 0 = a

0 + a = a

Examples

For the number 25, adding zero gives the same number back: $25 + 0 = 25$ .
If you start with zero and add a number, the result is that number: $0 + 150 = 150$ .
Maria has 8 books and receives 0 new books for her birthday. She still has a total of 8 books, demonstrating that $8 + 0 = 8$ .

Explanation

Section 3

Associative Property of Addition

Property

Associative Property of Addition: If $a$ , $b$ , $c$ are real numbers, then $(a+b)+c=a+(b+c)$

When adding three or more numbers, changing the grouping gives the same result.

Overview

Lesson overview

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

Overview

Section 1

Commutative Property of Addition

Property

Changing the order of the addends $a$ and $b$ does not change their sum.

a + b = b + a

Examples

The sum of $9 + 7$ is $16$ , which is the same as the sum of $7 + 9$ , also $16$ .
A recipe calls for 2 cups of flour and 1 cup of sugar. The total volume is $2+1=3$ cups, which is the same as adding the sugar first: $1+2=3$ cups.
Calculating $50 + 112$ gives $162$ , and reversing the order to $112 + 50$ also gives $162$ .

Explanation

Section 2

Identity Property of Addition

Property

The sum of any number $a$ and 0 is the number. Zero is called the additive identity.

a + 0 = a

0 + a = a

Examples

For the number 25, adding zero gives the same number back: $25 + 0 = 25$ .
If you start with zero and add a number, the result is that number: $0 + 150 = 150$ .
Maria has 8 books and receives 0 new books for her birthday. She still has a total of 8 books, demonstrating that $8 + 0 = 8$ .

Explanation

Section 3

Associative Property of Addition

Property

Associative Property of Addition: If $a$ , $b$ , $c$ are real numbers, then $(a+b)+c=a+(b+c)$

When adding three or more numbers, changing the grouping gives the same result.