Home » Math Theory » Algebra » Properties of Addition

## Introduction

The properties of addition are fundamental concepts in mathematics that help students develop a strong foundation in arithmetic and algebra. This article presents an overview of the topic, its key concepts, and real-life applications. We will explore the topic’s grade appropriateness, related math domain, Common Core Standards, and practical examples to help students and educators.

The properties of addition are introduced in elementary school, usually around the 1st or 2nd grade, and continue to be applied in higher-level mathematics courses.

## Math Domain

The properties of addition belong to the arithmetic domain, a branch of mathematics that deals with numbers and numerical operations.

## Applicable Common Core Standards

The relevant Common Core Standards for studying the properties of addition are:

1.OA.B.3: Apply properties of operations as strategies to add and subtract.

2.OA.B.2: Fluently add and subtract within 20 using mental strategies.

## Definition of the Topic

Addition is the process of adding one number to another number. The properties of addition are essential characteristics of the addition operation that hold for all numbers. These properties show different ways of adding but still getting the same sum. There are three primary properties of addition: commutative, associative, and identity.

## Key Concepts

Commutative Property

The commutative property of addition asserts that the order of numbers in an addition operation does not affect the sum. In other words, a + b = b + a.

Associative Property

The associative property of addition tells that the grouping of numbers in an addition operation does not affect the sum. In other words, (a + b) + c = a + (b + c).

Identity Property

The identity property states that adding 0 to any number does not change the number. In other words, a + 0 = a.

## Discussion with Illustrative Examples

Example 1 (Commutative Property):

According to the commutative property of addition, rearranging the addends does not influence or change the sum.

5 + 3 = 3 + 5

Both expressions are equal to 8, demonstrating that the order of numbers does not influence the sum.

Example 2 (Associative Property):

The associative property of addition shows that when three or more numbers are added, grouping these numbers will not change the sum.

(2 + 4) + 6 = 2 + (4 + 6)

(6) + 6 = 2 + (10)

12 = 12

Remember that parentheses tell us what we need to do first. In this case, we must first add the numbers inside the parenthesis.

Both expressions are equal, demonstrating that grouping numbers does not affect the sum.

Example 3 (Identity Property):

According to the identity property of addition, adding zero and any number, the sum is still the same number. Zero is called the identity element in addition.

7 + 0 = 7

Adding 0 to 7 does not change the value of 7.

Because of the commutative property, we can interchange the position of the addends.

Example 4 (More Properties of Addition):

Inverse Property

According to the inverse property of addition, for every “a,” there is an equivalent “-a.” Adding these additive inverses will always give us a sum of 0.

Examples:

a + (-a) = 0

b + (-b) = 0

Another property of addition explains the sum of all real numbers

When we say real numbers, these are whole numbers such as (0, 1, 2, 3…). Numbers with exponents or with decimals are not considered real numbers.

The closure property of addition teaches us that when two real numbers are added together, their sum will still be a real number.

Example: a + b = c

## Examples with Solution

Apply the properties of addition to simplify the expression: (3 + 5) + 2 + 0.

Solution

Using the associative property: (3 + 5) + 2 + 0 = 3 + (5 + 2) + 0.

Using the commutative property: 3 + (5 + 2) + 0 = 3 + (2 + 5) + 0.

Using the associative property: 3 + (2 + 5) + 0 = (3 + 2) + 5 + 0.

Calculating: (5) + 5 + 0 = 10.

Using the identity property: 10 + 0 = 10.

## Real-life Application with Solution

Example

Jane has five apples, and she receives three apples from her friend. Later, her mom gives her two more apples. How many apples does Jane have now?

Solution

Applying the properties of addition: (5 + 3) + 2 = 5 + (3 + 2).

Calculating: 5 + (3 + 2) = 5 + 5 = 10.

Jane has ten apples in total.

## Practice Test

1. Which shows an example of commutative property?

a. 2 + 2 = 2 + 1 + 1

b. 2 + 1 = 1 + 2

2. Which shows an example of the associative property of addition?

a. (2 + 3) + 1 = 2 + (3 + 1)

b. (2 + 3) + 1 = 3 + 3

3. Which shows an example of identity property?

a. 1 + 0 = 1

b. 1 + 1 = 2

4. Apply the properties of addition to simplify the expression: (8 + 2) + 4 + 0.

5. Use the properties of addition to find the sum: 6 + (7 + 3).

6. Apply the commutative property to rearrange the expression: 4 + 5 + 6.

7. Use the associative property to regroup the expression: (1 + 2) + 3.

### Why are the properties of addition important?

The properties of addition are important because they provide a foundation for understanding and working with numbers and simplifying and solving mathematical problems.

### Do the properties of addition apply to all numbers?

Yes, the properties of addition apply to all numbers, including whole numbers, integers, fractions, and decimals.

### Can the properties of addition be applied to subtraction?

The properties of addition do not directly apply to subtraction. However, there are similar properties for subtraction: the properties of subtraction are closely related to the properties of addition when considering subtraction as the inverse operation of addition.

### How do the properties of addition relate to algebra?

The properties of addition are also applicable in algebra, as they help simplify and solve algebraic expressions and equations involving variables and constants.

### Are the properties of addition applicable to multiplication?

While the properties of addition do not directly apply to multiplication, there are similar properties for multiplication called the properties of multiplication, which include commutative, associative, and identity properties for the multiplication operation.