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B.5 Properties of addition and subtraction

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What are the properties of addition and subtraction?

Properties of addition and subtraction are rules that describe how numbers work when we add or subtract them. These rules help us solve problems more efficiently and check our work for accuracy.

Examples:
  • Changing the order of numbers in addition does not change the sum.
  • Adding zero does not change a number’s value.
  • When we group numbers differently in addition, the sum stays the same.
Note

These properties make it easier to add and subtract in your head, estimate, or check your answers.

Commutative property of addition

The Commutative Property of Addition states that the order of the numbers does not affect the sum. You can add numbers in any order and get the same result.

Example:
  • 6 + 4 = 10
  • 4 + 6 = 10
Note

This property only works for addition, not subtraction. Try switching the numbers in subtraction and notice how the difference changes.

Associative property of addition

The Associative Property of Addition tells us that when three or more numbers are added, the way they are grouped does not change the sum.

Example:
  • (3 + 5) + 7 = 15
  • 3 + (5 + 7) = 15
Note

Parentheses show how numbers are grouped. You can regroup numbers to make mental math easier.

Identity property of addition

The Identity Property of Addition means that adding zero to any number does not change its value. Zero is called the “additive identity.”

Example:
  • 9 + 0 = 9
  • 0 + 9 = 9
Note

Zero keeps a number’s value the same when added. This is why it is called the identity element for addition.

Subtraction and its relationship to addition

Subtraction is the opposite of addition. It does not follow the commutative or associative properties, but it is closely related to addition because it can “undo” addition.

Example:
  • If 8 + 5 = 13, then 13 − 5 = 8 and 13 − 8 = 5.
Note

Think of subtraction as finding the missing part in an addition problem. Understanding this relationship helps when checking your answers.

Why the properties matter

Knowing these properties helps you solve problems efficiently, use mental math, and explain your reasoning when adding or subtracting numbers.

Example:
  • Using the Associative Property: (25 + 75) + 10 = 100 + 10 = 110
  • Using the Commutative Property: 32 + 18 = 18 + 32 = 50
Note

Mathematicians use properties to make sense of numbers. When you understand these patterns, math becomes easier and more predictable.