The Commutative and Associative Properties of Addition

Addition is one of the first operations that children learn. Two important ideas that help us think about addition questions are the commutative property and the associative property. These properties describe patterns in addition that make calculations easier, and they form the foundation for later work in algebra.

The Commutative Property of Addition

The word commutative comes from a Latin root meaning to exchange or switch (think “commute” you move/switch from home to work). The commutative property tells us that the order of numbers being added does not change the sum.

Key Idea: The order of addition does not matter.

The Associative Property of Addition

The word associative comes from to group or connect. The associative property tells us that when adding three or more numbers, it does not matter how the numbers are grouped.

Key Idea: The grouping of numbers in addition does not matter.

Why These Properties Matter

1. Mental Math:

   • Associative property helps us regroup for easier adding. For example,

     $7 + 9 = (7 + 3) + 6 = 10 + 6 = 16$

   • Commutative property lets us reorder numbers for convenience. For example,

     $25 + 47 = 47 + 25$

2. Algebra Connection:
   Later, students will use these properties with variables: For example: (a + b) + c = a + (b + c)

3. Fraction and Decimal Addition:
   These properties still hold!

   $$$$

[latex]\frac{1}{2} + \frac{3}{4} = \frac{3}{4} + \frac{1}{2} \quad \quad (0.2 + 0.3) + 0.5 = 0.2 + (0.3 + 0.5)[/latex]

4. Teaching Children:
   These properties can be introduced using manipulatives, visuals, and games where students notice that the total doesn’t change when numbers are switched or regrouped.

Mini-Alphabitia Connection

Recall that Alphabitia is a base-5 number system using the symbols A, B, C, D, and 0. The commutative and associative properties still hold there too!

Example (Base-5, Alphabitia):

$$B + D = D + B$$

$$(A + C) + D = A + (C + D)$$

This shows that these properties are universal—they work in any number system, not just base-10.

Discussion Prompts

• Give students counters or blocks and ask them to model

  $3+\mathrm{5\; and}$ $5 + 3$. What do they notice?

• Pose the question: If I’m adding three numbers, why might it be useful to group them differently?

• Ask: Do the commutative and associative properties work for subtraction? Why or why not?

Summary:

• The commutative property tells us that order does not matter in addition.

• The associative property tells us that grouping does not matter in addition.

• These properties make arithmetic easier, connect to algebra, and hold true in every number system.