The Commutative Property of Addition states that changing the order of the numbers being added does not change the sum. In other words:

a + b = b + a

No matter how you arrange the numbers, the result remains the same.

Examples

1. 3 + 5 = 5 + 3
   • Both equal 8.
2. 12 + 4 = 4 + 12
   • Both equal 16.
3. 20 + 6 = 6 + 20
   • Both equal 26.

Practice Problems

1. Show that 8 + 2 = 2 + 8.
2. Does 15 + 7 = 7 + 15? Prove it.
3. Why doesn’t 6 − 4 = 4 − 6?

Answers:

1. 8 + 2 = 10 and 2 + 8 = 10 ✅
2. 15 + 7 = 22 and 7 + 15 = 22 ✅
3. 6 − 4 = 2, but 4 − 6 = -2 ❌ (Subtraction isn’t commutative!)

The Associative Property of Addition states that when adding three or more numbers, the way they are grouped does not change the sum. In other words:

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

No matter how you arrange the parentheses, the result remains the same.

Examples

1. (2 + 3) + 4 = 2 + (3 + 4)
   • (5) + 4 = 2 + (7) → Both equal 9.
2. (10 + 5) + 6 = 10 + (5 + 6)
   • (15) + 6 = 10 + (11) → Both equal 21.
3. (1 + 8) + 2 = 1 + (8 + 2)
   • (9) + 2 = 1 + (10) → Both equal 11.

Practice Problems

1. Show that (7 + 4) + 5 = 7 + (4 + 5).
2. Does (20 + 10) + 30 = 20 + (10 + 30)? Prove it.
3. Why doesn’t (12 − 5) − 3 = 12 − (5 − 3)?

Answers:

1. (7 + 4) + 5 = 11 + 5 = 16; 7 + (4 + 5) = 7 + 9 = 16 ✅
2. (20 + 10) + 30 = 30 + 30 = 60; 20 + (10 + 30) = 20 + 40 = 60 ✅
3. (12 − 5) − 3 = 7 − 3 = 4; 12 − (5 − 3) = 12 − 2 = 10 ❌ (Subtraction isn’t associative!)

The Identity Property of Addition, also known as the Additive Identity, states that when you add 0 to any number, the sum is always the original number. In other words:

a + 0 = a
0 + a = a

Zero is called the “additive identity” because it does not change the value of a number when added to it.

Examples

1. 7 + 0 = 7
2. 0 + 15 = 15
3. 23 + 0 = 23
4. 0 + 100 = 100

In each case, the number remains the same after adding zero.

Practice Problems

1. What is 12 + 0?
2. If 0 + 45 = ___, what fills in the blank?
3. Explain why 0 + 78 = 78 using the Identity Property.

Answers:

1. 12
2. 45
3. Adding zero to any number does not change its value, so 0 + 78 = 78.

Demonstration with Counting Cubes

Let’s explore the commutative property using colored counting cubes. We’ll show that 2 + 3 gives the same result as 3 + 2 using visual representations.

Example 1: 2 Red Cubes + 3 Blue Cubes

🔴 🔴 + 🔵 🔵 🔵 = 🔴 🔴 🔵 🔵 🔵
(2) + (3) = (5)

Example 2: 3 Blue Cubes + 2 Red Cubes

🔵 🔵 🔵 + 🔴 🔴 = 🔵 🔵 🔵 🔴 🔴
(3) + (2) = (5)