COMMUTATIVE PROPERTY

Commutative Property

Definition Of Commutative Property

Commutative property is one of the basic properties of numbers. The word "commute" means "exchange" or "swap over"

Commutative property states that numbers can be added or multiplied in any order.
That is:
Commutative Property of Addition states that changing the order of addends does not change the sum. That is, a + b = b + a.
Commutative Property of Multiplication states that changing the order of factors does not change the product. That is, a x b = b x a

More About Commutative Property

Commutative property holds for both addition and multiplication.
That is: the operations addition and multiplication are commutative over the set of real numbers. That means, for any two real numbers x and
y, x + y = y + x and xy = yx.
Subtraction and division are not commutative.

Example of Commutative Property

2 + 3 = 3 + 2. Whether you add 3 to 2 or 2 to 3, you get 5 both ways.
4 x 7 = 7 x 4. Whether you multiply 4 by 7 or 7 by 4, the product is the same, i.e. 28.

Commutative Property in real-life

Counting a combination of different coins reminds you of commutative property.
Suppose you have 20 quarters and 10 dimes.
It doesn't matter whether you add the quarters first and then the dimes OR add the dimes first and then the quarters OR add a quarter and a dime alternately, finally the total is going to be $6.

Video Examples:commutative property

Solved Example onCommutative Property

Rewrite the expression "4 + m + n" using the commutative property.

Solution:

Commutative property means to exchange or swap things. So, the answer is any of the following.
4 + m + n = m + 4 + n OR
m + n + 4 OR
n + m + 4 OR
n + 4 + m OR
4 + n + m

Ques: Is 5 (3x) = (3x) 5 true?

Solution:

Yes it is. Because, just the positions of the terms 5 and 3x have been swapped. So, the equation above is true by the commutative property of multiplication.

Quick Summary

Changing the order of addends does not change the sum.
Changing the order of factors does not change the product.
Applies to addition and multiplication, not subtraction or division.

\[ a + b = b + a \\ a \times b = b \times a \]

⚠️ Common Mistakes

Applying the property to subtraction or division.
Forgetting that the property only applies to addition and multiplication within the same expression.

🌍 Real-World Uses

Calculating the total cost of groceries regardless of the order items are scanned at checkout. For example, milk + bread = bread + milk in terms of cost.
Arranging furniture in a room: the total space occupied remains the same whether you place the sofa first then the chair, or the chair first then the sofa.
Baking a cake: the final product is the same whether you add eggs before flour or flour before eggs (assuming the recipe allows flexibility in the order).

📋 Standards Alignment

CCSS.MATH.CONTENT.1.OA.B.3
CCSS.MATH.CONTENT.3.OA.B.5

🍎 Teacher Insights

Use real-world examples like combining coins to illustrate the property. Emphasize that it simplifies calculations.

🎓 Prerequisites

Basic Addition
Basic Multiplication
Understanding of Variables

Check Your Knowledge

Q1: Which equation demonstrates the commutative property of addition?

5 + (3 + 2) = (5 + 3) + 2 5 + 3 = 3 + 5 5 x 1 = 5 5 + 0 = 5

Q2: Which equation demonstrates the commutative property of multiplication?

4 x (2 x 3) = (4 x 2) x 3 4 x 2 = 2 x 4 4 x 1 = 4 4 x 0 = 0

Frequently Asked Questions

Q: Does the commutative property work for subtraction?
A: No, subtraction is not commutative. For example, 5 - 3 is not the same as 3 - 5.

Q: Does the commutative property work for division?
A: No, division is not commutative. For example, 10 / 2 is not the same as 2 / 10.