Properties (Commutative, Associative, Distributive, Identity, Zero) - Third Grade Mathematics

Multiplication possesses five extraordinary properties that make computing numbers logical, flexible, and surprisingly easy. Just as an architect relies on reliable tools to build a skyscraper, mathematicians rely on the Commutative, Associative, Distributive, Identity, and Zero properties to take apart complicated problems and solve them with elegance. In this chapter, you will master all five properties in depth.

Overview of the Five Multiplication Properties

Here is your master reference table for the five multiplication properties:

+-----------------------+-----------------------------+------------------------------------+
| Property Name         | Algebraic Rule              | Example with Numbers               |
+-----------------------+-----------------------------+------------------------------------+
| Commutative Property  | a x b = b x a               | 4 x 7 = 7 x 4 = 28                 |
| Associative Property  | (a x b) x c = a x (b x c)   | (2 x 3) x 4 = 2 x (3 x 4) = 24     |
| Distributive Property | a x (b + c) = (a x b)+(a x c)| 6 x 14 = (6 x 10) + (6 x 4) = 84   |
| Identity Property     | a x 1 = a                   | 9 x 1 = 9                          |
| Zero Property         | a x 0 = 0                   | 15 x 0 = 0                         |
+-----------------------+-----------------------------+------------------------------------+

1. Commutative Property of Multiplication

The Commutative Property states that the order of the factors does not change the product.
- 5 x 8 = 40

- 8 x 5 = 40


Think of "commuting" as traveling back and forth to school. Going from home to school is the exact same distance as traveling from school to home! If you ever forget what 8 x 3 is, you can simply recall 3 x 8 = 24.

2. Associative Property of Multiplication

The Associative Property states that when multiplying three factors, the way you group them with parentheses does not change the product.

Problem: Solve 2 x 5 x 7

Grouping Option 1:
(2 x 5) x 7
= 10 x 7
= 70

Grouping Option 2:
2 x (5 x 7)
= 2 x 35
= 70

Notice how much easier Grouping Option 1 was! By grouping (2 x 5) first to make a friendly 10, multiplying by 7 became instant mental math.

3. Distributive Property of Multiplication

The Distributive Property is the most famous property in all of third grade mathematics. It states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the products together.

Formula: a x (b + c) = (a x b) + (a x c)

Look at this array model showing 7 x 8 broken into friendly parts: Break 8 into (5 + 3):

                 <---- 5 ---->  <-- 3 -->
             ^   * * * * *      * * *
             |   * * * * *      * * *
       7     |   * * * * *      * * *
     rows    |   * * * * *      * * *
             |   * * * * *      * * *
             |   * * * * *      * * *
             v   * * * * *      * * *
                (7 x 5 = 35)   (7 x 3 = 21)

Now combine the two sections: 35 + 21 = 56! So 7 x 8 = 56. Whenever you face a challenging multiplication fact (like multiplying by 6, 7, 8, or 9), you can break one factor apart into friendlier numbers like 5 and 2, or 10 and something else!

4. Identity Property of Multiplication

The Identity Property states that the product of any number and 1 is that same number.
- 1 group of 6 cookies is 6 cookies (1 x 6 = 6).

- 8 groups of 1 pencil is 8 pencils (8 x 1 = 8).


Multiplying by 1 preserves the number's exact identity. The number stays true to itself.

5. Zero Property of Multiplication

The Zero Property states that the product of any number and 0 is always 0.
- 5 groups of 0 apples = 0 apples (5 x 0 = 0).

- 0 groups of 100 stars = 0 stars (0 x 100 = 0).


No matter how colossal a number is, multiplying by zero completely flattens it to zero: 1,000,000 x 0 = 0.

Chapter Practice Exercises

  1. Name the property shown in each equation: a. 8 x 6 = 6 x 8 b. 12 x 0 = 0 c. (3 x 2) x 5 = 3 x (2 x 5) d. 9 x 1 = 9 e. 4 x (10 + 2) = (4 x 10) + (4 x 2)
  2. Use the Distributive Property to solve 6 x 13: Break 13 into (10 + 3). Show all steps.
  3. Use the Associative Property to solve in your head: 5 x (8 x 2). Show how you regrouped the factors to make mental math easy.
  4. Fill in the missing number: a. 7 x ___ = 0 b. 45 x ___ = 45 c. 9 x 4 = 4 x ___ d. (2 x 4) x 3 = 2 x (___ x 3)
  5. A student writes: 5 x (6 + 4) = (5 x 6) + 4. Is this equation correct? If not, correct the right side using the Distributive Property.

Solutions and Step-by-Step Answers

  1. Properties: a. Commutative Property of Multiplication (order reversed). b. Zero Property of Multiplication (product is zero). c. Associative Property of Multiplication (grouping changed). d. Identity Property of Multiplication (multiplying by 1 keeps identity). e. Distributive Property of Multiplication (multiplying across the sum).
  2. For 6 x 13: Break 13 into 10 + 3: 6 x (10 + 3) = (6 x 10) + (6 x 3) = 60 + 18 = 78.
  3. For 5 x (8 x 2):
  4. First use commutative property inside or regroup: (5 x 2) x 8.
  5. 5 x 2 = 10.
  6. 10 x 8 = 80. Grouping 5 and 2 creates a friendly 10, making the calculation instant.
  7. Missing values: a. 7 x 0 = 0 b. 45 x 1 = 45 c. 9 x 4 = 4 x 9 d. (2 x 4) x 3 = 2 x (4 x 3)
  8. No, the student's equation is incorrect. The student forgot to distribute the 5 to the second addend (4). Correct equation: 5 x (6 + 4) = (5 x 6) + (5 x 4) = 30 + 20 = 50.