Commutative, Associative or Distributive? | 3rd Grade Math - Free Printable

Problem Overview:

The task is to identify the property being used in each given equation. The properties to consider are:

1. Commutative Property: This property states that the order of the numbers can be changed without affecting the result.
- For addition: \( a + b = b + a \)
- For multiplication: \( a \times b = b \times a \)

2. Associative Property: This property states that the grouping of numbers can be changed without affecting the result.
- For addition: \( (a + b) + c = a + (b + c) \)
- For multiplication: \( (a \times b) \times c = a \times (b \times c) \)

3. Distributive Property: This property involves distributing a number over a sum or difference.
- For multiplication over addition: \( a \times (b + c) = (a \times b) + (a \times c) \)
- For multiplication over subtraction: \( a \times (b - c) = (a \times b) - (a \times c) \)

We will analyze each equation and determine which property it exemplifies.

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Solution:

#### A: \( 7 + (8 + 4) = (7 + 8) + 4 \)
- Here, the grouping of the numbers is being changed from \( 7 + (8 + 4) \) to \( (7 + 8) + 4 \).
- This is an example of the Associative Property of Addition.

#### B: \( 8 + 0 + 6 = 6 + (8 + 0) \)
- In this equation, the order of the numbers is being changed, and the grouping is also altered.
- The key observation is that the order change (\( 8 + 0 + 6 \) to \( 6 + (8 + 0) \)) indicates the Commutative Property of Addition, while the grouping change indicates the Associative Property of Addition.
- However, since the primary focus is on the order change, this is primarily an example of the Commutative Property of Addition.

#### C: \( 8 + 6 + 0 = 6 + 8 + 0 \)
- Here, the order of the numbers is being changed from \( 8 + 6 + 0 \) to \( 6 + 8 + 0 \).
- This is an example of the Commutative Property of Addition.

#### D: \( 8 + (2 + 4) = (8 + 2) + 4 \)
- Similar to equation A, the grouping of the numbers is being changed from \( 8 + (2 + 4) \) to \( (8 + 2) + 4 \).
- This is an example of the Associative Property of Addition.

#### E: \( 5 \times (2 + 4) = (5 \times 2) + (5 \times 4) \)
- Here, the number 5 is being distributed over the sum \( 2 + 4 \).
- This is an example of the Distributive Property of Multiplication over Addition.

#### F: \( 6 \times (2 \times 3) = (6 \times 2) \times 3 \)
- The grouping of the numbers is being changed from \( 6 \times (2 \times 3) \) to \( (6 \times 2) \times 3 \).
- This is an example of the Associative Property of Multiplication.

#### G: \( 8 \times 2 = 2 \times 8 \)
- Here, the order of the numbers is being changed from \( 8 \times 2 \) to \( 2 \times 8 \).
- This is an example of the Commutative Property of Multiplication.

#### H: \( 18 \times 1 = 1 \times 18 \)
- The order of the numbers is being changed from \( 18 \times 1 \) to \( 1 \times 18 \).
- This is an example of the Commutative Property of Multiplication.

#### I: \( 8 \times (3 + 4) = (8 \times 3) + (8 \times 4) \)
- Here, the number 8 is being distributed over the sum \( 3 + 4 \).
- This is an example of the Distributive Property of Multiplication over Addition.

#### J: \( (8 \times 2) \times 4 = 8 \times (2 \times 4) \)
- The grouping of the numbers is being changed from \( (8 \times 2) \times 4 \) to \( 8 \times (2 \times 4) \).
- This is an example of the Associative Property of Multiplication.

#### K: \( 8 \times (6 + 9) = (8 \times 6) + (8 \times 9) \)
- Here, the number 8 is being distributed over the sum \( 6 + 9 \).
- This is an example of the Distributive Property of Multiplication over Addition.

#### L: \( 8 \times (6 - 9) = (8 \times 6) - (8 \times 9) \)
- Here, the number 8 is being distributed over the difference \( 6 - 9 \).
- This is an example of the Distributive Property of Multiplication over Subtraction.

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Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{A: Associative Property of Addition} & \text{B: Commutative Property of Addition} \\
\text{C: Commutative Property of Addition} & \text{D: Associative Property of Addition} \\
\text{E: Distributive Property of Multiplication over Addition} & \text{F: Associative Property of Multiplication} \\
\text{G: Commutative Property of Multiplication} & \text{H: Commutative Property of Multiplication} \\
\text{I: Distributive Property of Multiplication over Addition} & \text{J: Associative Property of Multiplication} \\
\text{K: Distributive Property of Multiplication over Addition} & \text{L: Distributive Property of Multiplication over Subtraction} \\
\end{array}
}
\]