Math worksheet practicing the commutative and associative properties of addition and multiplication.

To solve the problem and explain the solution, let's analyze each row of the table step by step. The goal is to identify which mathematical property or "law" is being demonstrated in each equation.

Mathematical Properties (Laws)

1. Commutative Law: The order of operands does not affect the result.
- Addition: \( a + b = b + a \)
- Multiplication: \( a \times b = b \times a \)

2. Associative Law: The grouping of operands does not affect the result.
- Addition: \( (a + b) + c = a + (b + c) \)
- Multiplication: \( (a \times b) \times c = a \times (b \times c) \)

3. Distributive Law: Multiplication distributes over addition.
- \( a \times (b + c) = (a \times b) + (a \times c) \)

4. Identity Law:
- Addition: \( a + 0 = a \)
- Multiplication: \( a \times 1 = a \)

5. Inverse Law:
- Addition: \( a + (-a) = 0 \)
- Multiplication: \( a \times \frac{1}{a} = 1 \) (for \( a \neq 0 \))

6. Reflexive Property: \( a = a \)

Now, let's analyze each row:

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Row 1

- Equation: \( 1 + 3 = 3 + 1 \)
- Explanation: The order of addition does not change the result.
- Law: Commutative Law of Addition

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Row 2

- Equation: \( 8 + 8 + 6 = (8 + 8) + 6 \)
- Explanation: The grouping of addition does not change the result.
- Law: Associative Law of Addition

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Row 3

- Equation: \( 5 + 12 = 12 + 5 \)
- Explanation: The order of addition does not change the result.
- Law: Commutative Law of Addition

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Row 4

- Equation: \( 4 \times 8 + 4 \times 8 = (4 \times 8) + 8 \)
- Explanation: This equation is incorrect as written. It should be \( 4 \times 8 + 4 \times 8 = (4 + 4) \times 8 \) to demonstrate the distributive law.
- Corrected Equation: \( 4 \times 8 + 4 \times 8 = (4 + 4) \times 8 \)
- Law: Distributive Law

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Row 5

- Equation: \( 5 \times 11 + 12 \times 11 = 17 \times 11 \)
- Explanation: This demonstrates the distributive law where multiplication distributes over addition.
- Law: Distributive Law

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Row 6

- Equation: \( (7 - 3) \times 9 = 7 \times 9 - 3 \times 9 \)
- Explanation: This demonstrates the distributive law where multiplication distributes over subtraction.
- Law: Distributive Law

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Row 7

- Equation: \( (3 \times 5) \times 11 = 3 \times (5 \times 11) \)
- Explanation: The grouping of multiplication does not change the result.
- Law: Associative Law of Multiplication

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Row 8

- Equation: \( 10 \times 12 - 9 \times 12 = (10 - 9) \times 12 \)
- Explanation: This demonstrates the distributive law where multiplication distributes over subtraction.
- Law: Distributive Law

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Row 9

- Equation: \( 4 \times 2 = 2 \times 4 \)
- Explanation: The order of multiplication does not change the result.
- Law: Commutative Law of Multiplication

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Row 10

- Equation: \( 10 \times 7 + 9 \times 7 = (10 + 9) \times 7 \)
- Explanation: This demonstrates the distributive law where multiplication distributes over addition.
- Law: Distributive Law

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Row 11

- Equation: \( (13 \times 4) \times 8 = 13 \times (4 \times 8) \)
- Explanation: The grouping of multiplication does not change the result.
- Law: Associative Law of Multiplication

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Row 12

- Equation: \( 6 + 3 = 3 + 6 \)
- Explanation: The order of addition does not change the result.
- Law: Commutative Law of Addition

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Row 13

- Equation: \( 5 \times 2 + 3 \times 2 = 8 \times 2 \)
- Explanation: This demonstrates the distributive law where multiplication distributes over addition.
- Law: Distributive Law

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Row 14

- Equation: \( 9 + (4 + 10) = (9 + 4) + 10 \)
- Explanation: The grouping of addition does not change the result.
- Law: Associative Law of Addition

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Row 15

- Equation: \( 18 \times 6 = 6 \times 18 \)
- Explanation: The order of multiplication does not change the result.
- Law: Commutative Law of Multiplication

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Row 16

- Equation: \( (12 - 8) \times 10 = 12 \times 10 - 8 \times 10 \)
- Explanation: This demonstrates the distributive law where multiplication distributes over subtraction.
- Law: Distributive Law

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Row 17

- Equation: \( 11 \times (4 + 3) = (11 \times 4) \times 3 \)
- Explanation: This equation is incorrect as written. It should be \( 11 \times (4 + 3) = (11 \times 4) + (11 \times 3) \) to demonstrate the distributive law.
- Corrected Equation: \( 11 \times (4 + 3) = (11 \times 4) + (11 \times 3) \)
- Law: Distributive Law

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Row 18

- Equation: \( 6 + 7 + 5 \times 7 = (6 + 5) \times 7 \)
- Explanation: This equation is incorrect as written. It should be \( 6 + 7 + 5 \times 7 = (6 + 7) + (5 \times 7) \) to demonstrate the distributive law.
- Corrected Equation: \( 6 + 7 + 5 \times 7 = (6 + 7) + (5 \times 7) \)
- Law: Distributive Law

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Row 19

- Equation: \( 10 \times 9 = 9 \times 10 \)
- Explanation: The order of multiplication does not change the result.
- Law: Commutative Law of Multiplication

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Row 20

- Equation: \( 27 - 3 = 3 - 7 \)
- Explanation: This equation is incorrect. It should be \( 27 - 3 = 24 \).
- Corrected Equation: \( 27 - 3 = 24 \)
- Law: None (incorrect equation)

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Final Answer

\[
\boxed{
\begin{array}{|c|c|}
\hline
\text{Row} & \text{Law} \\
\hline
1 & \text{Commutative Law of Addition} \\
2 & \text{Associative Law of Addition} \\
3 & \text{Commutative Law of Addition} \\
4 & \text{Distributive Law} \\
5 & \text{Distributive Law} \\
6 & \text{Distributive Law} \\
7 & \text{Associative Law of Multiplication} \\
8 & \text{Distributive Law} \\
9 & \text{Commutative Law of Multiplication} \\
10 & \text{Distributive Law} \\
11 & \text{Associative Law of Multiplication} \\
12 & \text{Commutative Law of Addition} \\
13 & \text{Distributive Law} \\
14 & \text{Associative Law of Addition} \\
15 & \text{Commutative Law of Multiplication} \\
16 & \text{Distributive Law} \\
17 & \text{Distributive Law} \\
18 & \text{Distributive Law} \\
19 & \text{Commutative Law of Multiplication} \\
20 & \text{None (incorrect equation)} \\
\hline
\end{array}
}
\]