Properties of Addition Worksheets with Answer Key - Free Printable

Problem Analysis:

The task involves identifying the correct application of two fundamental properties of addition:
1. Commutative Property: This property states that the order of the numbers being added does not affect the sum. Mathematically, it is expressed as:
\[
a + b = b + a
\]
2. Associative Property: This property states that the grouping of the numbers being added does not affect the sum. Mathematically, it is expressed as:
\[
(a + b) + c = a + (b + c)
\]

We will solve each part of the problem step by step.

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Part 1: Commutative Property

#### Question (a): \( 4 + 5 = \)
- The commutative property tells us that \( 4 + 5 = 5 + 4 \).
- Among the options:
- \( 5 + 2 + 2 \) (Incorrect, as it changes the numbers being added.)
- \( 5 + 4 \) (Correct, as it simply swaps the order of 4 and 5.)
- \( 6 + 3 \) (Incorrect, as it changes the numbers being added.)
- \( 2 + 7 \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( 5 + 4 \)

#### Question (b): \( 6 + 7 = \)
- The commutative property tells us that \( 6 + 7 = 7 + 6 \).
- Among the options:
- \( 6 + 4 + 3 \) (Incorrect, as it changes the numbers being added.)
- \( 1 + 5 + 7 \) (Incorrect, as it changes the numbers being added.)
- \( 7 + 6 \) (Correct, as it simply swaps the order of 6 and 7.)
- \( 2 + 4 + 7 \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( 7 + 6 \)

#### Question (c): \( 3 + 9 = \)
- The commutative property tells us that \( 3 + 9 = 9 + 3 \).
- Among the options:
- \( 3 + 8 + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 9 + 3 \) (Correct, as it simply swaps the order of 3 and 9.)
- \( 3 + (7 + 2) \) (Incorrect, as it introduces grouping, which is related to the associative property, not the commutative property.)
- \( 5 + 4 + 2 + 1 \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( 9 + 3 \)

#### Question (d): \( 5 + 8 = \)
- The commutative property tells us that \( 5 + 8 = 8 + 5 \).
- Among the options:
- \( 8 + 4 + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 12 + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 8 + 5 \) (Correct, as it simply swaps the order of 5 and 8.)
- \( 7 + 6 \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( 8 + 5 \)

#### Question (e): \( 3 + 2 = \)
- The commutative property tells us that \( 3 + 2 = 2 + 3 \).
- Among the options:
- \( 1 + 1 + 3 \) (Incorrect, as it changes the numbers being added.)
- \( 2 + 1 + 1 + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 4 + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 2 + 3 \) (Correct, as it simply swaps the order of 3 and 2.)

Correct Answer: \( 2 + 3 \)

#### Question (f): \( 7 + 9 = \)
- The commutative property tells us that \( 7 + 9 = 9 + 7 \).
- Among the options:
- \( 9 + 7 \) (Correct, as it simply swaps the order of 7 and 9.)
- \( 7 + 7 + 2 \) (Incorrect, as it changes the numbers being added.)
- \( 6 + 10 \) (Incorrect, as it changes the numbers being added.)
- \( 7 + (3 + 6) \) (Incorrect, as it introduces grouping, which is related to the associative property, not the commutative property.)

Correct Answer: \( 9 + 7 \)

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Part 2: Associative Property

#### Question (a): \( 3 + (12 + 5) = \)
- The associative property tells us that \( 3 + (12 + 5) = (3 + 12) + 5 \).
- Among the options:
- \( (15 + 4) + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 4 + (11 + 5) \) (Incorrect, as it changes the numbers being added.)
- \( (3 + 12) + 5 \) (Correct, as it regroups the numbers without changing their order.)
- \( 3 + (11 + 6) \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( (3 + 12) + 5 \)

#### Question (b): \( (10 + 6) + 1 = \)
- The associative property tells us that \( (10 + 6) + 1 = 10 + (6 + 1) \).
- Among the options:
- \( 2 + (8 + 7) \) (Incorrect, as it changes the numbers being added.)
- \( 7 + (2 + 8) \) (Incorrect, as it changes the numbers being added.)
- \( 10 + (6 + 1) \) (Correct, as it regroups the numbers without changing their order.)
- \( 2 + (8 + 4) + 3 \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( 10 + (6 + 1) \)

#### Question (c): \( 6 + (7 + 8) = \)
- The associative property tells us that \( 6 + (7 + 8) = (6 + 7) + 8 \).
- Among the options:
- \( 1 + (5 + 15) \) (Incorrect, as it changes the numbers being added.)
- \( (2 + 4) + 15 \) (Incorrect, as it changes the numbers being added.)
- \( (6 + 7) + 8 \) (Correct, as it regroups the numbers without changing their order.)
- \( 3 + (13 + 5) \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( (6 + 7) + 8 \)

#### Question (d): \( 1 + (3 + 5) = \)
- The associative property tells us that \( 1 + (3 + 5) = (1 + 3) + 5 \).
- Among the options:
- \( (1 + 3) + 5 \) (Correct, as it regroups the numbers without changing their order.)
- \( 3 + 1 + 4 + 1 \) (Incorrect, as it changes the numbers being added.)
- \( 2 + (2 + 5) \) (Incorrect, as it changes the numbers being added.)
- \( 5 + (4 + 0) \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( (1 + 3) + 5 \)

#### Question (e): \( 9 + (12 + 14) = \)
- The associative property tells us that \( 9 + (12 + 14) = (9 + 12) + 14 \).
- Among the options:
- \( 10 + (11 + 14) \) (Incorrect, as it changes the numbers being added.)
- \( (9 + 12) + 14 \) (Correct, as it regroups the numbers without changing their order.)
- \( 12 + 7 + 7 + 9 \) (Incorrect, as it changes the numbers being added.)
- \( 10 + (6 + 6 + 13) \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( (9 + 12) + 14 \)

#### Question (f): \( (4 + 6) + 10 = \)
- The associative property tells us that \( (4 + 6) + 10 = 4 + (6 + 10) \).
- Among the options:
- \( 15 + 5 + 10 \) (Incorrect, as it changes the numbers being added.)
- \( 4 + (6 + 10) \) (Correct, as it regroups the numbers without changing their order.)
- \( 10 + 10 \) (Incorrect, as it changes the numbers being added.)
- \( (3 + 7) + 10 \) (Incorrect, as it changes the numbers being added.)

Correct Answer: \( 4 + (6 + 10) \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Commutative Property:} & \text{(a) } 5+4, \text{(b) } 7+6, \text{(c) } 9+3, \text{(d) } 8+5, \text{(e) } 2+3, \text{(f) } 9+7 \\
\text{Associative Property:} & \text{(a) } (3+12)+5, \text{(b) } 10+(6+1), \text{(c) } (6+7)+8, \text{(d) } (1+3)+5, \text{(e) } (9+12)+14, \text{(f) } 4+(6+10)
\end{array}
}
\]