Properties of Operation in Integers worksheet - Free Printable

Let's solve this step by step. The task is to match each expression in Column A with the correct property of integers listed in Column B.

We’ll go through each item in Column A and identify which property it illustrates.

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🔹 Definitions of Properties (Quick Review):

- a. Commutative Property of Addition:
$ a + b = b + a $

- b. Commutative Property of Multiplication:
$ a \times b = b \times a $

- c. Associative Property:
$ (a + b) + c = a + (b + c) $ or $ (a \times b) \times c = a \times (b \times c) $

- d. Inverse Property of Addition:
$ a + (-a) = 0 $

- e. Inverse Property of Multiplication:
$ a \times \frac{1}{a} = 1 $, for $ a \neq 0 $

- f. Identity Property:
- Addition: $ a + 0 = a $
- Multiplication: $ a \times 1 = a $

- g. Closure Property:
The sum or product of two integers is also an integer.

- h. Distributive Property:
$ a(b + c) = ab + ac $

- i. Zero Property of Addition:
$ a + 0 = a $ — same as identity, but sometimes distinguished

- j. Zero Property of Multiplication:
Any number multiplied by zero is zero: $ a \times 0 = 0 $

> Note: "Zero Property of Addition" (i) and "Identity Property" (f) are closely related. But usually:
> - Identity Property of Addition: $ a + 0 = a $
> - Zero Property of Addition is not standard; likely a mislabeling. But here it's listed separately, so we'll assume it means the same as Identity Property.

But let’s proceed carefully.

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Now, let's analyze each item:

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1. $ 5 + 4 = 9 $

→ This is just a basic addition. It shows that adding two integers gives another integer.
✔ Closure Property (g)

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2. $ 2 + 3 = 3 + 2 $

→ Order changed, result same.
✔ Commutative Property of Addition (a)

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3. $ 27 + (-27) $

→ This equals 0. The sum of a number and its opposite is zero.
✔ Inverse Property of Addition (d)

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4. $ 4 + (2 + 3) = (4 + 2) + 3 $

→ Grouping changed.
✔ Associative Property (c)

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5. $ 3 + (6 + 2) = (6 + 2) + 3 $

→ Left side: grouping, right side: order changed.
Wait:
Left: $ 3 + (6+2) = 3 + 8 = 11 $
Right: $ (6+2) + 3 = 8 + 3 = 11 $
But the order of numbers is different on both sides:
$ 3 + (6+2) $ vs $ (6+2) + 3 $ → So it's commutative, not associative.

Actually, this is showing that adding 3 and (6+2) is same as (6+2) + 3 → swapping positions of 3 and (6+2).
So this is Commutative Property of Addition (a)

✔ a

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6. $ 0 + (-8) = -8 $

→ Adding zero doesn’t change the number.
✔ Identity Property of Addition (f)

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7. $ -5 \times \frac{1}{5} = 1 $

→ Multiplying a number by its reciprocal gives 1.
Note: $ \frac{1}{5} $ is not an integer, but the property still applies.
This is Inverse Property of Multiplication (e)

✔ e

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8. $ (13)(-3) = (-3)(13) $

→ Order of multiplication changed.
✔ Commutative Property of Multiplication (b)

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9. $ 5 = (5)(1) $

→ Multiplying by 1 gives the same number.
✔ Identity Property of Multiplication (f)

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10. $ 4(5 - 2) = 4(5) - 4(2) $

→ Distributing 4 over subtraction.
This is Distributive Property (h)

✔ h

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11. $ 3 + [4 + (-4)] = 3 + 0 $

→ $ 4 + (-4) = 0 $ → additive inverse
Then $ 3 + 0 = 3 $ → identity
But the key idea is that $ 4 + (-4) = 0 $, so it's illustrating Inverse Property of Addition (d)

✔ d

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12. $ -3 + 5 = 2 $ is an integer

→ Sum of two integers is an integer.
✔ Closure Property (g)

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13. $ -3(6 + 5) = -18 + (-15) $

→ Left: $ -3 \times 11 = -33 $
Right: $ -18 + (-15) = -33 $
So $ -3(6+5) = -3×6 + (-3)×5 $ → Distributive Property

✔ h

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14. $ (-3)(-2)(-1)(0) = 0 $

→ Any number times zero is zero.
✔ Zero Property of Multiplication (j)

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15. $ \frac{5}{6} \times \frac{6}{5} = 1 $

→ Reciprocals multiply to 1.
Even though fractions are involved, the concept is Inverse Property of Multiplication (e)

✔ e

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✔ Final Answers:

| Column A | Answer |
|---------|--------|
| 1. $ 5 + 4 = 9 $ | g |
| 2. $ 2 + 3 = 3 + 2 $ | a |
| 3. $ 27 + (-27) $ | d |
| 4. $ 4 + (2+3) = (4+2)+3 $ | c |
| 5. $ 3 + (6+2) = (6+2)+3 $ | a |
| 6. $ 0 + (-8) = -8 $ | f |
| 7. $ -5 \times \frac{1}{5} = 1 $ | e |
| 8. $ (13)(-3) = (-3)(13) $ | b |
| 9. $ 5 = (5)(1) $ | f |
| 10. $ 4(5 - 2) = 4(5) - 4(2) $ | h |
| 11. $ 3 + [4 + (-4)] = 3 + 0 $ | d |
| 12. $ -3 + 5 = 2 $ is an integer | g |
| 13. $ -3(6+5) = -18 + (-15) $ | h |
| 14. $ (-3)(-2)(-1)(0) = 0 $ | j |
| 15. $ \frac{5}{6} \times \frac{6}{5} = 1 $ | e |

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✔ Final Matching:

```
1. g
2. a
3. d
4. c
5. a
6. f
7. e
8. b
9. f
10. h
11. d
12. g
13. h
14. j
15. e
```

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📌 Notes:

- Some properties are repeated (like `a`, `d`, `f`, `e`, `h`), which is allowed.
- Identity Property (f) includes both addition and multiplication.
- Zero Property of Addition (i) was not used — because all examples involving zero were either:
- Identity (e.g., $ a + 0 = a $) → use f
- Zero Product → use j
- So i wasn't needed. That’s okay — not every option needs to be used.

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✔ Final Answer List (to write on worksheet):

```
1. g
2. a
3. d
4. c
5. a
6. f
7. e
8. b
9. f
10. h
11. d
12. g
13. h
14. j
15. e
```