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 $

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Now let’s match each:

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1. 5 + 4 = 9

This is just a simple addition. But no property is being shown, since it's not showing order, grouping, or inverse. However, it does show closure because the result is an integer.
But wait — this is just a fact, not illustrating a property like commutativity or associativity. Let's see if it's identity? No. It's not rearranging terms.
Actually, this is just an example of closure, since 5 and 4 are integers, and their sum is an integer.
✔ Answer: g. Closure Property

> Wait! But 5+4=9 is just arithmetic. But the closure property says that when you add two integers, you get another integer. So yes, this shows closure.

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2. 2 + 3 = 3 + 2

Order changed → Commutative Property of Addition
✔ Answer: a

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3. 27 + (–27)

This equals 0. This is adding a number and its opposite → Inverse Property of Addition
✔ Answer: d

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

Grouping changed → Associative Property of Addition
✔ Answer: c

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

Here, the order of 3 and (6+2) is swapped. So it's commutative property of addition.
Note: (6+2) is just a number (8), so we’re swapping 3 and 8.
So $ 3 + 8 = 8 + 3 $ → Commutative Property of Addition
✔ Answer: a

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6. 0 + (–8) = –8

Adding zero doesn't change the number → Identity Property of Addition
✔ Answer: f

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7. –5 × 1/5 = 1

Multiplying a number by its reciprocal gives 1 → Inverse Property of Multiplication
✔ Answer: e

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8. (13)(–3) = (–3)(13)

Order of multiplication changed → Commutative Property of Multiplication
✔ Answer: b

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9. 5 = (5)(1)

Multiplying by 1 keeps the number same → Identity Property of Multiplication
✔ Answer: f

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10. 4(5 – 2) = 4(5) – 4(2)

Distributing 4 over subtraction → Distributive Property
✔ Answer: h

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11. 3 + [4 + (–4)] = 3 + 0

Here, 4 + (–4) = 0 → additive inverse → then 3 + 0 = 3
So it's using Inverse Property of Addition (4 and –4 cancel)
✔ Answer: d

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12. –3 + 5 = 2 is an integer

The sum of two integers is an integer → Closure Property
✔ Answer: g

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13. –3(6 + 5) = –18 + (–15)

Left side: –3 × 11 = –33
Right side: –18 + (–15) = –33
So: –3 × (6 + 5) = (–3×6) + (–3×5) → Distributive Property
✔ Answer: h

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14. (–3)(–2)(–1)(0) = 0

Any number times zero is zero → Zero Property of Multiplication
✔ Answer: j

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15. 5/6 × 6/5 = 1

Reciprocal multiplication → Inverse Property of Multiplication
✔ Answer: e

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✔ Final Answers:

| Number | Answer |
|--------|--------|
| 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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✔ Matching List:

```
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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✔ Summary of Properties Used:

- Commutative Addition (a): 2, 5
- Commutative Multiplication (b): 8
- Associative (c): 4
- Inverse Addition (d): 3, 11
- Inverse Multiplication (e): 7, 15
- Identity (f): 6, 9
- Closure (g): 1, 12
- Distributive (h): 10, 13
- Zero Property of Multiplication (j): 14

> Note: Some properties appear multiple times, which is allowed.

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✔ Final Answer Key (to fill in blanks):

```
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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