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(bc) = (ab)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:
$ a + 0 = a $, $ a \times 1 = a $

- g. Closure Property:
The result of an operation on integers is also an integer.

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

- i. Zero Property of Addition:
$ a + 0 = a $ → same as Identity Property of Addition

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

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Now, let’s analyze each problem:

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1. 5 + 4 = 9

This is just a simple addition. It shows that adding two integers gives another integer.
✔ Closure Property
→ Answer: g

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2. 2 + 3 = 3 + 2

Order of addition changes — but sum stays the same.
✔ Commutative Property of Addition
→ Answer: a

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3. 27 + (−27)

Adding a number and its opposite gives zero: $ 27 + (-27) = 0 $
✔ Inverse Property of Addition
→ Answer: d

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

Grouping changed — parentheses moved.
✔ Associative Property of Addition
→ Answer: c

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

Here, both grouping and order change. But note:
Left side: $ 3 + (6+2) = 3 + 8 = 11 $
Right side: $ (6+2) + 3 = 8 + 3 = 11 $
But the order of 3 and (6+2) is swapped → so it's commutative.

Wait! Let's see:
- $ 3 + (6+2) = (6+2) + 3 $ → yes, the sum is commuted.
So it's 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

> Note: "Zero Property of Addition" (i) might seem similar, but that's not standard terminology. Usually, Identity Property of Addition is used.
> Also, “Zero Property” usually refers to multiplication (j), not addition.

So this is f, not i.

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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 doesn’t change the value.
✔ Identity Property of Multiplication
→ Answer: f

> So both #6 and #9 use f, which is allowed since repetition is permitted.

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10. 4(5 − 2) = 4(5) − 4(2)

Distributing 4 over subtraction:
$ 4(5 - 2) = 4×5 - 4×2 $ → Yes, distributive property
✔ Distributive Property
→ Answer: h

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11. 3 + [4 + (−4)] = 3 + 0

Inside the brackets: $ 4 + (-4) = 0 $, then $ 3 + 0 = 3 $
This uses the Inverse Property of Addition (4 and -4 cancel), and then Identity Property.

But the key part is that $ 4 + (-4) = 0 $, which is inverse.
So the main property illustrated here is Inverse Property of Addition
→ Answer: d

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12. −3 + 5 = 2 is an integer

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

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13. −3(6 + 5) = −18 + (−15)

Left: $ -3(11) = -33 $
Right: $ -18 + (-15) = -33 $
So it's showing: $ -3(6 + 5) = -3×6 + (-3)×5 $ → Distributive Property
✔ Answer: h

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14. (−3)(−2)(−1)(0) = 0

Any number multiplied by 0 is 0 → Zero Property of Multiplication
✔ Answer: j

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15. 5/6 × 6/5 = 1

A number times its reciprocal equals 1 → 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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📝 Summary:

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

> Note: i. Zero Property of Addition was not used. That would be like $ a + 0 = a $, but that's already covered under Identity Property (f). So i is not needed.

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✔ Final Matched Answers:

```
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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