Properties of Numbers & Equality Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem by identifying the property of numbers being applied. Here are the 12 problems and their corresponding properties, with explanations.

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1. $ 4 + (2 - 3) = (4 + 2) - 3 $

- Property: Associative Property of Addition
- Explanation: This property states that when adding three or more numbers, the way the numbers are grouped does not change the sum. Although subtraction is involved here, it can be thought of as addition of a negative:
$ 4 + (2 + (-3)) = (4 + 2) + (-3) $.
So, grouping changes, but the result stays the same.

> ✔ Answer: Associative Property of Addition

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2. $ 8 + 0 = 8 $

- Property: Identity Property of Addition
- Explanation: Adding zero to any number leaves the number unchanged. Zero is the additive identity.

> ✔ Answer: Identity Property of Addition

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3. $ 2(4 + 6) = 20 $

- Note: The left side simplifies to $ 2 \times 10 = 20 $, so this is just showing an example.
- But the key is that $ 2(4 + 6) $ is expanded using distribution: $ 2 \cdot 4 + 2 \cdot 6 = 8 + 12 = 20 $
- Property: Distributive Property
- Explanation: Multiplication distributes over addition: $ a(b + c) = ab + ac $

> ✔ Answer: Distributive Property

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4. $ 16 + (-16) = 0 $

- Property: Inverse Property of Addition
- Explanation: A number plus its opposite (additive inverse) equals zero.

> ✔ Answer: Inverse Property of Addition

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5. $ 5 \cdot \frac{1}{5} = 1 $

- Property: Inverse Property of Multiplication
- Explanation: A number multiplied by its reciprocal (multiplicative inverse) equals 1.

> ✔ Answer: Inverse Property of Multiplication

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6. $ 8 \cdot 9 = 9 \cdot 8 $

- Property: Commutative Property of Multiplication
- Explanation: The order of multiplication does not affect the product.

> ✔ Answer: Commutative Property of Multiplication

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7. $ -7 + 6 = 6 + (-7) $

- Property: Commutative Property of Addition
- Explanation: The order of addition does not affect the sum.

> ✔ Answer: Commutative Property of Addition

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8. $ -90 \cdot 1 = -90 $

- Property: Identity Property of Multiplication
- Explanation: Multiplying any number by 1 gives the same number. 1 is the multiplicative identity.

> ✔ Answer: Identity Property of Multiplication

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9. $ 1,773 + 425 = 425 + 1,773 $

- Property: Commutative Property of Addition
- Explanation: The order of the addends doesn't matter.

> ✔ Answer: Commutative Property of Addition

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10. $ 16 \cdot (2 \cdot 4) = (16 \cdot 2) \cdot 4 $

- Property: Associative Property of Multiplication
- Explanation: When multiplying three numbers, how they are grouped doesn’t affect the product.

> ✔ Answer: Associative Property of Multiplication

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11. $ \frac{2}{3} \cdot 1 = \frac{2}{3} $

- Property: Identity Property of Multiplication
- Explanation: Any number times 1 remains unchanged.

> ✔ Answer: Identity Property of Multiplication

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12. $ -3(5 + 1) = -18 $

- First simplify: $ -3(6) = -18 $, which is correct.
- But the expression shows multiplication distributed over addition:
$ -3(5 + 1) = (-3)(5) + (-3)(1) = -15 + (-3) = -18 $
- Property: Distributive Property
- Explanation: Multiplication distributes over addition.

> ✔ Answer: Distributive Property

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✔ Final Answers:

| Problem | Property |
|--------|----------|
| 1 | Associative Property of Addition |
| 2 | Identity Property of Addition |
| 3 | Distributive Property |
| 4 | Inverse Property of Addition |
| 5 | Inverse Property of Multiplication |
| 6 | Commutative Property of Multiplication |
| 7 | Commutative Property of Addition |
| 8 | Identity Property of Multiplication |
| 9 | Commutative Property of Addition |
| 10 | Associative Property of Multiplication |
| 11 | Identity Property of Multiplication |
| 12 | Distributive Property |

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