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

Let's solve and complete the "Properties of Numbers Notes" worksheet step by step. We’ll fill in the blanks in the Definition column and compute the Examples.

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✔ 1. Commutative Property

- Definition: When adding or multiplying, the order does not matter.
- Examples:
- $ 2 + 3 = 5 $
- $ 3 + 2 = 5 $
- $ 4 \cdot 5 = 20 $
- $ 5 \cdot 4 = 20 $

✔ Explanation: The commutative property means that changing the order of numbers in addition or multiplication doesn’t change the result.

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✔ 2. Associative Property

- Definition: When adding or multiplying, it doesn't matter how the numbers are grouped.
- Examples:
- $ 3 + (2 + 1) = 3 + 3 = 6 $
- $ (3 + 2) + 1 = 5 + 1 = 6 $
- $ 2 \cdot (4 \cdot 6) = 2 \cdot 24 = 48 $
- $ (2 \cdot 4) \cdot 6 = 8 \cdot 6 = 48 $

✔ Explanation: The associative property states that grouping numbers differently (using parentheses) doesn’t affect the sum or product.

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✔ 3. Distributive Property

- Definition: When multiplying by a sum, you can multiply each term separately and add.
- Example:
- $ 3(2 + 4) = 3 \cdot 2 + 3 \cdot 4 = 6 + 12 = 18 $

✔ Explanation: This property allows us to distribute multiplication over addition: $ a(b + c) = ab + ac $.

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✔ 4. Identity Property

- Definition:
- The sum of any number and zero is that number.
- The product of any number and one is that number.
- Examples:
- $ 4 + 0 = 4 $
- $ 8 \cdot 1 = 8 $

✔ Explanation:
- Additive Identity: Adding zero leaves a number unchanged.
- Multiplicative Identity: Multiplying by one leaves a number unchanged.

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✔ 5. Inverse Property

- Definition:
- Additive Inverse: The sum of any number and the negative of that number is zero.
- Multiplicative Inverse: The product of any number and its reciprocal is one.
- Examples:
- $ 7 + (-7) = 0 $
- $ 3 \cdot \frac{1}{3} = 1 $

✔ Explanation:
- Every number has an additive inverse (opposite) that adds to zero.
- Every non-zero number has a multiplicative inverse (reciprocal) that multiplies to one.

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✔ Final Completed Table

| PROPERTY | DEFINITION | EXAMPLES |
|----------------------|----------------------------------------------------------------------------|----------|
| Commutative Property | When adding or multiplying, the order does not matter. | $ 2+3=5 $, $ 3+2=5 $<br>$ 4\cdot5=20 $, $ 5\cdot4=20 $ |
| Associative Property | When adding or multiplying, it doesn’t matter how the numbers are grouped. | $ 3+(2+1)=6 $, $ (3+2)+1=6 $<br>$ 2\cdot(4\cdot6)=48 $, $ (2\cdot4)\cdot6=48 $ |
| Distributive Property | When multiplying by a sum, you can multiply each term separately and add. | $ 3(2+4)=6+12=18 $ |
| Identity Property | The sum of any number and zero is that number.<br>The product of any number and one is that number. | $ 4+0=4 $, $ 8\cdot1=8 $ |
| Inverse Property | Additive Inverse: The sum of any number and its opposite is zero.<br>Multiplicative Inverse: The product of any number and its reciprocal is one. | $ 7+(-7)=0 $, $ 3\cdot\frac{1}{3}=1 $ |

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✔ Summary of Answers:

- Commutative: order
- Associative: grouped
- Distributive: term
- Identity: zero, one
- Inverse: zero, one

This completes the worksheet! Let me know if you'd like this formatted for printing or as a PDF.