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 solve the Example problems.

---

✔ 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 order of numbers in addition or multiplication doesn’t change the result.

---

✔ 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 grouping (parentheses) doesn’t affect the sum or product.

---

✔ 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: You distribute the multiplier to each term inside the parentheses.

---

✔ 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: Adding zero or multiplying by one leaves the number unchanged.

---

✔ 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:
- Additive inverse "cancels" the number to give zero.
- Multiplicative inverse (reciprocal) gives a product of 1.

---

✔ Final Completed Table

| PROPERTY | DEFINITION | EXAMPLES |
|----------------------|------------------------------------------------------------------------------------------------|----------|
| Commutative | When adding or multiplying, the order does not matter. | $2+3=5$, $3+2=5$<br>$4\cdot5=20$, $5\cdot4=20$ |
| Associative | 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 | When multiplying by a sum, you can multiply each term separately and add. | $3(2+4)=6+12=18$ |
| Identity | Sum with zero is the number; Product with one is the number. | $4+0=4$, $8\cdot1=8$ |
| Inverse | Additive: sum with negative is zero.<br>Multiplicative: product with reciprocal is one. | $7+(-7)=0$, $3\cdot\frac{1}{3}=1$ |

---

🎯 Summary of Key Concepts:

| Property | Key Idea |
|---------------------|--------|
| Commutative | Order doesn’t matter |
| Associative | Grouping doesn’t matter |
| Distributive | Multiply across a sum |
| Identity | Zero for addition, One for multiplication |
| Inverse | Additive: opposite → 0; Multiplicative: reciprocal → 1 |

This completes the worksheet! Let me know if you'd like a printable version or a study guide.