Commutative Property and Associative Property

🌟 Definition of Commutative Property:

The Commutative Property says you can swap the numbers when you add or multiply, and the answer stays the same.

$$ a+b=b+a $$

$$ a \times b = b \times a $$

🔔 Important: The Commutative Property does not work for subtraction or division.

💡 Addition Example:

• Start with:
  $77 + 246 + 23$
• Use the Commutative Property to swap 246 and 23:
  $77 + 23 + 246$
• Add 77 and 23 first:
  $100 + 246$
• Final answer:
  $346$

💡 Multiplication Example:

• Start with:
  $25 \times 19 \times 4$
• Use the Commutative Property to swap 19 and 4:
  $25 \times 4 \times 19$
• Multiply 25 and 4 first:
  $100 \times 19$
• Final answer:
  $1900$

🌟 Definition of Associative Property:

The Associative Property allows you to group numbers differently when you add or multiply, without changing the final answer.

$$(a+b)+c = a+(b+c)$$

$$(a \times b) \times c = a \times (b \times c)$$

🔔 Important: The Associative Property does not work for subtraction or division.

Use the Associative Property when you want to group numbers differently to make calculations easier.

You can only use it when the expression has only addition or only multiplication.

💡 Addition Example:

• Start with:
  $(127+48)+52$
• Use the Associative Property to regroup $48$ and $52$ because they add up to $100$, making the addition simpler:
  $127+(48+52)$
• Add inside the brackets first:
  $127+100$
• Final answer:
  $227$

💡 Multiplication Example:

• Start with:
  $5 \times (2 \times 128)$
• Use the Associative Property to regroup $5$ and $2$ because they multiply to $10$, making the multiplication simpler:
  $(5 \times 2) \times 128$
• Multiply inside the brackets first:
  $10 \times 128$
• Final answer:
  $1280$

Sometimes, you can combine the Commutative and Associative Properties to make calculations even easier!

• Use the Commutative Property to rearrange numbers.
• Then use the Associative Property to regroup numbers smartly.

This helps you pick easier pairs to add or multiply first.

💡 Addition Example:

• Start with:
  $32 + (115 + 68)$
• Step 1: Use the Commutative Property to swap $115$ and $68$:
  $32 + (68 + 115)$
• Step 2: Use the Associative Property to regroup $32$ and $68$, because $32 + 68$ makes $100$, an easy number to add:
  $(32 + 68) + 115$
• Step 3: Add inside the brackets first:
  $100 + 115$
• Final answer:
  $215$

💡 Multiplication Example:

• Start with:
  $20 \times (48 \times 5)$
• Step 1: Use the Commutative Property to swap $48$ and $5$:
  $20 \times (5 \times 48)$
• Step 2: Use the Associative Property to regroup $20$ and $5$, because $20 \times 5$ makes $100$, an easy number to multiply:
  $(20 \times 5) \times 48$
• Step 3: Multiply inside the brackets first:
  $100 \times 48$
• Final answer:
  $4800$