Commutative • Associative • Identity • Inverse • Distributive
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Commutative Property
Changing the order of the numbers does not change the result.
Addition: \( a + b = b + a \)
Multiplication: \( a \cdot b = b \cdot a \)
Example: \( 4 + 9 = 9 + 4 \)
Does not work for subtraction or division.
Associative Property
Changing the grouping of the numbers does not change the result.
Addition: \( (a + b) + c = a + (b + c) \)
Multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)
Example: \( (2 + 3) + 5 = 2 + (3 + 5) \)
Identity Property
Adding 0 or multiplying by 1 leaves a number unchanged.
Additive Identity: \( a + 0 = a \)
Multiplicative Identity: \( a \cdot 1 = a \)
Example: \( 31 + 0 = 31 \)
Inverse Property
Every number has an opposite (for addition) and a reciprocal (for multiplication) that return the identity.
Additive Inverse: \( a + (-a) = 0 \)
Multiplicative Inverse: \( a \cdot \dfrac{1}{a} = 1 \) (\( a \neq 0 \))
Distributive Property
Multiplication distributes over addition (or subtraction).
$$a(b + c) = ab + ac$$
$$a(b – c) = ab – ac$$
Example: \( 3(x + 4) = 3x + 12 \)
Important Notes
• Commutative and Associative properties work for addition and multiplication only. • They do not hold for subtraction or division. • Always be ready to give a counter-example when a property fails.
Order can be changed for addition and multiplication without changing the result.
Rewrite using the commutative property of addition: \( 7 + 12 \)
$$12 + 7$$
Answer: \( 12 + 7 \)
Rewrite using the commutative property of multiplication: \( 5 \cdot 9 \)
$$9 \cdot 5$$
Answer: \( 9 \cdot 5 \)
True or False: \( 8 – 3 = 3 – 8 \)
False. Subtraction is not commutative.
\( 8 – 3 = 5 \) but \( 3 – 8 = -5 \).
Answer: False
Give an example that shows the commutative property of addition.
Any correct example works, such as:
$$15 + 7 = 7 + 15$$
Sample Answer: \( 15 + 7 = 7 + 15 \)
Give an example to show why the commutative property does not work with subtraction.
Any counter-example works. One common choice:
$$10 – 4 = 6 \quad \text{but} \quad 4 – 10 = -6$$
Since the results are different, subtraction is not commutative.
Sample Answer: \( 10 – 4 \neq 4 – 10 \)
Give an example to show why the commutative property does not work with division.
Any counter-example works. One common choice:
$$12 \div 3 = 4 \quad \text{but} \quad 3 \div 12 = \dfrac{1}{4}$$
Sample Answer: \( 12 \div 3 \neq 3 \div 12 \)
Which of the following shows the commutative property of multiplication? A) \( (2 \cdot 5) \cdot 3 = 2 \cdot (5 \cdot 3) \) B) \( 2 \cdot 5 = 5 \cdot 2 \) C) \( 2 + 5 = 5 + 2 \)
A is the associative property of multiplication. C is the commutative property of addition. B correctly changes the order of the factors.
Answer: B
Explain in your own words why the commutative property works for multiplication but not for division. Give a numerical example for each.
Multiplication is commutative because the order of the factors does not change the product: \( 6 \cdot 4 = 24 \) and \( 4 \cdot 6 = 24 \).
Division is not commutative because changing the order changes the result: \( 8 \div 2 = 4 \) but \( 2 \div 8 = \dfrac{1}{4} \).
Sample Answer shown above
Grouping can be changed for addition and multiplication without changing the result.
Rewrite using the associative property of addition: \( (5 + 3) + 9 \)
$$5 + (3 + 9)$$
Answer: \( 5 + (3 + 9) \)
Rewrite using the associative property of multiplication: \( (2 \cdot 4) \cdot 7 \)
$$2 \cdot (4 \cdot 7)$$
Answer: \( 2 \cdot (4 \cdot 7) \)
True or False: \( (10 – 4) – 2 = 10 – (4 – 2) \)
False. Subtraction is not associative.
\( (10 – 4) – 2 = 4 \) but \( 10 – (4 – 2) = 8 \).
Answer: False
Which property is shown? \( (6 + 2) + 9 = 6 + (2 + 9) \)
The grouping of the addends changed; the order stayed the same.
Answer: Associative Property of Addition
Give a counter-example that shows why the associative property does not work with division.
One possible counter-example:
$$(24 \div 6) \div 2 = 4 \div 2 = 2$$
$$24 \div (6 \div 2) = 24 \div 3 = 8$$
Since \( 2 \neq 8 \), division is not associative.
Sample Answer shown above
Adding zero or multiplying by one leaves a number unchanged.
What property is illustrated? \( 31 + 0 = 31 \)
Adding zero does not change the value of the number.
Answer: Additive Identity Property
What property is illustrated? \( 15 \cdot 1 = 15 \)
Multiplying by one does not change the value of the number.
Answer: Multiplicative Identity Property
Fill in the blank: \( 47 + \underline{\hspace{1cm}} = 47 \)
The additive identity is 0.
Answer: 0
What is the multiplicative identity for real numbers?
The number 1 is the multiplicative identity because \( a \cdot 1 = a \) for any real number \( a \).
Answer: 1
Which property is shown? \( -8 + 0 = -8 \)
Answer: Additive Identity Property
Every real number has an additive inverse (opposite) and (except zero) a multiplicative inverse (reciprocal).
What is the additive inverse of 9?
The number that adds with 9 to give 0 is −9.
Answer: −9
What is the multiplicative inverse of 5?
The number that multiplies with 5 to give 1 is \( \dfrac{1}{5} \).
Answer: \( \dfrac{1}{5} \)
What property is illustrated? \( 6 + (-6) = 0 \)
Answer: Additive Inverse Property
What is the multiplicative inverse of \( -\dfrac{3}{4} \)?
The reciprocal of \( -\dfrac{3}{4} \) is \( -\dfrac{4}{3} \).
Answer: \( -\dfrac{4}{3} \)
What property is illustrated? \( \dfrac{2}{5} \cdot \dfrac{5}{2} = 1 \)
Answer: Multiplicative Inverse Property
Multiplication distributes over addition and subtraction.
Use the distributive property to expand: \( 4(x + 3) \)
$$4 \cdot x + 4 \cdot 3 = 4x + 12$$
Answer: \( 4x + 12 \)
Use the distributive property to expand: \( 5(y – 2) \)
$$5 \cdot y – 5 \cdot 2 = 5y – 10$$
Answer: \( 5y – 10 \)
What property is illustrated? \( 3(a + b) = 3a + 3b \)
Answer: Distributive Property
Use the distributive property to expand: \( -2(3x – 5) \)
$$-2 \cdot 3x – (-2) \cdot 5 = -6x + 10$$
Answer: \( -6x + 10 \)
Factor using the distributive property: \( 8x + 12 \)
The greatest common factor is 4:
$$4(2x + 3)$$
Answer: \( 4(2x + 3) \)
Name the property illustrated by each equation. Be as specific as possible (include “of addition” or “of multiplication” when needed).
What property is illustrated? \( 31 + 0 = 31 \)
Answer: Additive Identity Property
What property is illustrated? \( 9 \cdot 4 = 4 \cdot 9 \)
Answer: Commutative Property of Multiplication
What property is illustrated? \( (2 + 5) + 8 = 2 + (5 + 8) \)
Answer: Associative Property of Addition
What property is illustrated? \( 7(x + 3) = 7x + 21 \)
Answer: Distributive Property
What property is illustrated? \( \dfrac{3}{8} \cdot \dfrac{8}{3} = 1 \)
Answer: Multiplicative Inverse Property
What property is illustrated? \( -5 + 5 = 0 \)
Answer: Additive Inverse Property
Name the property used in each step of the simplification: \( 3(x + 2) + 5 = 3x + 6 + 5 \) (Step 1) \( = 3x + 11 \) (Step 2)
Step 1: Distributive Property
Step 2: Combining like terms (or Associative / Commutative properties of addition)
Answer: Distributive; then combining like terms
Give an example to show why the commutative property does not work with subtraction, and then state the correct name of the property that does allow us to change the order of addition.
Counter-example for subtraction: \( 9 – 4 = 5 \) but \( 4 – 9 = -5 \).
The property that allows changing order for addition is the Commutative Property of Addition.
Sample Answer shown above
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