Properties of Operations

Properties of Operations

Properties of Operations: Vocabulary

• Commutative Property

• Associative Property

• Additive Identity Property

• Multiplicative Identity Property

• Multiplicative Property of Zero

• counterexample

Commutative Property

• the ORDER in which numbers are ADDED or MULTIPLIED does not change the SUM or PRODUCT

• Addition Symbols: x + y = y + x

• Addition Example:  3 + 5 = 5 + 3

• Multiplication Symbols: a * b = b * a

• Multiplication Example:  2 * 4 = 4 * 2

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Multiple Choice

Which of the following shows the COMMUTATIVE PROPERTY?

• 3 * 6 = 6 * 3

• 2 * 4 = 3 * 4

Multiple Choice

Which of the following show the COMMUTATIVE PROPERTY?

• 4 + 9 = 5 + 8

• 10 + 12 = 12 + 10

Associative Property

• the way in which numbers are GROUPED when they are added or multiplied does not change the SUM or PRODUCT

• Addition Symbols: a + (b + c) = (a + b) + c

• Addition Example: 2 + (3 + 8) = (2 + 3) + 8

• Multiplication Symbols: a * (b * c) = (a * b) * c

• Multiplication Example: 3 * (4 * 5) = (3 * 4) * 5

Multiple Choice

Which of the following shows the ASSOCIATIVE property?

• 2 + ( 3 + 5) = (2 + 3) + 5

• 2 + 4 +8 = (2 + 4) + 8

Multiple Choice

Which of the following shows the ASSOCIATIVE property?

• 2 * 3 = 3 * 2

• (3 * 4) * 9 = 3 * (4 * 9)

Property

• A property is a statement that is true for any number.

Additive Identity Property

• When 0 is added to any number, the sum is the number.

• Symbols: 0 + x = x

• Example: 0 + 9 = 9

Multiplicative Identity

• When any number is multiplied by 1, the product is the number.

• Symbols: x * 1 = x

• Example: 1 * 5 = 5

Multiplicative Property of Zero

• When any number is multiplied by 0, the product is 0.

• Symbols: x * 0 = 0

• Example: 3 * 0 = 0

Multiple Choice

Name the property shown by the statement:

2 * (5 * n) = (2 * 5) * n

• Associative Property of Multiplication

• Associative Property of Addition

• Commutative Property of Multiplication

• Additive Identity Property

Multiple Choice

Name the property shown by the statement:

42 + x + y = 42 + y + x

• Associative Property of Addition

• Additive Identity

• Commutative Property of Addition

Multiple Choice

Name the property shown by the statement:

3x + 0 = 3x

• Additive Identity

• Multiplicative Identity

• Associative Property of Addition

Counterexample

• You may wonder if any of the properties apply to subtraction or division.

• If you can find a counterexample, an example that shows that a conjecture is false, the property does not apply.

State whether the following conjecture is true or false. If false, provide a counterexample.

• Division of whole numbers is commutative.

• Write two division expressions using the Commutative Property.

•  $15\ \div\ 3\ =\ 3\ \div\ 15$  <-- conjecture 

•  $5\ \ne\ \frac{1}{5}$  <-- Divide 

• The conjecture is false. We found a counterexample. That is  $15\ \div\ 3\ \ne\ 3\ \div\ 15$ .  So division is NOT commutative.

State whether the following conjecture is true or false. If false, provide a counterexample.

• The difference of two different whole numbers is always less than both of the numbers.

• False

• Counterexample:

   $8\ -\ 2\ =\ 6;\ is\ 6\ <\ 2?\ No\$  

Simplify each expression. Justify each step.

•  $\left(7\ +\ g\right)\ +\ 5$  

• (7 + g) + 5 = (g + 7) + 5  ---- Commutative Property of Addition 

• (7 + g) + 5 = g + (7 + 5) --- Associative Property of Addition

• = g + 12 --- Simplify 

Simplify each expression. Justify each step.

• (m * 11) * m = (11 * m) * m

• (m * 11) * m = (11 * m) * m --- Commutative Property of Multiplication

• (m * 11) * m = 11 * (m * m) --- Associative Property of MMultiplication

•  $11m^2$  -- Simplify