Properties of Operations Properties are rules that will help you solve problems with accuracy and efficiency!!

Properties of Operations This lesson gives a “mathematical property” name to the things you already know how to do. Our objective? Classify mathematical properties as they are used to show equivalent expressions. We are naming things you already do!

Real-World Link Angelica and Nari are baking cookies. They tell us that Angelica baked 6 sheets of 10 cookies. Nari baked 10 sheets with 5 cookies on each one. Yes, we KNOW they made the same amount…….remember, we are classifying expressions as to the type of property that TELLS us these are equal! Angelica: 6 x 10 = 60 Nari: 10 x 6 = 60 The key to identifying properties lies in noticing what has CHANGED in the expression, still leaving it equal. So……when multiplying, we can change the ORDER of the factors and not change the value. Who remembers which property states this? commutative p. 473

Vocabulary Commutative Property: The order in which two numbers are added or multiplied does not change their sum or product = · 6 = 6 · 4 a + b = b + a ab = ba Associative Property: The way that numbers are grouped when added or multiplied does not change the sum or product. 3 + (9 + 4) = (3 + 9) + 4 a + (b + c) = (a + b) + c 8 · (5 · 7) = ( 8 · 5 ) · 7 a(bc) = (ab)c WARNING Just because an expression as parentheses does NOT always mean it is associative! 5 + (4 + 9) = 5 + (9 + 4) ?????????????????

Vocabulary Identity Properties: Multiplying by 1 or adding 0 does not change the value of a number. Identity Property of Addition = 13 a + 0 = a Identity Property of Multiplication 7(1) = 7 a(1) = a 0 is the additive identity. 1 is the multiplicative identity! Properties: Statements that are true for any number. Notice they only apply to certain operations! Equivalent Expressions: Statements that have the same value. **Zero Property of Multiplication: Any number multiplied by 0 is 0. Don’t just say, “Zero property!!!”

Examples p. 474 Seeing What Has Changed… (5 + 8) = ( ) + 8 Did the order change? No, but the grouping is different. Associative Property of + 2. (20 – 12) – 3 ???? 20 – (12 – 3) WAIT…..have we even mentioned subtraction????? 8 – 3 ???? 20 – 9 5 ≠ 11 These are not equivalent! = 34 Identity Property of Addition ID +

Examples p. 474 Seeing What Has Changed… ÷ 5 ???? 5 ÷ 20 WHAT? No, no, no. Got it? a. 5 x (6 x 3 ) = ( 5 x 6) x 3 Did the order change? No, the grouping is different. Associative Property of x b. 27 ÷ 3 ???? 3 ÷ 27 No! These properties do NOT apply to subtraction and division!

Properties and Problem Solving p. 475 We are asked to write two equivalent statements using the Associative Property about the total positions on the Kansas Jayhawks basketball team. Associative means we will change the grouping. We will not change the order! 15 + (4 + 3) = 15 + (4 + 3) Guards15 Forwards4 Centers3

Financial Literacy We are asked to write two equivalent statements using the Commutative Property to find the sum of Brandi’s earnings. $7 to babysit $12 to clean the garage =

Geometry It doesn’t matter whether we write the 15 first or the 12 first. The product is the same….half of the product is still half of the product!

Financial Literacy Vicki earned $6 an hour while working for 11 hours over the weekend. She put one-third of what she earned in a savings account. Write equivalent expressions using the commutative property of multiplication. You may also want to apply the associative property in a different set of equivalent statements

Guided Practice 1. ( ) + 43 and 35 + ( ) Yes; Associative Prop. of + 2. ( 25 – 9) – 5 and 25 – (9 – 5) 16 – 5 is NOT equal to 25 – 4 NO! These are not equivalent Commutative Property of +

Guided Practice = Commutative Property must be used to find the total (sum) of the scores. The directions say “mentally.” Please write your equivalent expressions as you find you solution (sum). They have asked for the Associative Property. You may do both. Eventscore Vault8.95 Uneven bars ( ) = ( ) + 18 $45 = $45 Associative of = = 45 Commutative of + $15 $18 $12

Partner Challenge! Name that property: (Use the letter choices)5 x 1 = x n = n x = 8 3.(3∙ 2) ∙ 5 = 3 ∙ (2 ∙ 5) = (5 + n) = (2 + 5) + n 6.(5-5) + 3 = 3 ( tricky!) 7.(3-2) x 5 = = + 9.(a+b) + c = (b+a) + c (be careful!) 10.( + ) + = + ( + ) E B F D A C F E A A C

What Have We Learned? There are rules to describe the way you combine numbers. These rules are called “properties.” These properties help you to solve some types of problems mentally.

Look for properties in all processes!