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 sheetsof 10 cookies. Nari baked 10 sheets with 5 cookies on each one.

Yes, we KNOW they made the same amount…….remember, we are classifyingexpressions 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 liesin noticing what has CHANGED in theexpression, still leaving it equal.

So……when multiplying, we can change the ORDER of the factorsand not change the value.

Who remembers which property states this?commutative

p. 473

VocabularyCommutative Property: The order in which two numbers

are added or multiplied does not change their sum or product. 7 + 9 = 9 + 7 4 · 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

WARNINGJust 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 + 0 = 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….

1. 15 + (5 + 8) = ( 15 + 5) + 8 Did the order change?

No, but the grouping isdifferent.Associative Property of +

2. (20 – 12) – 3 ???? 20 – (12 – 3)

WAIT…..have we even mentionedsubtraction?????

8 – 3 ???? 20 – 9 5 ≠ 11 These are not equivalent!

3. 30 + 0 = 34Identity Property of Addition ID +

Examples p. 474 Seeing What Has Changed….

4. 20 ÷ 5 ???? 5 ÷ 20 WHAT? No, no, no.

205 ≠

520

Got it?

a. 5 x (6 x 3 ) = ( 5 x 6) x 3

Did the order change?

No, the grouping isdifferent.

Associative Property of x

b. 27 ÷ 3 ???? 3 ÷ 27273 ≠

327

91 ≠19

No! These propertiesdo NOT apply to subtraction anddivision!

Properties and Problem Solving p. 475

We are asked to write two equivalent statementsusing the Associative Propertyabout the total positions on the Kansas Jayhawksbasketball team.

Associative means we will change the grouping. We willnot change the order!

15 + (4 + 3) = 15 + (4 + 3)

Guards 15

Forwards 4

Centers 3

Financial Literacy

We are asked to write two equivalent statementsusing the Commutative Property to find the sum of Brandi’s earnings.

$7 to babysit

$12 to clean the garage

7 + 12 = 12 + 7

Geometry You will recall that the formula for the area of atriangle involves multiplication.

A = base(height)

In our text example, the base is 15 ft. The height is 12 ft. Finding half of thisproduct gives us the area of the triangular space created by the slide and the ladder.

A = 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 theproduct!

12

(15 ) (12 )=12 (12)¿

Financial Literacy

Vicki earned $6 an hour while working for 11 hoursover the weekend. She put one-third of what she earned ina savings account. Write equivalent expressions using thecommutative property of multiplication. You may also want to apply the associative property in a differentset of equivalent statements

Comm. of x (6 ·11) 66() $22= $22

Assoc. of x (6 ·11) 66() = 11(2) $22 = $22

Guided Practice 1. (35 + 17) + 43 and 35 + (17 + 43)

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 +

Commutative Property of +

Guided Practice

8.95 + 9.2 = 9.2 + 8.95Commutative Property must be used to find the total (sum)of the scores.

The directions say “mentally.” Pleasewrite your equivalent expressions asyou find you solution (sum). They have askedfor the Associative Property. You may do both.

Event score

Vault 8.95

Uneven bars 9.2

12 + (15 + 18) = (12 + 15) + 18 $45 = $45 Associative of +

12 + 15 + 18 = 18 + 15 + 12 45 = 45 Commutative of +

$15 $18

$12

Partner Challenge!• Name that property: (Use the letter choices)5 x 1 = 5 1. 6 x n = n x 62. 8 + 0 = 83. (3 2) 5 = 3 (2 5) ∙ ∙ ∙ ∙4. 12 + 4 = 4 + 125. 2 + (5 + n) = (2 + 5) + n6. (5-5) + 3 = 3 (tricky!)7. (3-2) x 5 = 58. + = +

9. (a+b) + c = (b+a) + c (be careful!)10. ( + ) + = + ( + )

EB

F

DA

C

FE

AA

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!