Properties of Numbers

and the Distributive

Property

Sections 2-1 and 2-2

Mental Math Warm - Up

1. 18 + (2+7) =

2. (18+2) + 7 =

3. (95) 4 =

4. 9 (54) =

27

27

180

180

Commutative Properties of Addition and Multiplication

The order of the values does NOT change the sum or

product.

Commutative Property of Addition and Multiplication (Cont.)

Arithmetic

7 + 3 = 3 + 7

Algebra

a + b = b + a

2 4 = 4 2 a b = b a

Example 1

Associative Property of Addition and Multiplication

The grouping of the values does NOT change the sum or

product.

Associative Property of Addition and Multiplication (Cont.)

Arithmetic(2 + 7) + 3 = 2 + (7 +

3)

Algebra

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

(9 4)5 = 9(4 5) (ab)c = a(bc)

At the Movies…

At the movies you buy an adult ticket, small popcorn, and a small drink. What was your total cost?

What property/properties could you use?

$9

Associative Property (5 + 3) +1 5 + (3 +1)

Tickets Popcorn Candy Drinks

Child - $3 Small - $3 Small - $2 Small - $1

Adult - $5 Medium - $4 Medium - $3 Medium - $2

Senior - $3 Large - $5 Large - $5 Large - $3

Example 2

(100 2) 3 Commutative Property

100 (2 3) Associative Property

= 100 6 Multiply within parentheses

= 600 Solve

Use mental math and properties to solve the following problem: 100 3 2

Identity Property of Addition

The sum of any number and zero is the original number.

7 + 0 = 7

0 + x = x

Identity Property of MultiplicationThe product of any number and

1 is the original number.

15 1 = 15

1 b = b

Example 3

Name each property shown.3 4 = 4 3

z 1 = z

+ ★ = ★ +

5(xy) = (5x)y

Commutative Property of Multiplication

Identity Property of Multiplication

Commutative Property of Addition

Associative Property of Multiplication

Distributive Property

To multiply a sum or difference, multiply each number within the parentheses by the number outside the parentheses.

3(2+6) = 3(2) + 3(6) (2-6)3 = 2(3) – 6(3)

a(b+c) = a(b) + a(c) (b-c)a = ba – ca

Distributive Property (cont.)

Example 4

Use the Distributive Property to find 20(102) mentally.

20(102) = 20(100 + 2) Think of 102 as 100 + 2

20(100 + 2) = 20(100) + 20(2) Use Distributive Property

= 2000 + 40 Multiply

= 2,040 Add

Example 5

You have 3 friends and want to give each friend 6 cupcakes, 5 cookies, and 4 brownies. How many baked goods will you have all together?

3(6+5+4)

3(6) + 3(5) + 3(4)

18 + 15 + 12

= 45 baked goods

We can also use mathematical models to demonstrate the distributive property.

Example 5 (cont.)

= Cookies = Brownies = Cupcakes

Friend 1: Friend 2: Friend 3:

Total:

= 45 baked goods

Practice Problems

1. 3(6+7) = ?

2. 4(22+3) = ?

3. 6(b+5) = ?

4. 9(2h-1) = ?

5. 3(5-3w) = ?

3(6) + 3(7) = 18 + 21 = 394(22) + 4(3) = 88 + 12 = 1006(b) + 6(5) = 6b + 30

9(2h) – 9(1) = 18h - 9

3(5) – 3(3w) = 15 – 9w

Practice Problems (Cont.)

6. Suppose your friend wrote 7(2m+t) = 14m + t. What error did your friend make? What is the correct answer?

He or she forgot to distribute the 7 to the variable t.

The correct answer should be:7(2m) + 7(t) = 14m + 7t

Picture Citations:

Light Bulb Guy Picture (Slide 2) - http://www.cascadewellnessclinic.com/articles/2003art/0311art.shtml

Shopping Bag Picture (Slide 5) – http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/structure/st275ma.htm

Distributive Arrow picture (Slide 14) - http://www.coolmath.com/reference/math-dictionary-D.html

Distributive Property Balance (Slide 13) - http://www.leslienettling.com/6thNewsletters/6thSep8-06.htm

Content Citations

Badur, Michalina Central Michigan University student

Davison, David M., Marsha S. Landau, Leah McCracken, and Linda Thompson. Prentice Hall Pre-algebra: Tools for a Changing World. Upper Saddle River, NJ: Prentice Hall, 2001. Print.