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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.