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The Foundation
Copyright Scott Storla 2015
Simplify
12  4  3 
12

12  4  3 
12
Wolfram Alpha
10w


5w  2  2 5w
Solve
3  k k 5k 1
 

9
6 36 3
Prime Factor the Polynomial
9k 4  18k 3  16k 2  32k
Copyright Scott Storla 2015
Properties
Copyright Scott Storla 2015
A property allows us to use a general idea in
specific situations.
For instance a property of fire is that it needs
oxygen to burn.
We use this property when we blow on a
struggling campfire or extinguish a frying pan fire
with a cover.
Numbers and operations have properties too.
Copyright Scott Storla 2015
The Commutative Properties
The Commutative Property of Addition
The Commutative Property of Multiplication
The order of the terms doesn’t affect the sum.
The order of the factors doesn’t affect the product.
Example: 3  4  4  3
Example: 3  4  4  3
Note: Subtraction is not commutative.
Note: Division is not commutative.
The Associative Properties
The Associative Property of Addition
The Associative Property of Multiplication
The grouping of the terms doesn’t affect the sum.
The grouping of the factors doesn’t affect the product.
Example: 3   4  5    3  4   5
Example: 3   4  5    3  4   5
Note: Subtraction is not associative.
Note: Division is not associative.
The Distributive Property of Multiplication Over Addition
A sum with a common factor can be rewritten as the product of the
common factor and the sum of the remaining factors.
Example: 3(4)  3(2)  3  4  2  and 3  4  2   3(4)  3(2)
The Identity Properties
The Additive Identity
The Multiplicative Identity
0 is the additive identity. Adding 0 to an expression
doesn’t change the value of the expression.
1 is the multiplicative identity. Multiplying an expression
by 1 doesn’t change the value of the expression.
Example: 0  3 is equivalent to 3.
Example: 1 5  is equivalent to 5.
The Inverse Properties
The Additive Inverse
The Multiplicative Inverse
The expression which when added to the original
expression gives a sum of 0.
The expression which when multiplied to the original
expression gives a product of 1.
Example: The additive inverse of 8 is 8 .
Example: The multiplicative inverse of 2 is 1/2.
Copyright Scott Storla 2015
Where it’s appropriate use a property to justify the step.
Simplify
15k  14 j  12k  2 j
____________________________________
Wrote subtraction as adding an opposite.
15k  14 j  12k  2 j
____________________________________
The commutative property of addition.
15k  12k  14 j  2 j
The
distributive property.
____________________________________
 15  12  k  14  2  j
____________________________________
Added.
27k  12 j
___________________________________
The
commutative property of addition.
12 j  27k
____________________________________
Wrote adding an opposite as subtraction.
12 j  27k
Copyright Scott Storla 2015
The Foundation
Copyright Scott Storla 2015
The Commutative properties are about order.
Property – The Commutative Property of Addition
English: The order of the terms doesn’t affect the sum.
Example: 3  4  4  3
Note: Subtraction is not commutative.
Property – The Commutative Property of Multiplication
English: The order of the factors doesn’t affect the product.
Example: 3  4  4  3
Note: Division is not commutative.
Copyright Scott Storla 2015
The Associative properties are about grouping.
Property – The Associative Property of Addition
English: The grouping of the terms doesn’t affect the sum.
Example: 3   4  5    3  4   5
Note: Subtraction is not associative.
Property – The Associative Property of Multiplication
English: The grouping of the factors doesn’t affect the product.
Example: 3   4  5    3  4   5
Note: Division is not associative.
Copyright Scott Storla 2015
Describe which property is being used to transform
the upper expression to the lower expression. Don’t
simplify the expression.
2   3  2 
2   2  3 
2   2  3 
 2  2   3
t  8 
8t
4  1  3  1
 1 1 4  3 
The commutative property of addition.
The associative property of addition.
The commutative property of multiplication.
The commutative property of multiplication.
1  1 4   3
 1 1  4  3
The associative property of multiplication.
Copyright Scott Storla 2015
Describe which property is being used to transform
the upper expression to the lower expression. Don’t
simplify the expression.
 4y  5    y  6 
 y  6    4y  5 
 3 x  5  2x  1
 2x  1 3 x  5 
2(1)  2( 1)  5( 1)
2(1)  2( 1)  5( 1)
2  3k 
2  3 k
The commutative property of addition.
The commutative property of multiplication.
The associative property of addition.
The associative property of multiplication.
Copyright Scott Storla 2015
The Distributive property of multiplication over addition
Property – The Distributive Property of Multiplication
over Addition
English: A sum of terms, each with a common factor,
can be rewritten as the product of the
common factor and the sum of the
remaining factors.
Example: 3(2)  3(5)  3(2  5)
When we write 3(2)  3(5) as 3(2  5) we say
we have "factored out" the 3.
Copyright Scott Storla 2015
Using the distributive property to add or subtract
natural numbers
Simplify using the distributive property.
35
463
3 1  5 1
4(1)  6(1)  3(1)
5(1)  4(1)  1(1)
(1)(4  6  3)
(1)(5  4  1)
1 3  5 
1 8
8 1
8
(1)(13)
5  4 1
(1)(10)
(13)(1)
(10)(1)
13
10
Copyright Scott Storla 2015
The Identity Properties
Property – The Additive Identity
English: 0 is the additive identity. Adding a term of 0 to
an expression doesn’t change the value of the
expression.
Example: 3  0 is equivalent to 3
Property – The Multiplicative Identity
English: 1 is the multiplicative identity. Multiplying an
expression by 1 doesn’t change the value of the
expression.
Example: 1(5) is equivalent to 5.
Copyright Scott Storla 2015
The Inverse properties
Property – The Additive Inverse
English: The expression which when added to the
original gives a sum of 0.
Example: The additive inverse of 8 is 8 .
Property – The Multiplicative Inverse
English: The expression which when multiplied to the
original expression gives a product of 1.
Example: The multiplicative inverse of 2 is 1/2.
Note: 0 does not have a multiplicative inverse.
Copyright Scott Storla 2015
Describe which property is being used to transform
the upper expression to the lower expression. Don’t
simplify the expression.
2  2
0
1
The additive inverse.
3
8
The multiplicative identity.
3
8
1

5

 x  2


5

1  x  2 
The multiplicative inverse.
1  x  2 
x2
The multiplicative identity.
Copyright Scott Storla 2015
Fill in the property which allows each step.
21 
k

12 
____________________________________
The associative property of multiplication
 2 1
 1 2k


____________________________________
The multiplicative inverse
1k
____________________________________
The multiplicative identity
k
Copyright Scott Storla 2015
Justifying the Product to a Power Property

2 3

2
____________________________________
The definition of an exponent
 2 3  2 3 
____________________________________
The associative property of multiplication
2
3 2
 3 
___________________________________
The commutative property of multiplication
(2  2 3 )
 3
____________________________________
The associative property of multiplication
(2  2)

3 3

____________________________________
The definition of an exponent
22
 
3
2
Copyright Scott Storla 2015
Properties
Copyright Scott Storla 2015