Readings for Session 10 – Properties of Addition and Standard Algorithm
Properties for Addition
of Whole Numbers
Commutative Property of
Multiplication:
The
Commutative Property of
Multiplication of Whole Numbers says that the order of the
factors does not change the product.
General Property:
ab =
ba
Numeric Example:
3
× 5 = 15 = 5
× 3
Algebraic Example:
(3x)(4x)
= (4x)(3x)
Associative Property of
Multiplication:
The
Associative Property of
Multiplication of
Whole Numbers says that how the factors are grouped does not
change the product.
General Property: (ab)c
= a(bc)
Numeric Example:
(2
× 6)
× 8
= 12
× 8
=
96
= 2 ×
48
= 2
× (6
×
8)
Algebraic Example:
2
∙ (3x)
= (2
∙ 3)x
=
6x
Notice that in this case, regrouping allows us to simplify the
expression.
Example:
We show how the associative and commutative properties
for multiplication of whole numbers are used to simplify an
algebraic expression.
(3x)(4x)
= 3(x
∙
4)x
Associative Property of Multiplication
=
3(4
∙
x)
x
Commutative Property of Multiplication
=
(3
∙ 4)(x
∙
x)
Associative Property of Multiplication
=
12x2
Identity Property for
Multiplication:
The
Identity Property for
Multiplication of
Whole Numbers says that when a value is multiplied by one
the product is that value; i.e., multiplication by one does not
change the value of a number. One is called the
multiplicative identity.
General Property: 1
∙
a
= a
∙
1
= a
Numeric Example:
1
∙ 5
= 5
∙ 1
= 5
Algebraic Example:
1(4x) = (4x)
∙
1 = 4x
Distributive Properties
of Multiplication:
The
Distributive Property of
Multiplication over Addition
of Whole Numbers (the
Distributive Property of
Multiplication over Subtraction
of Whole Numbers)
shows us how multiplying a value times a sum (difference) may be
broken into the sum (difference) of separate products.
General Property:
a(b
+ c) =
ab +
ac
or
a(b
– c) =
ab –
ac
Numeric Example:
4(145) = 4(100 + 40 + 5)
=
4(100) + 4(40) + 4(5)
=
400
+ 160
+ 20
=
580
Algebraic Example: 5(3x + 9) = 5(3x) + 5(9)
=
(5
∙
3)x + 5(9)
=
15x
+ 45
Note that the Associative Property of Multiplication is used in
the second step.
Example: The distributive property allows us to more easily perform computations mentally.
7(29) = 7(30 – 1)
=
7(30) – 7(1)
=
210 – 7
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