Answer:
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
