MTH 231 Section 2.4 Multiplication and Division of Whole Numbers

Multiplication Some of the conceptual models mentioned in the section: 1.Multiplication as repeated addition 2.Array model 3.Rectangular area model 4.Skip-count model

Repeated Addition

Array

Rectangular Area

Skip-Count

Multiple Models

Properties of Whole-Number Multiplication Like addition, multiplication is: 1.Closed 2.Associative 3.Commutative However, there are three new properties we need to discuss.

4. Multiplicative Identity Property There is a “special” element in the whole numbers. This element has the property that any whole number multiplied by it gives back the number you started with: a x 1 = a and 1 x a = a for all whole numbers a

5. Multiplication-by-Zero Property Any whole number multiplied by 0 gives a result of 0 b x 0 = 0 and 0 x b = 0 for all whole numbers b

6. Distributive Property If a, b, and c are any three whole numbers: a x (b + c) = (a x b) + (a x c) and (a + b) x c = (a x c) + (b x c) The official title of the property, “distributive property of multiplication over addition”, is reflected in the fact that both operations are present.

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Division of Whole Numbers Division is inherently more difficult to model than multiplication, yet there are fewer models: 1.Repeated-subtraction 2.Partition 3.Missing-factor

Repeated-Subtraction In this model, elements in a set are subtracted away in groups of a specified size. This model is also called division by grouping.

Partition In this model, elements in a set are separated into groups of a specified size.

Missing Factor In this model, division is recognized as the inverse of multiplication.

Division By Zero Consider the following questions: 1.John has 12 pieces of candy. He wants to give each of his friends 0 pieces. How many friends will receive 0 pieces of candy? (repeated-subtraction) 2.John has 12 pieces of candy. He wants to divide them in groups of 0 pieces. How many groups of 0 pieces can John make? (partition) 3.Find a whole number c such that 0 x c = 12. (missing-factor)

Division With Remainders Sticking with the missing-factor model, we now consider those situations where a whole number c cannot be found: Find a whole number c such that 5 x c = 7. The other models further support the idea that, in some cases, a remainder is needed to extend the division operation.

The Division Algorithm Let a and b be whole numbers with b not equal to zero (Why?). Then there exist whole numbers q and r such that a = q x b + r, with 0 < r < b. a is called the dividend. b is called the divisor. q is called the quotient. r is called the remainder.

7 Divided By 5, 3 Ways