MTH 231

Section 2.4Multiplication and Division of Whole

Numbers

Multiplication

• Some of the conceptual models mentioned in the section:

1.Multiplication as repeated addition2.Array model3.Rectangular area model4.Skip-count model

Repeated Addition

Array

Rectangular Area

Skip-Count

Multiple Models

Properties of Whole-Number Multiplication

• Like addition, multiplication is:1.Closed2.Associative3.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.

Images

More Images

Division of Whole Numbers

• Division is inherently more difficult to model than multiplication, yet there are fewer models:

1.Repeated-subtraction2.Partition3.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

2

5

175

2157

2517

R