Download Section 2.4

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
1
5 7
5
2
7  1 5  2
7  5  1R 2