mth 231 section 2.4 multiplication and division of whole numbers
TRANSCRIPT
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
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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
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