Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-3
Page 1 of 3
3.3 Multiplication and Division of Whole Numbers
In Focal Points, “Understanding properties of multiplication and the relationship between
multiplication and division is a part of algebra readiness that develops at grade 3” (p.
15).
Multiplication of Whole Numbers
Repeated-Addition Model
Example: Mary has 3 balloons, Tom has 3 balloons, Gary has 3 balloons, Kathryn
has 3 balloons, and Jerome has 3 balloons. How many balloons do they have all
together?
Mary
Tom
Gary
Kathryn
Jerome
The Array and Area Models
Example: Li has planted 4 rows of 5 tomato plants in her garden. How many
tomato plants were planted in her garden?
Multiplication: For any whole numbers a and n ≠ 0, na = a+a
+
a +...+
a.
n terms
If n = 0, then 0a = 0.
Cartesian-Product Model
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-3
Page 2 of 3
Example: Christy is making ice cream sundaes for a birthday party. If she has
chocolate, vanilla, or strawberry ice cream to choose from, and toppings of hot
fudge, caramel, butterscotch, or mixed fruit, how many different ice cream sundaes
can be made?
Multiplication: For finite sets A and B, if n(A) = a and n(B) = b,
then ab = n(A × B).
The expression ab, is the product of a and b, and a and b are factors.
Properties of Whole-Number Multiplication
Theorem 3-5: Properties of Multiplication of Whole Numbers
Closure property of multiplication of whole numbers: For whole numbers a and b,
ab is a unique whole number.
Commutative property of multiplication of whole numbers: For whole numbers a
and b, ab = ba.
Associative property of multiplication of whole numbers: For whole numbers a, b,
and c, (ab)c = a(bc).
Theorem 3-6: Multiplication Properties of 1 and 0
Identity property of multiplication of whole numbers: There is a unique whole
number 1 such that for any whole number a, a1 = a = 1a.
Zero multiplication property of whole numbers: For any whole number a,
a0 = 0 = 0a.
The Distributive Property of Multiplication over Addition and Subtraction
Theorem 3-7: Distributive Property of Multiplication over Addition for Whole
Numbers
For any whole numbers a, b, and c, a(b + c) = ab + ac.
Theorem 3-8: Distributive Property of Multiplication over Subtraction for
Whole Numbers
For any whole numbers a, b, and c with b > c, a(b – c) = ab – ac.
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-3
Page 3 of 3
Division of Whole Numbers
Set (Partition) Model
Example: Maura has 18 cookies to give to Charlie and two of his friends. How
many should each of the 3 friends receive?
Missing-Factor Model
Example: 3c = 18
Division: For any whole numbers a and b, with b ≠ 0, a ÷ b = c if, and only if, c is
the unique whole number such that bc = a.
Repeated-Subtraction Model
Example: Eighteen cookies are to be packaged in boxes that hold 6 cookies each.
How many boxes are needed?
The Division Algorithm
Given any whole numbers a and b with b ≠ 0, there exist unique whole numbers q
(quotient) and r (remainder) such that a = bq + r with 0 ≤ r < b.
Division of whole numbers is not closed.
Relating Multiplication and Division as Inverse Operations
Division by 0 and 1
Order of Operations
When no parenthesis are present, multiplication and division are preformed in the
order they occur, and then the additions and subtractions are preformed in the order
they occur.