Section 2.3
Multiplication and Division of Whole Numbers
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
Using Models and Sets to Define Multiplication
Multiplication - joining equivalent sets
3 sets with 2 objects in each set
3 x 2 = 6 or 2 + 2 + 2 = 6
Repeated Addition
Multiplication using a rectangular array
3 rows
2 in each row
3x2=6
How are addition, subtraction,
multiplication, and division
connected?
• Subtraction is the inverse of additioin.
• Division is the inverse of multiplication.
• Multiplication is repeated addition.
• Division is repeated subtraction.
•
“Amanda Bean’s Amazing Dream”
Using Models and Sets to Define Multiplication
Multiplication by joining
segments of equal length on a
number line
Number of
segments
being joined
4 x 3 = 12
Length of
one
segment
Using Models and Sets to Define Multiplication
Multiplication using the Area of a Rectangle
width
length
Area model of a polygon
Can be a continuous region
Definition of Multiplication as Repeated Addition
In the multiplication of whole numbers, if there are
m sets with n objects in each set, then the total
number of objects (n + n + n + . . . + n, where n is
used as an addend m times) can be represented
by m x n, where m and n are factors and m x n is
the product.
Example:
5 sets with 3 elements in each set suggest
that 5 x 3 can be interpreted as 3 + 3 + 3 +
3+3
Definition of Multiplication for whole numbers using set
language
The number of elements in the union of a
disjoint equivalent sets, each containing b
elements.
Example:
3 sets with 2 elements in each set: 3
x2
Definition of Cartesian Product
The Cartesian product of two sets A and B, A X B
(read “A cross B”) is the set of all ordered pairs (x,
y) such that x is an element of A and y is an
element of B.
Example:
A = { 1, 2, 3 } and B = { a, b },
A x B = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }
Note that sets A and B can be equal
The Cartesian Product (another example)
In a particular game of chance, a player’s turn
consists of rolling a die twice. What are the possible
results a player could get on a turn? How many
results are there?
Solution: Each die can be modeled by a set of six
numbers: S = {1, 2, 3, 4, 5, 6}. The 36 resulting pairs
of numbers represent the Cartesian product, S x S.
Number on first roll
Number on second roll
1
1
2
3
4
5
6
2
3
4
5
6
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
Problem Solving: Color Combinations for Invitations
Suppose that you are using construction paper to make
invitations for a club function. The construction paper
comes in blue, green, red, and yellow, and you have
gold, silver, or black ink. How many different color
combinations of paper and ink do you have to choose
from?
Use a tree diagram or an array of ordered pairs to match
each color of paper with each color of ink.
Blue
Green
Red
Yellow
Gold
Silver
Black
(B, G)
(B, S)
(B, Bk)
(GR, G)
(GR, S)
(GR, Bk)
(R, G)
(R, S)
(R, Bk)
(Y, G)
(Y, S)
(Y, Bk)
4 x 3 = 12 combinations
Properties of Multiplication of Whole Numbers
Closure property
For whole numbers a and b, a x b is a unique whole number
Identity property
There exists a unique whole number, 1, such that 1 x a = a x 1 =
a for every whole number a. Thus 1 is the multiplicative identity
element.
Commutative property
For whole numbers a and b, a x b = b x a
Associative property
For whole numbers a, b, and c, (a x b) x c = a x (b x c)
Zero property
For each whole number a, a x 0 = 0 x a = 0
Distributive property of multiplication over addition
For whole numbers a, b, and c, a x (b + c) = (a x b) + (a x c)
Models of Division
• Think of a division problem you might
give to a fourth grader.
Models of Division
How many groups (subsets)?
You have a total of 52 cards, with 13 cards in each
stack. How many stacks of 13 cards are there?
This is the Repeated Subtraction or
Measurement Interpretation of Division
Modeling Division (continued)
How many in each group (subset)?
There is a total of 52 cards. Four people want to
play a card game that requires that the whole deck
be dealt. How many cards will each person
receive?
This is the Sharing or
Separating Interpretation of
division
Division as the Inverse of Multiplication
Factor
Factor
Product
9 x 8 = 72
Product Factor Factor
72 ÷ 8 = 9
So the answer to the division equation, 9, is
one of the factors in the related multiplication
equation.
This relationship suggest the following
definition:
Definition of Division
• In the division of whole numbers a
and b. b≠0, a ÷ b = c if and only if c is a
unique whole number such that c x b
= a. In the equation, a ÷ b = c, a is the
dividend, b is the divisor, and c is the
quotient.
Division as Finding the Missing Factor
When asked to find the quotient 36 ÷ 3 =？
Turn it into a multiplication problem:？x 3 = 36
Think of 36 as the product and 3 as one of the factors
Then ask,
What factor multiplied by 3 gives the product 36
？
Why Division by Zero is Undefined
When you look at division as finding the
missing factor it helps to give understanding
why zero cannot be used as a divisor.
3 ÷ 0 = ？No number multiplied by 0 gives 3.
There is no solution!
0 ÷ 0 = ？Any number multiplied by 0 gives 0.
There are infinite solutions!
Thus, in both cases 0 cannot be used as a divisor.
However, 0 ÷ 3 = ？ has the answer 0. 3 x 0 = 0
Does the Closure, Identity, Commutative,
Associative, Zero, and Distributive Properties
hold for Division as they do for Multiplication?
Division does not have the same
properties as multiplication
The End
Section 2.3
Linda Roper