Section 2.3

Multiplication and Division of Whole Numbers

Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

Multiplication - joining equivalent sets

3 sets with 2 objects in each set3 x 2 = 6 or 2 + 2 + 2 = 6

Repeated Addition

Multiplication using a rectangular array

3 rows2 in each row

3 x 2 = 6

Using Models and Sets to Define Multiplication

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”

Multiplication by joining segments of equal length on

a number line

4 x 3 = 12

Length of one

segment

Number of segments

being joined

Using Models and Sets to Define Multiplication

Multiplication using the Area of a Rectangle

width

lengthArea model of a polygon

Can be a continuous region

Using Models and Sets to Define Multiplication

Definition of Multiplication as Repeated AdditionIn 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.

5 sets with 3 elements in each set suggest that 5 x 3 can be interpreted

as 3 + 3 + 3 + 3 + 3

Example:

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.

3 sets with 2 elements in each set: 3 x 2

Example:

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 second roll

11 22 33 44 55 66

Number

on first roll

11(1, 1)

(1, 2)

(1, 3)

(1, 4)

(1, 5)

(1, 6)

22(2, 1)

(2, 2)

(2, 3)

(2, 4)

(2, 5)

(2, 6)

33(3, 1)

(3, 2)

(3, 3)

(3, 4)

(3, 5)

(3, 6)

44(4, 1)

(4, 2)

(4, 3)

(4, 4)

(4, 5)

(4, 6)

55(5, 1)

(5, 2)

(5, 3)

(5, 4)

(5, 5)

(5, 6)

66(6, 1)

(6, 2)

(6, 3)

(6, 4)

(6, 5)

(6, 6)

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.

Problem Solving: Color Combinations for Invitations

GoldGold SilverSilver BlackBlackBlueBlue (B, G) (B, S) (B, Bk)

GreenGreen (GR, G) (GR, S) (GR, Bk)

RedRed (R, G) (R, S) (R, Bk)

YellowYellow (Y, G) (Y, S) (Y, Bk)

4 x 3 = 12 combinations

Properties of Multiplication of Whole Numbers

Closure propertyFor whole numbers a and b, a x b is a unique whole number

Identity propertyThere 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 propertyFor whole numbers a and b, a x b = b x a

Associative propertyFor whole numbers a, b, and c, (a x b) x c = a x (b x c)

Zero propertyFor each whole number a, a x 0 = 0 x a = 0

Distributive property of multiplication over additionFor 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

This is the Repeated Subtraction or Measurement Interpretation of

Division

You have a total of 52 cards, with 13 cards in each stack. How many stacks of 13 cards are there?

How many groups (subsets)?

Modeling Division (continued)

This is the Sharing or Separating Interpretation

of division

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?

Division as the Inverse of Multiplication

Factor Factor Product

9 x 8 = 72

÷72 8 = 9

Product Factor Factor

This relationship suggest the following definition:

So the answer to the division equation, 9, is one of the factors in the related multiplication equation.

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

Think of 36 as the product and 3 as one of the factors

What factor multiplied by 3 gives the product 36 ？

When asked to find the quotient 36 ÷ 3 = ？Turn it into a multiplication problem: ？ x 3 = 36

Then ask,

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

Why Division by Zero is Undefined

Division does not have the same properties as multiplication

Does the Closure, Identity, Commutative, Associative, Zero, and Distributive

Properties hold for Division as they do for Multiplication?

The EndSection 2.3

Linda Roper