Chapter 3

Whole NumbersSection 3.3

Multiplication and Division of Whole Numbers

Multiplication

Operationally multiplication is defined in terms of addition.

For any whole numbers a and b where a 0,

a b = b + b + b + … + b

a terms

* Note: if a = 0 we define a b = 0

This concept for multiplication is called repeated-addition. How would we apply a repeated-addition definition to compute the following?

5 9 =

9 + 9 + 9 + 9 + 9 = 45

Since multiplication is just repeated addition the multiplication of whole numbers is also closed. That is to say:

If a and b are whole numbers then a b is a whole number.

Like addition and subtraction there are several different ways to model multiplication using both set (count) and measurement ideas.

5 terms of 9

5 an 9 are called factors of the number 45

45 is the product of 5 and 9

Repeated Sets (count)

What multiplication problems are modeled by the following?

4 3 = 12

Repeated Measures (number line)

What multiplication problems are shown below?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

7 2 = 14

Array (count)

What multiplication problems are represented when dots are in rows and columns?

7 4 = 28

3 4 = 12

2 7 = 14

4 7 = 28

Area (measures)

What is the multiplication problem represented by finding the area of the rectangle below?

12 5 = 60

Counting Trees

What multiplication problem is represented by forming all possible pairs first with the letters in the set {a,b} then with a a symbol in the set {,,}

5

12

a

b

a,

a,

a,

b,

b,

b,

2 3 = 6

Division

When division is first introduced it is considered to be the inverse operation to multiplication like subtraction is the inverse operation to addition. Knowing certain multiplication facts enables you to know a corresponding division fact.

For example knowing that 3 7 = 21 what two division problems do we know?

21 3 = 7 and 21 7 = 3

In the problem 21 3 = 7 the number 21 is called the dividend, 3 is called the divisor and 7 is called the quotient.

Division, like subtraction is not a closed operation for whole numbers. For example, if I want to find 17 5 = 3.4 which is not a whole number. Since division is introduced before fractions and decimals we have a way to characterize the answer to a division problem using only whole numbers. We called it the Division Algorithm. It uses the idea that the answer to a division problem does not have one number as an answer but two.

In the example 17 5 we would say the answer is given by two whole numbers 3 and 2. The number 3 is called the quotient (like before) and the number 2 is called the remainder. They are related as follows: 17 = 5 3 + 2

More generally: dividend = divisor quotient + remainder

Visual Models for Division

The models for division are very similar to multiplication.

Repeated Sets

What division problem is represented below?

20 6 = 3 remainder 2

Repeated Measures (number line)

What division problem is represented below?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1515 6 = 2 remainder 3

Partition of a Set (count)

Dividers are placed between groups of elements of a set to divide it in parts. What problem is represented below?

13 4 = 3 remainder 1

Partition of a Measure

A certain lengths are used to build up another length. What is the division problem for the illustration below?

29 inches

8 inches 8 inches 8 inches29 8 = 3 remainder 5

Array (count)

This is similar to multiplication. What problem is demonstrated below?

19 3 = 6 remainder 1

Division as repeated subtraction

One way to think of division is how many times you can subtract one number from another. What division problem is illustrated below?

87 – 15 = 72

72 – 15 = 57

57 – 15 = 42

42 – 15 = 27

27 – 15 = 12

87 15 = 5 remainder 12

Repeated Multiplication, Exponents and Order of Operations

It is often convenient to multiply the same number by itself a certain number times. We call this operation exponentiation. We write the number of times we multiply as an exponent: 34 = 3 3 3 3 = 81

The order the operations of addition and multiplication are done in will make a difference to the answer you get.

3 + 4 5 = 7 5 = 35 (operations done left to right)

3 + 4 5 = 3 + 20 = 23 (operations done right to left)

In other words we can not just let the reader (or problem solver) interpret how they want 3 + 4 5 evaluated. We use a standard order for operations that states that when grouping or parenthesis are not used multiplication is done before addition. If we want the addition done first we need to use parenthesis.

(3 + 4) 5 = 7 5 = 35 (addition is done first)

3 + 4 5 = 3 + 20 = 23 (multiplication is done first)

From this we get the following order in which numerical expressions are evaluated:

1. parenthesis

2. exponents

3. multiply and divide left to right

4. add and subtract left to right

This concept plays an important role in understanding Algebra. The conventions used to evaluate an expression like 3 + 4x when the value of x is 5 are an abstracted way of thinking of the same problem we had above.