Transcript Section 3.3
Chapter 3
Whole Numbers
Section 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,
ab=b+b+b+…+b
* Note: if a = 0 we
define a b = 0
a terms
This concept for multiplication is called repeated-addition. How would we
apply a repeated-addition definition to compute the following?
5 an 9 are called
factors of the
number 45
5 9 = 9 + 9 + 9 + 9 + 9 = 45
45 is the product of 5 and 9
5 terms of 9
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.
Repeated Sets (count)
What multiplication problems are modeled by the following?
4 3 = 12
3 4 = 12
Repeated Measures (number line)
What multiplication problems are shown below?
7 2 = 14
2 7 = 14
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Array (count)
What multiplication problems are represented when dots are in rows and columns?
7 4 = 28
4 7 = 28
Area (measures)
What is the multiplication problem represented by finding the area of the
rectangle below?
12 5 = 60
5
12
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 {,,}
a,
a
a,
a,
b,
b
b,
b,
23=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?
15 6 = 2 remainder 3
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
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
29 8 = 3 remainder 5
8 inches
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
87 15 = 5 remainder 12
42 – 15 = 27
27 – 15 = 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.