
Division of directed numbers  Mixed
operations of directed numbers 
 Quick quiz 
Division
of directed numbers
The
operation of division is often explained by referring to its inverse
operation, multiplication. In our discussion of Multiplication
of negative numbers we saw that,
3
x (4) = 12.
We
can rearrange this example into two division problems,
(12)
÷ 3 = (4) or (12) ÷ (4) = 3.
Because
division is the inverse operation of multiplication, the same
rules of sign that apply for multiplication must also apply for
division. 
Some
division problems are represented very easily by the annihilation
model, using partition or quotition meanings for division.
Example
1: (6) ÷ (2) = 3
We
can think of this division problem as a quotition division problem.
In quotition division we know the number in each group and we want
to find out how many groups there are.
6, how many groups of 2



6,
how many groups of 2? 3 groups of 2

We
chose to think of this problem as quotition division because we can
talk about '6 how many groups of 2'. It does not make sense to say
'6 shared into 2 equal groups' which is partition division.
Example
2: (12) ÷ 4 = (3)
This
division problem can be represented as partition division .We know
the number of groups and we want to know how many are in each group.
12 shared into 4 equal groups




12
divided into 4 equal groups, gives 3 in each group

Example
3: 18 ÷ (9) = (2)
How
can we think of this division problem?
Thinking

18,
how many groups of 9?
There are 2 groups of 9 
By
expressing the problem as quotition division, we are saying that
there are 2 groups of 9. This is no help for teaching or understanding! 
18
shared into 9 equal groups?
There are 2 in each of the 9 groups 
By
expressing the problem as partition division, we are saying that
there are 9 groups of 2. This is no help for teaching or understanding! 
We
cannot picture this problem as either quotition or partition division.
We need to rely on what we know about multiplication and the fact
that it is the inverse operation of division.
Using
the annihilation model in Multiplication
of negative numbers, we saw that a negative number multiplied
by a positive number or a positive number multiplied by a negative
number gives a negative number.
By
looking at number properties we also saw that a negative number
multiplied by a negative number gives a positive result.

Looking
at Example 3, 18
÷ (9) again,
18
is equal to 9 multiplied by the answer to 18 ÷ (9). Since
18 is positive, the answer must be negative. It is in fact, 2.
Mixed
operations of directed numbers
Example
4: movie, 4 x 5 + 18 ÷ 2
Example
4 movie, 5  (1  7)/2
Quick quiz
1. 
Using
the annihilation model solve the following problems: 

a)
16 ÷ 4 

b)
40 ÷ 8 

c)
39 ÷ 13 

d)
12 ÷ 6 


2. 
Without
using a model, solve the following problems: 

a)
5(18 ÷ 9) 

b)
5 + 4(12/3) 

c)
(6 x 5) + (49 ÷
7) 

d)
(128)/(4) + (144)/(3) 


3. 
A
company advises its shareholders that it had an overall loss
of $15 million for the first quarter of the financial year,
a profit of $2 million for the second quarter, a loss of $3
million for the third quarter and a profit of $4 million for
the last quarter. What was the average quarterly profit (or
loss)?

To
view the quiz answers, click here.
'Talking
through' questions
The
'talking through' questions and answers below have been provided to
enable you to see how an 'expert' might tackle these questions. The
annihilation model has been used in the explanations where appropriate.
1.
Solve ( 48) ÷ 4 ÷ ( 3) 
We
will start with ( 48) ÷ 4 which can be thought of
as 48 negative chips shared into 4 equal groups. There will
be 12 negative chips in each group. Hence, ( 48) ÷
4 =  12
Now
we have ( 12) ÷ ( 3) which can be thought of
as 12 negative chips, how many groups of 3 negative chips? There
will be 4 such groups.
Hence
( 12) ÷ ( 3) = 4

2.
Solve [( 90) ÷ ( 9)] ÷ [( 60) ÷
30] 
Brackets
must be done first
We
will start with ( 90) ÷ ( 9). We can think
of this as ( 90) how many groups of ( 9)? The answer
is 10 such groups. So ( 90) ÷ ( 9) = 10
Next
we will do ( 60) ÷ 30
This
can be thought of as 60 negative chips shared into 30 equal
groups. The answer is 2 negative chips per group.
Hence,
( 60) ÷ 30 = ( 2)
The
problem now becomes 10 ÷ ( 2). We know that (
5) x ( 2) = 10, so 10 ÷ ( 2) = ( 5)
Hence
[( 90) ÷ ( 9)] ÷ [( 60) ÷
30] =  5

