
Using the annihilation model  Addition
and Subtraction of directed numbers 
 Quick quiz 
Using the annihilation model
Subtraction
is modelled as take away using the annihilation model.
We
have already discussed some ideas relating to subtraction in the earlier
sections. The
difference between the negative sign in front of a number and the
subtraction sign was discussed in Key
Ideas.
Using
the annihilation model in Addition of Negative
Numbers we saw that addition of a negative number is the same
as subtraction of the positive value of that number. So 4  7 is equivalent
to 4 + (7).
Example
1: (4)  (2)
We
can represent (4)  (2) by starting with 4 negative chips and taking
away 2 negative chips. This leaves 2 negative chips.
Here
we have taken away 2 negative chips.
Using the model again we can show that we could have got the same
result by adding 2 positive chips.
Example
2: Using the model to show (4) (2) = (4) + (+2)
So
we can see that 4  (2) is equivalent to 4 + (+2).
Taking away a negative number is the same as adding the positive value
of the number
Example
3: (+3)  (+5)

+3

(+5)

= 
? 
I
want to take away +5 from +3, but I haven't got enough positive
chips to do this. One way I could approach the problem is to
represent +3 in another way, as I have done below. +3 can be
represented by many arrangements of positive and negative chips
so long as when all positives and negatives annihilate one another
there are 3 positive chips left.
To
represent +3 I have added another 5 positive chips and 5 negative
chips. I now have 8 positive chips and 5 negative chips (which
still represents +3). I now have enough positive chips to take
away 5.




+3

3
 (+ 5)

= 2

Instead of taking away 5 positive chips
we could have added 5 negative chips
to get the same result. So we can see again that taking away +5 is
the same as adding 5. We call this method of subtraction adding
opposites.
Example
4: Using the model to show (+3)  (+5)
is also equal to (2)
Subtraction
of Integers  Adding opposites
If
a and b are any integers, then a  b = a + (b)

These
explanations may seem unnecessarily complicated for the fairly straightforward
examples we have used here; however the point is to illustrate how
and why the rules which we apply in the addition and subtraction of
directed numbers work. In order to be able to teach these concepts
we need to understand the principles underlying the general rules
which are often stated with little justification in school textbooks.
Addition
and Subtraction of directed numbers
Example
5: movie  using the annihilation model, 12  (4)
Example
6: movie  using the number line model, 5  3 + 7
Quick quiz
1. 
Using
the annihilation model, or another model if you prefer, find answers
to the following examples: 

a)
6  (+3) 

b)
7  4 

c)(8)
 (5) 

d)
8 + 5 


2. 
Without
using a model find answers to the following examples: 

a)
4  16 

b)
10  19 

c)
22  13 

d)
7  (14) 

e)
9  (5) + 15  7  (4) 

f)
12  (4) + (3)  9 + (11)  5 

g)
6 + 8  (9) + 7 (10)


h)
5  (17)  25 + (13) + 26 


3. 
Lance
wants to check the final balance of all the transactions listed
on his monthly savings account statement. What is his final
balance?

1235.99

583.00

299.00


25.75


60.00

784.21

201.00




4. 
The
local shopping centre has just been expanded and now has a ground
level, 5 levels above this and a car park which is on levels
B1, B2 and B3 below ground level.
a)
From level B2, how many levels do you travel to get to level
B1?
b)
From level, B2, how many levels do you travel to get to ground
level?
c)
If I am shopping on level 5 of the shopping centre and I have
parked on level B3, how many levels do I need to travel to get
to my car?
d)
If I park on level B2, how many levels do I need to travel to
get to level 2?



5. 
Write
each of the problems above as an addition or subtraction of
negative numbers. The first is done for you.
a)
(2)  (1) = ? or (2) + ? = (1)
(2)  (1) = 1 or (2) + 1 = (1)





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.
Find the answer to 9  ( 4)  8 
This
reads, positive nine take away negative four take away positive
eight.
Using
the annihilation method, we have 9 positive chips and we wish
to take away 4 negative chips. However, we do not have any negative
chips to take away, so we must add 4 negative chips and 4 positive
chips to 9. This gives 13 positive chips and 4 negative chips.
When we take four negative chips away from this we are left
with 13 positive chips. 13 positive chips take away 8 positive
chips leaves us with 5 positive chips.
So,
9  ( 4)  8 = 13  8 = 5

