The Token Model

Using tokens for addition

In Bela's Building of Fun, the lift attendant is bored. To entertain himself, he keeps a box containing lots of positive (red) and negative (green) tokens. Each time he presses the '+' button, he takes a positive token from the box and puts it in his pocket. Similarly, each time he presses the '–' button, he takes a negative token and puts it in his pocket.

He starts on the floor (Floor 0) with an empty pocket. After one hour, he checks his pocket and finds 5 positive and 3 negative tokens. On which floor is he now?

He must have pressed '+' five times and '–' 3 times and +5+−3= .

So, he is at Floor +2 now.

Here is another way to do the calculation.

Subtract

Lets remove 3 blue-dots from them. Remove blue-dots by clicking and adding negative-dots.

As you see, when you remove three from 5 you are left with 2.

This is represented in math as 5-3=2

A positive token and a negative token cancel each other, because the value of this pair of tokens together is . These two tokens in his pocket meant that he pressed '+' once and '–' once, respectively, and these cancel each other. We say that a positive and a negative token make a 'zero pair'. When you remove all the zero pairs, you are left with two positive tokens,

so +5+−3=+2.

We can perform any such addition using tokens!

Example: Add + 5 and – 8.

Subtract

Lets remove 8 blue-dots from them. Remove blue-dots by clicking and adding negative-dots.

As you see, when you remove eight from 5 you are left with .

From the picture, we see that we can remove five zero pairs, and we are then left with –3. Therefore +5+−8=.

Using tokens for subtraction

We have seen how to perform addition of integers with positive tokens and negative tokens. We can also perform subtraction using tokens!

Example: Let us subtract: (+5) – (+4).

This is easy to do. From 5 positives take away 4 positives to see the result.

+5−+4=

Example: Let us subtract: (–7) – (–5).

Is −7−−5 the same as −7++5?

−7−−5=

Example: Let us subtract: (+5)–(+6).

Put down 5 positives.

But there are not enough tokens to take out 6 positives!

To get around this issue, we can put out an extra zero pair (a positive and a negative), knowing that this does not change the value of the set of tokens.

Now, we can take out 6 positives!

See what is left:

We conclude that +5−+6=.