The Token Model

Tokens and Zero Pairs

Imagine we have two types of tokens:

• Positive Token (+): Represents $+1$ (e.g., Green/Red).
• Negative Token (-): Represents $-1$ (e.g., Red/Blue).

The Zero Pair

One positive token and one negative token cancel each other out. Their sum is zero.

Addition with Tokens

Task: Add $(+3) + (-2)$

1. Place 3 positive tokens: (+) (+) (+)
2. Place 2 negative tokens: (-) (-)
3. Group them into zero pairs.
   • Pair 1: (+) and (-) $\rightarrow$ Cancel.
   • Pair 2: (+) and (-) $\rightarrow$ Cancel.
4. What is left? One (+).
   • Result: $+1$.

Subtraction with Tokens

Task: Subtract $(+3) - (-2)$

1. Start with 3 positive tokens: (+) (+) (+)
2. We need to “take away” 2 negative tokens.
3. Problem: We don’t have any negative tokens to take away!
4. Solution: Add Zero Pairs. Adding zero pairs doesn’t change the value.
   • Add 2 zero pairs: (+) (-) and (+) (-).
   • Now we have: (+) (+) (+) and (+) (-) (+) (-).
5. Now, take away the 2 negative tokens (-) (-).
6. What is left?
   • Original (+) (+) (+) PLUS the two positives from the zero pairs (+) (+).
   • Total: 5 Positives.
   • Result: $+5$.

This proves why $3 - (-2) = 3 + 2 = 5$.