Using the token interpretation, find the values of: (a) (b) (c) (d)

Question

Homi · Accepted Answer

We use positive tokens (yellow) for positive numbers and negative tokens (red) for negative numbers.

Step 1 — Multiplying $3 \times (-2)$

The first number is 3. It tells us to add tokens 3 times. The second number is -2. It tells us to add 2 negative tokens each time. We start with an empty board. We add 2 red tokens. Then we add 2 more red tokens. Finally, we add 2 more red tokens. We have a total of 6 red tokens.

$3 \times (-2)$ $= \text{add 2 negative tokens, 3 times}$ $= \text{6 negative tokens}$

$\boxed{\mathbf{-6}}$

Diagram 1

Step 2 — Multiplying $(-5) \times (-2)$

The first number is -5. It tells us to remove tokens 5 times. The second number is -2. It tells us to remove 2 negative tokens each time. We need to remove 2 red tokens, 5 times. This means we remove a total of $5 \times 2 = \mathbf{10}$ red tokens. We start with an empty board. To remove tokens, we first add zero pairs. A zero pair has one yellow token and one red token. Its value is zero. We add 10 zero pairs to the board. Now we remove the 10 red tokens. The 10 yellow tokens are left on the board.

$(-5) \times (-2)$ $= \text{remove 2 negative tokens, 5 times}$ $= \text{add 10 zero pairs, then remove 10 negative tokens}$ $= \text{10 positive tokens}$

$\boxed{\mathbf{10}}$

Diagram 2

Step 3 — Multiplying $(-4) \times (-1)$

The first number is -4. It tells us to remove tokens 4 times. The second number is -1. It tells us to remove 1 negative token each time. We need to remove 1 red token, 4 times. This means we remove a total of $4 \times 1 = \mathbf{4}$ red tokens. We start with an empty board. We add 4 zero pairs to the board. Now we remove the 4 red tokens. The 4 yellow tokens are left on the board.

$(-4) \times (-1)$ $= \text{remove 1 negative token, 4 times}$ $= \text{add 4 zero pairs, then remove 4 negative tokens}$ $= \text{4 positive tokens}$

$\boxed{\mathbf{4}}$

Diagram 3

Step 4 — Multiplying $(-7) \times 3$

The first number is -7. It tells us to remove tokens 7 times. The second number is 3. It tells us to remove 3 positive tokens each time. We need to remove 3 yellow tokens, 7 times. This means we remove a total of $7 \times 3 = \mathbf{21}$ yellow tokens. We start with an empty board. We add 21 zero pairs to the board. Now we remove the 21 yellow tokens. The 21 red tokens are left on the board.

$(-7) \times 3$ $= \text{remove 3 positive tokens, 7 times}$ $= \text{add 21 zero pairs, then remove 21 positive tokens}$ $= \text{21 negative tokens}$

$\boxed{\mathbf{-21}}$

Diagram 4

Answer

(a) $3 \times (-2) = \mathbf{-6}$ (b) $(-5) \times (-2) = \mathbf{10}$ (c) $(-4) \times (-1) = \mathbf{4}$ (d) $(-7) \times 3 = \mathbf{-21}$

Question 2

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