The token bag multiplies

The idea

Starting with an empty bag, a positive multiplier means placing the multiplicand's tokens into the bag that many times; a negative multiplier means removing them that many times, after putting in zero pairs.

Tokens into an empty bag

Put 2 green tokens into an empty bag, 4 times. That is 4 × 2. The bag now holds 8 green tokens: 8.

What about 4 × (−2): put 2 red tokens into the empty bag, 4 times?

• 8 green tokens
• 8 red tokens (−8)
• 6 tokens in all

Fill, or empty, the bag

Do what the multiplier says. Take out only works when the bag has those tokens, so you may need zero pairs. Do the first round yourself, then repeat it.

Put in, or take out

A positive multiplier puts the tokens in. A negative multiplier takes them out, and for that the bag needs tokens: put in zero pairs first, then take out.

For (−4) × (−2), the 2 red tokens were taken out of zero pairs, so 8 greens stayed.

Starting with an empty bag, a positive multiplier means placing the multiplicand’s tokens into the bag that many times. A negative multiplier means removing them that many times, after putting in zero pairs.

• Multiplication | What we do | Bag
• 4 × 2 | put in 2 green, 4 times | 8
• 4 × (−2) | put in 2 red, 4 times | −8
• (−4) × 2 | take out 2 green, 4 times | −8
• (−4) × (−2) | take out 2 red, 4 times | 8

Notes

Starting with an empty bag, a positive multiplier means placing the multiplicand’s tokens into the bag that many times. A negative multiplier means removing them that many times, after putting in zero pairs.

Check yourself

Use the red tokens. Work out 4 × (−6).

Answer: -24

4 times 6 red tokens is 24 red tokens, so 4 × (−6) = −24.

How do we work out (−3) × (−5) with tokens?

Why do we put in zero pairs before taking tokens out of an empty bag?

Use tokens to work out (−5) × 3: take out 3 green tokens, 5 times.

Answer: -15

Each round puts in 3 zero pairs and takes out 3 greens, leaving 3 reds. After 5 rounds there are 15 reds: −15.

• Take out 5 red tokens, 3 times (zero pairs go in first) — correct. Yes! The multiplier −3 says take out; the multiplicand −5 is red tokens.
• Put in 5 red tokens, 3 times. That is 3 × (−5). A negative multiplier takes tokens out.
• Take out 5 green tokens, 3 times. The multiplicand is −5, so the tokens are red.
• Zero pairs make the answer bigger. A zero pair is worth 0, so it never changes the value of the bag.
• A zero pair adds nothing to the value, but gives tokens to take out — correct. Yes! The value stays 0, and now there are tokens to remove.
• We can take tokens out of an empty bag without them. An empty bag has nothing to take out. Zero pairs give us tokens without changing the value.