Multiplying with tokens

The token model handled addition and subtraction. Can it give meaning to MULTIPLICATION too, including multiplication by a negative number? It can, with one rule to remember: for every new operation, start with an EMPTY bag.

Begin with what you know. Put 2 positive tokens into the empty bag, and do that 4 times. How many positives are in the bag? Eight. So 4 × 2 = 8, and we can read the multiplication as 'place the multiplicand into the bag, multiplier times'.

Name the parts while they are in front of you: in 4 × 2, the 4 is the MULTIPLIER, the 2 is the MULTIPLICAND, and 8 is the PRODUCT.

Now the new case. What could 4 × (−2) mean? Read it the same way: place 2 NEGATIVES into an empty bag, 4 times.

Negatives are red tokens, so place 2 red tokens into the bag, 4 times. There are now 8 red tokens in the bag, which means −8.

So 4 × (−2) = −8 — and nothing new had to be assumed. The same reading of 'multiplier times, place the multiplicand' gave the answer.

Use it on more. 4 × (−6) means placing 6 negatives into the bag 4 times, giving 24 negatives: −24. And 9 × (−7) means placing 7 negatives 9 times, giving −63.