I previously tackled the difference between the ” – ” symbol used to represent a negative and a subtraction problem. This post gets into multliplication through the context of buying a Wendy’s frosty.
One comment to preface what is presented here. Negative numbers are abstract and challenging for many students. Multiplying by negatives is even more so. The approach presented here for multiplying gets a little complex, which is inherent to the topic. In other words, this is an involved process as “there is no royal road to geometry.” What I present here is a path I develop over time, as seen in the sequence below.
First, review of a couple building blocks. Multiplication can be represented at groups of items. 2 x 3 can be represented as 2 groups of 3 $1bills, e.g., you bought 2 Frosties for $3 each.
Negative in terms of money can mean you owe money. Hence, -3 means you owe $3, e.g., you order a frosty and owe $3.
If you change your mind and cancel the frosty, the $3 you owe is cancelled and you get your money back. +$3.
If you order 2 frosties, you owe $3 and another $3, which is -3 + -3. (You owe $6 or -6.)
Cancel those two frosties and you get your money back. -(-6) is now +6
2 x -3 means you means you have 2 negative 3s – repeated addition, or -3 + -3.
If you ordered the 2 frosties and owe $6 then cancel, you get your $6 back or +6.
In mathy terms, cancelling the 2 x -3 is written as -(2 x -3) or -2 x -3.
The – for the -2 can be held out front to focus on 2 orders of frosties or 2 x -3. That was covered previously. Then the extra negative cancels that order so you get your money back. And so you have multiplying two negatives!
All images were generated on this jamboard.