When we teach that the multiplication of two negatives always produces a positive, we need to prove to pupils that this is so always and we don’t really want them taking our word for it without a good reason to back up this counter intuitive statement. Logic is a perfect ally to us here. After all if the outcome is either positive or negative (okay it could also be zero but we have this type of result all sewn up as one of the numbers must be equal to zero in the first place) then if we can show that it is not negative then it has to be positive.
Let us show the result of a simple multiplication on a number grid. for example we can look at 18 x 19.
The grid has allowed us to brake the problem into four separate multiplications that have simply to be added at the end to arrive at the result. However this multiplication can be represented in many more ways than 18 x 19 to get the same result. For example not only is 18 equal to 10 + 8 as in the grid above it is equally legal to call it 20 – 2 and the same rule can be applied to 19 making this 20 – 1. So let us put the same problem using these numbers into a similar grid.
At this point we can make sure that pupils are still happy that 18 x 19 = (20 – 2) x (20 -1) before we go any further. If they are confident that this is so then they will not argue that the result of 342 will be the same for both. In the second version however we are left with an unknown answer, but all is not lost. If we combine all the results we do know that are in the grid we get 400 – 40 = 360, then 360 – 20 gives us 340. We also know that the unknown is a number which we must combine with 340 to give us the result 342, in other words +2. And where has the result +2 come from? It has come from (-2) x (-1) and hence the multiplication of two negatives has to be positive. Finally in this section here is another example of a different multiplication.
An algebraic alternative.
Proof that (-a)(-b) = ab