1) There is no minus times minus. There is no verb times verb. There is no multiplier times multiplier.
2) There is no negative times negative. There is no adjective times adjective. There is no multiplicand times multiplicand.
An equation needs a noun or the object of the sentence.
An equation needs an adjective to describe how many objects there are before they are operated upon.
The adjective describing the number of objects is called the multiplicand.
The X or multiplication sign is an abbreviation stating there will now be repeated verbs or actions. In multiplication. these verbs can be EITHER addition or subtraction.
The adverbial that describes the number of repeated verbs or actions is called the multiplier.
So let us take the example of
- -3 x -2 which is NOT negative three times negative two. Neither is it minus 3 times minus two.
The multilpicand comes first as it is anchored to the object and must be stated first so we know what we are talking about ASAP.
The correct multiplication algorithm is mk = the sum of m either added or taken away from zero k times
Therefore, starting from zero, here are the combinations:
a) -3 x -2 is negative three taken away two times
b) +3 x -2 is positive three taken away two times
c) -2 x +4 is negative two added four times
d) +3 x +5 is positive three added five times
In a) above, tell a child the way to take away negative three is the same as adding positive three. To do this two times means, starting from zero, adding three twice gives you positive six. You can also use a model of unit bumps and unit holes, in which putting a positive bump into a negative hole makes a zero. ie additive OR subtractive inverses.
In b) above, starting from ground zero, if you take away 3 unit bumps of dirt from flat ground and do this twice, you end up with six holes or negative six.
In c) you add two holes to ground level zero four times to get 8 holes or negative 8. A child may realise taking away unit bumps of dirt is the same as adding unit holes! (Very easy to grasp as we can all play with a bucket and spade!)
In d) you add three bumps to ground level zero five times to get 15 bumps or positive 15.
Whenever you do not have enough bumps or holes to take them away from ground level zero, just dig some more holes so you keep adding equal numbers of inverse or opposite units as positive/negative or bumps/holes.
So if you only have 2 holes and you need to take away 4 holes, just add 2 more holes AND two more bumps before you take away the 4 holes. What remains? Two bumps! -2 - -4 = +2
At the end of this game, you simply cancel out all the bumps and holes (fill the holes) and observe what remains.
Whatever remains is your answer to the expression and the equation is balanced!
Children playing this game are simply following the instructions contained in the multiplication algorithm.
An algorithm* is "a precisely-defined sequence of rules telling how to produce specified output information from given input information in a finite number of steps" or a recipe for computation."
After playing with bumps and holes (positive presence of dirt and negative presence of dirt) and adding and taking away, then children can later move onto number line models.
Just as < and > do NOT mean less than and more than, i wish so called mathematicians would STOP the minus times minus and negative times negative talk as it simply does not map to the logic of the multiplication algorithm, which at the arithmetic stage, requires both a multiplicand AND a multiplier.
Jonathan Crabtree P.S. I have simply corrected the multiplication algorithm (or enforced logic upon it!) and made integer multiplication into a game. It's fun to play and answers Peter's quest for a simple demonstration.
* Source: Knuth 1974 and Steen 1990 quoted on p. 103 of Adding it Up, Helping Children learn Mathematics, written by the National Research Council, Published by the National Academy Press. 2005 edition