What's minus five times minus two?

OK, finally a bit of mathematics to get our webteeth into...

I caught a bit of "The Weakest Link" on Friday, or more accurately I should say it was on in the background and I wasn't paying much attention, honest. But then I heard Anne Robinson ask the above question (more or less, as far as I can recall now) and I looked up with interest. Y'see, TWL usually goes for much more basic addy/subtracty/timesy stuff, and never with negative numbers. Well, I say more basic... of course the magnitude of the numbers involved is usually much greater, so in a sense there's more of a calculation to be done mentally than with -5 times -2, but all the same, conceptually, this was harder.

So, how did the contestant do? (They were down to the last two, and the lad asked this question was clearly not the brightest of the bunch in the previous round and had only survived by the classic TWL squeeze-play where the strongest link gets voted off and does the walk of shame through gritted teeth.) Well, what fascinated me was that he looked absolutely stunned when he heard the question. Not in the sense of, "I don't know the answer", but more like "this question doesn't make any sense".

It really was quite a look, and I thought to myself, OK, this guy has never, ever encountered multiplication within the integers. (I would think that, being a maths teacher and all.) In a way he had the common sense to think that this is nonsensical, in as far as it met his experience of mathematics.

So here's the thing: how best to get across the concept? Classically, (assuming we're OK with addition and subtraction in the integers), we move from 5 times 2 (OK, got that) to 5 times -2 (OK, still with you, sort of), then we have to dance a little (who wants to mention commutativity?) to get -2 times 5 being the same thing (well, OK, I suppose so) ... and then comes the final piece of the puzzle. Again, classically I suppose we have to say that we get -5 times -2 being negative negative ten, which has to be ten (cue some hand-waving). But though I think I can get a class through this with no apparent trouble, I'm now wondering how many are like this contestant, and saying "eh????" to themselves. And I'm open to better offers... concrete, real-life applications?

OK, OK, you want to know what answer he gave. He said, after much thought, "zero". And in a way I admired him for it. I think he was reasoning, "this doesn't make any sense, so I can't give any answer other than zero". Or maybe he thought it was a trick question.

"No, -5 times -2 is 10" said our Anne, correcting him. And he still looked astonished. The sum is so easy to those of use who know the rules, I suppose, but what about those who don't know they're playing a game?

Please don't think I was laughing at this lad, by the way. This is what teaching is all about.

(Saved for another time: correct usage (minus versus negative; times versus multiply). Things could get nasty!)