![]() ![]() I didn’t see multiplication (or division (even though I know that in some sense they are the same things)) by two negatives (even worse, division by a negative) as a legitimate idea because I never thought of the negative as a symbol meaning opposite. I’m a hardcore questioner of conventional wisdom. ![]() So far the only answer that shows this as incorrect is that the -x*-x=+x is a rule, not a relationship, and that the idea that multiplication is just the repeated number of other numbers is only partially true.Īctually I agree. ![]() (3+-3)*2=0 thus 3*2+(-3*2)=0 (again this is based off of another discovered relationship) 3*2=2*-3 (this only works a discovered relationship overrides the operator and even if it does then (-3)*2=2*(-3) would then force the negative to be applied first before the multiplication(like -1^2=-2 but (-1)^2=2) even if you were to flip the operators you must put a zero at the beginning in order for it to work ) However according to the repeating rule how could you even repeat a nonexistent number by anything? I have, since then, been trying to have someone prove that you can even multiply a negative by a positive. negative numbers only exist within concept and measurement(you can have negative velocity and money)(like the i axis but more relatable). I understand that 2 * -3 = -6 because -3+-3=-6 however how could you even do the opposite to begin with? negative numbers don’t exist in physically. One Day I wondered to myself how does -1 * -1 = 1? in fact how do you multiply -x by anything. Learning new models engenders the kind of rich thinking that math class is supposed to be about learning new mantras engenders the uncritical thinking of the cult-follower. But even if they weren’t – even if the use of mantras led to error-free computation with negatives – I’d still favor the “mental model” approach. Good mental models are more effective than mantras like “two negatives make a positive,” I believe. So the opposite of that is “happy” again.įor adding and subtracting with negatives, I tend to favor a debt model.įor multiplying and dividing with negatives, I think a slightly more abstract approach is necessary – it’s all about the properties of multiplication. What’s the opposite of “the opposite of happy”? What does make sense is a slight variant, less catchy but far more true: “The opposite of the opposite is just the thing itself.” In fact, “two negatives make a positive” doesn’t really make much sense anywhere. In fact, that’s one of my major complaints with “two negatives make a positive”: it is such a swift, over-arching generalization that students wind up applying it in places where it doesn’t make much sense. It’s not even true with negative numbers, where -10 + -30 does NOT equal +40 (although I have seen students claim that it does, citing “two negatives make a positive” as their justification). Rain on your wedding day plus grand larceny on your wedding day does not make for a winning combination, despite what “two negatives make a positive” would suggest. We can all think of many, many cases where two negatives don’t make a positive. ![]() “Two negatives make a positive” is one of those math slogans that drives me crazy, because it is so pithy, so memorable, so easy to apply… while also being so vague and non-mathematical that I’m amazed students find it useful at all. Then, if you listen carefully, you will hear something else: the low rumble of my teeth grinding together with tectonic force. They all know, for example, that 5 – (-2) = 7. My 6th- and 7th-grade students are pretty effective at calculating with negative numbers. ![]()
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