![]() This thinking works well for up and down, and all directions because the opposite of up is in fact down. I would think using this idea would be context dependent and would limit where the idea of negatives would be allowed to be used.įor instance, if I said I wanted two groups of the opposite of two apples what would that mean? Maybe somebody would grab me two oranges, or two glass apples, rather than make my two apples disappear. Is the opposite of a marble a missing marble? Does that idea make sense? Though, perhaps the opposite of food is no food.īut take marble for instance. I do not think the opposite of apples are missing apples. I never thought that the opposite of an apple was a missing apple (is it?). For instance, if I had two apples I saw negatives as literally making those apples disappear. I always read the symbol as literally negative quantities, such as removing objects from existence. 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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