Algebra Tiles and NegativesDate: 11/05/2001 at 11:31:46 From: David Herman Subject: Integers How can we use a model (algebra tiles) to demonstrate that a negative times a negative = a positive? I tried to show repeated addition but that doesn't work. For example, -3(-5). How do you represent -3 sets of -5 at a 7th grade level? Date: 11/05/2001 at 12:38:30 From: Doctor Ian Subject: Re: Integers Hi David, The truth is, models only go so far, and it's not clear that there is a way to model the fact that -1 * -1 = +1 that doesn't create more confusion than it clears up. One thing you might try is modeling 'this times that' in the following way: When 'this' is positive, add copies. When 'this' is negative, take away copies. When 'that' is positive, use items. When 'that' is negative, use holes. So, 3 times 4: 1. Start with 4 items * * * * 2. Add copies * * * * * * * * * * * * -3 times 4: 1. Start with 4 items * * * * 2. Take away copies, leaving holes o o o o o o o o o o o o 3 times -4: 1. Start with 4 holes o o o o 2. Add copies o o o o o o o o o o o o -3 times -4 1. Start with 4 holes o o o o 3. Take away holes... by adding copies! * * * * * * * * * * * * But even this isn't very satisfying, because it's not any easier to think of removing holes by adding copies than it is to remember that -1 * -1 = +1. (You could use electrical charges as a model -- removing a negative charge has the same effect as adding a positive one -- but many students find electricity even more confusing than math.) And even if it were satisfying, by trying to reduce this to something that can be modeled with physical items, you would be missing the chance to make a very important point about one of the ways in which mathematics grows. If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way: (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab For example, -2 * -3 = (-1)(2)(-1)(3) = (-1)(-1)(2)(3) = (-1)(-1) * 6 So the real question is, (-1)(-1) = ? and the answer is that the following convention has been adopted: (-1)(-1) = +1 This convention has been adopted for the simple reason that any other convention would cause something to break. For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers: (-1)(1 + -1) = (-1)(1) + (-1)(-1) (-1)(0) = -1 + -1 0 = -2 As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth." Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1. If that seems too abstract, try to think of it in terms of a murder mystery. We've considered all the other possible suspects, and we know that none of them could have committed the murder; and we have no reason for believing that '1' didn't commit the murder; and _someone_ did it; so '1' must be guilty. Also, if you just want to teach your students a good way to remember that a negative times a negative must be positive, they can use this trick: -a(b + -b) = (-a)(b) + (-a)(-b) \________/ \_____/ \______/ 0 neg must be pos For more on the topic, see the Dr. Math FAQ: Negative x Negative = Positive http://mathforum.org/dr.math/faq/faq.negxneg.html I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 11/15/2001 at 11:53:09 From: David Herman Subject: Re: Integers Thank you for your response to my question i thought that I was going to go insane. I also thought that I was the only person having problems explaining why a negative times a negative was a positive. It helps me to rest a little easier knowing that there are a lot of people that have wrestled with this question. |
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