Jean in CA Posted September 2, 2010 Share Posted September 2, 2010 I know the algorithm is (-a) x (-b)= ab; in other words, multiplying 2 negative numbers equals the positive product of the 2 numbers, but why? Can someone explain this conceptually, and offer a few "real life" examples of this? Thanks! Quote Link to comment Share on other sites More sharing options...

Mom22ns Posted September 2, 2010 Share Posted September 2, 2010 MUS explains a negative as opposite. So if you have one number and put a negative sign in front of it, you have the opposite of that number (on a number line for example) which is -number. If you have two negatives, then it is opposite twice. The first opposite is negative, the next one turns it back to positive. I'm not sure I explained that well enough to help. Quote Link to comment Share on other sites More sharing options...

Caitilin Posted September 2, 2010 Share Posted September 2, 2010 Imagine it on that "evil" number line. :) -3 once is still -3. -3 x 2, or -3 twice, is -6. But when you do -3 x -2 you are doing the opposite of -3 x 2, or the opposite of -6, so what you get is +6 instead. Does that make sense? Quote Link to comment Share on other sites More sharing options...

Dana Posted September 2, 2010 Share Posted September 2, 2010 You can also see it as a pattern: (-3)(3)=-9 (-3)(2)=-6 (-3)(1)=-3 (-3)(0)=0 If we continue the pattern, we continue to decrease our second number and our product continues to increase by three. So... (-3)(-1)=3 (-3)(-2)=6 etc. Quote Link to comment Share on other sites More sharing options...

silver Posted September 2, 2010 Share Posted September 2, 2010 I'm guessing this would be too advanced to use to show your children, but maybe it will help you? We know that: 1 - 1 = 0 Rewrite (it makes a future step easier to see)=> 1 + (-1)*1 = 0 add (-1)*(-1) to both sides => 1+ (-1)*1 + (-1)*(-1) = (-1)*(-1) So now we have: 1 + (-1)*1 + (-1)*(-1) = (-1)*(-1) factor the (-1) out => 1 + (-1)*(1-1) = (-1)*(-1) 1-1 is 0 so => 1 + (-1)*0 = (-1)*(-1) 0 times any number is 0 => 1 = (-1)*(-1) So we have that (-1)*(-1) is (+1). Quote Link to comment Share on other sites More sharing options...

tarana Posted September 3, 2010 Share Posted September 3, 2010 I'm guessing this would be too advanced to use to show your children, but maybe it will help you? We know that: 1 - 1 = 0 Rewrite (it makes a future step easier to see)=> 1 + (-1)*1 = 0 add (-1)*(-1) to both sides => 1+ (-1)*1 + (-1)*(-1) = (-1)*(-1) So now we have: 1 + (-1)*1 + (-1)*(-1) = (-1)*(-1) factor the (-1) out => 1 + (-1)*(1-1) = (-1)*(-1) 1-1 is 0 so => 1 + (-1)*0 = (-1)*(-1) 0 times any number is 0 => 1 = (-1)*(-1) So we have that (-1)*(-1) is (+1). Great Explanation! Quote Link to comment Share on other sites More sharing options...

Jean in CA Posted September 3, 2010 Author Share Posted September 3, 2010 Thanks for all the responses. On paper, I can understand how to manipulate the numbers and why the product of 2 negative integers is positive. However, it is in the realm of real life that I am trying to figure out when we need to multiply 2 negative numbers together! So for example, in subtraction, a negative number can represent loss, such as withdrawals from a bank account. So that if I withdraw one hundred dollars, that can be represented as -100. Let's say I withdraw $100 every month for 3 months. This can be represented mathematically as 3 x (-100) or a net withdrawal of -300. But then, how do you explain something like -3 X (-100)? If we continue the "month" analogy... could you say that moving forward in time is postive and moving backward is negative, so that 3 months ago is -3. The next part is where I get stuck. What kind of story problem would result in -3 X (-100)? Would you say how much more money did I have at the beginning of 3 months? But then, I would always think in the positive.... just 3X100=300. Why do we even need to multiply 2 negative integers? Ack! Quote Link to comment Share on other sites More sharing options...

Gooblink Posted September 3, 2010 Share Posted September 3, 2010 Here's an answer to a similar question: http://mathforum.org/library/drmath/view/72312.html :) Quote Link to comment Share on other sites More sharing options...

silver Posted September 3, 2010 Share Posted September 3, 2010 The problem with coming up with word problems that result in multiplying two negative numbers is that most people wouldn't solve a problem that way. With the example you started to formulate about bank withdrawals, if you asked: You withdraw $100 a month and have been doing so for a year. You currently have $900 in your bank account, how much did you have three months ago? Well, most people would not write out "$900+(-3 months)*(-$100/month)" to solve the problem. Most people would multiply $100 by 3 and add it to $900. Some examples that aren't word problems that might result in multiplying negative numbers would be graphing an equation like y=-2x+4. But the sort of problem that would require coming up with this formula and using it probably belongs in an algebra course. Quote Link to comment Share on other sites More sharing options...

Jean in CA Posted September 3, 2010 Author Share Posted September 3, 2010 Silver, I think I'm finally understanding... multiplying 2 negative integers might not correspond to a real-world model that makes sense (because most people naturally think in the positive). I have not taught my kids algebra yet (soon!) but you've helped me to make sense of my dilemma. Thanks! Quote Link to comment Share on other sites More sharing options...

oakmom Posted September 3, 2010 Share Posted September 3, 2010 I like the explanation in the Pre-Algebra for Visual Learners demo video. Here's a link: http://www.mugginsmath.com/store.asp The program is near the bottom. Quote Link to comment Share on other sites More sharing options...

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