The product of integers question about why....

Hello all,

I saw a tweet this weekend and I wondered how many of you have encountered this situation.  

The product of a (+) and (-) integer is negative: 3(-2) = 3 (-2)s = -2,-2,-2, = -6. None of the explanations of (-)(-) = (+) that I know are convincing to me in this intuitive way. How are you convinced that (-)(-) = (+)?




I'm easy to convince.   Taking away a bad thing is like giving me a good thing.   I learned to draw two extra vertical lines in there and then finish with a curvy smiley face mouth that looks like I'm really happy and drunk (that's what appealed to me in seventh grade, anyway ;)) …. 
    If I've been using 2 cups of bird seed a day for the past seven days.... how would I figure out how much bird seed I had a week ago?   Would I have more, or less?   

Brooke, I just watched an instructional video that said when multiplying two negative numbers, you factor out a -1 from each and they cancel each other leaving you with a postitive number.  See here:

I have never seen it explained/demonstrated this way.



What would happen in the example if it was 2(-2)?

2(-2) = 2(-2)s = (-2)(-2) = +4.  Hmm.

This issue is they are changing the 3(-2) = 3(-2)s but it should be equal to (-2) + (-2) + (-2). Multiplication is repeated addition. To correct my example: 2(-2) = 2(-2)s = (-2) + (-2) = -4. Hope this might shed some light.


Another way of explaining this is if someone borrows money that person owes money.  Debt is a negative number.  Then the person who lent the money says, I cancel your debt.  It is like the borrower got extra money.  A way of explaining this to remember is:

If something good happens to someone good that is good

If something bad happens to someone good that is bad

If something good happens to someone bad that is bad

If something bad happens to someone bad that is good.

Though I have had some students object to the final statement, it helps them remember how it works. 

Some of my students analogized it to "reverse!"  -- and I say "yes, the negative sign means opposite..."  and ... even when it's being eccentric (negative exponents anyone?)  it means "opposite."   And .... if the opposite of yes is no... the opposite of no is yes... so if you take the opposite twice it's like doing a 180 degree turn twice... you're back facing where you started. 
    Honestly, while negative times negative confuses *some* people, I have many more who do think -2 + -9 is positive because they've overgeneralized "two negatives make a positive."   It sometimes helps to say "Two wrongs don't make a right!"   and to use number lines and the easier connections to concrete ideas that make that make sense.  THose folks may be the ones who are perfectly comfortable with math being some arbitrary rules that don't necessarily make *sense.*