Given integers $(-a)$ and $(-b)$ why is the product positive? I was given the assignment to find a way to explain why the product of two negative numbers is positive. If a middle school or high school student were to ask why is it true, I must show some logic explanation as to how we can get a positive number.   
$$(-a)\times(-b)=(a)\times(b)$$ 
I wanted to talk about a video of a guy running backwards but the video itself playing backwards. When I hit play, it actually shows the guy running forward. But I can not find a logic way to explain it. 
Any ideas or a way to explain this to a middle school student?
I have to convince the student that this is a fact.
 A: Proof of $\quad -(-x)=x$
$\forall x\; \exists y\quad x+y=y+x=0$   (From Z.F.C.)
Then we can say, $\quad x=-y\quad$  and likewise $\quad y=-x$
$-x=y\quad $  then ,$-(-x)=-(y)=-y\quad$ and we know $-y=x$
$-(-x)=-(y)=-y=x\quad \Box\quad$ (Note:This $y$ is unique.) 
$\quad$
$\forall a$, $-a=(-1)a$,
Because ,$\quad 0=0a=(1+(-1))a=1a+(-1)a=a+(-1)a \quad \Box \quad$(this $(-1)a$ is unique.)
Then if we write $a=-1$;
$(-1)(-1)=-(-1)=1$
Then;
$(-a)(-b)=(-1)a(-1)b=(-1)(-1)ab=ab\quad\Box$
A: Strategy: rewrite $ab$ several times using a few laws of arithmetic (each in two variants, equivalent because $xy=yx$):
(1a) $x=x+0$, (1b) $0+x=x$; (2a) $0=x0$, (2b) $0x=0$; (3a) $x\left( y+z\right)=xy+xz$, (3b) $\left( x+y\right)z=xz+yz$.
We'll also repeatedly recall $u-v$ is an abbreviation for $u+\left( -v\right)$.
Now for the rewrite: $$ab=ab+\left( -a\right)\left( b-b\right)=\left(a -a\right)b + \left( -a\right)\left( -b\right)=\left( -a\right)\left( -b\right).$$The first $=$ sign uses (1) and (2a); similarly, the third $=$ sign uses (1b) and (2b). The second $=$ sign is more complicated: it expands a product so we have three terms, and factorises the leftmost two of them, viz. (3a) and (3b).
