students usually are unwilling to accept that $(-a) \times (-b)= a \times b$, or $(-1)\times (-1) =1$ in elementary school. Any good way to teach them such knowledge?
BTW: This $(-1)\times (-1) =1$ can not be proved, it is just extension of ring, so the quesion asking formal proof is in wrong way. My question is not duplicated.
You might like to teach the students that there were hundreds of years of history where people refused to believe in negative numbers, and then after accepting them debated what multiplying one negative number by another would mean.
But the argument is not so hard to make if you have already accepted that $1\times-1=-1$:
$$ \begin{align} 0&=(1+-1)\times(1+-1)\\ &=1\times1+2\times1\times(-1)+(-1)\times(-1)\\ &=1-2+(-1)\times(-1)\\ &=-1+(-1)\times(-1) \end{align} $$
(See "A History of Abstract Algebra" by Israel Kleiner for more on the debates on negative numbers.)