I usually use geometric block to do multiplication. $4\times 5$ is the number of $1\times 1$ blocks inside a rectangle with sides $4$ and $5$ that is $20$ $1\times 1$ blocks. in the case of $0.5\times 0.5$ we have a square with side $0.5$ and we want to know the number of $1\times 1$ blocks inside that.
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like this: As...
abstract algebra - Why is negative times negative = positive ...
Why is $\infty\times 0$ indeterminate? - Mathematics Stack Exchange