Good page. Helped me a lot with the geometric representation.
Those 3 basic forms
are also the basis for mental arithmetic. Lets take your example:
If you wanted to multiply 38 * 42 you could use that form with y=10.
Answer 16*100 - 4 = 1596 like magic.
For 63 * 57
63*57 = 36 * 100 -9 = 3591
With a little practice and algebra, you can do lots of arithmetic in your head.
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.