Multiplication that answers why

I know that all of us in my generation learned to multiply the same way.  5 x 7 = 35, Mrs. Cross would say, then she would tell us, "put down the 5 and carry the 3".  We all followed her directions.  Then she would continue "5 x 3 = 15 + 3 = 18".  Then we all put down the 18.  "See class", she would say "5 x 37 is 185".
There were some of us who were thinking, of course, that's the way it has to be.  Another group was thinking, alright, if that's the process I'll follow it.  And then there was the group that was thinking "WHAT IN THE HECK ARE YOU DOING".
Many, if not all of those, last two groups grew up thinking, "I can't do math".   If Mrs Cross had taken a different look at the same problem, maybe we would have more people in the first group.
The problem is 37 x 5 = ?, however that is (30 + 7) x 5 = ?  Now what we need to teach is the distributive law of multiplication over addition.  The 5 is multiplied by the 30 and the 7.
                                5 x 30 = 150 + 5 x 7 = 35 and 150 + 35 = 185
Now we know why we did what we did, and how we got to 185.  What is more important is that the distributive law is very important in Algebra.  So by learning it here, we have a natural transfer to Algebra latter on.  From this point we will move to multiplying a 2–digit number by a 2–digit number.
Lets try 48 x 63 = ?   We can look at it like Mrs. Cross and say, " 3 x 8 = 24, put down the 4 and carry the 2",  and so on.
Or we can look at it as (40 + 8) (60 + 3) where each number in the first parenthesis needs to be multiplied by each in the second.  So this generates:
                       40 x 60  +  40 x 3   +  8 x 60  +  8 x 3 =
                        2400     +   120      +   480     +   24  =  2400 + 600 + 24 = 3024
This also teaches us another important pattern that is easy to follow and will be a major part of Algebra latter on.  It also allows us to work with multiplying with 0's which is an important concept to understand.  Like any process it takes practice, but it also teaches us WHY something is happening.   And by understanding the why the student has more power over the process and more confidence in solving the problem.