A while ago we were discussing my failure to explain why the rule for the
number of decimals in a product works.  I found an approach that worked.
It used the commutative and associative properties they had studied
earlier and the observation that multiplying by 10 moves the decimal one
place left which they had learned previously, plus the new fact that 
multiplying by 0.1 moves the decimal one place right, which is easily
established.
Start by multiplying the original problem by 1, once for each decimal
point in the original problem.  So
     .2 x .03  becomes .2 x .03 x 1 x 1 x 1
Change each of the added ones into (10 x 0.1).  The example becomes
   .2 x .03 x (1x0.1) x (1x0.1) x (1x0.1)
Associate and communicate to turn the original numbers into whole numbers.
   2 x 3 x 0.1 x 0.1 x 0.1
multiply the whole numbers
    6 x 0.1 x 0.1 x 0.1
The multiply by the 0.1's, becomes
        .006
The result is always to multiply the decimals as integers ignoring the
decimal point, then move the decimal point once left for each decimal
point in the original problem.
It was clear enough we settled it verbally in the car.  MUCH more successful
than my previous attempts using subdivided rectangles.
 
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