Step 1)  Correct or avoid statements which are factually incorrect.

For example, many textbooks state the following:

    "In multiplication and division, the product or quotient can not have more significant digits than the minimum number of significant digits used in the calculation."

Problem:  This statement has been shown to be factually incorrect.  Just as doctors follow the creed "first, do no harm", educators should follow the creed "first, do not mislead".
 

Step 2)  Emphasize that rounding rules are simply conventions.

    Rounding rules provide a convenient way to handle the propagation of errors when dealing with numbers that do not have explicitly stated uncertainties.  It may be obvious to teachers that these rules are only approximate, but many students interpret "rule" to mean "something that always works".  These rounding conventions are called "rules" only because they provide a "set of instructions".
 

Step 3)  Decide which rounding rule to teach for multiplication and division.

    I strongly feel that the alternate rule is superior to the standard rule.  No matter which rule is taught, however, it is equally important to following the next (and last) step.
 

Step 4)  Tell students what can happen and what to expect when they use the rounding rules.

Example: Standard Rule for Addition and Subtraction

    The standard rule for addition and subtraction works exceedly well. For the addition and/or subtraction of fewer than 10 numbers, the rule always predicts the minimum number of digits needed to preserve information content. The rule never allows valuable information to be discarded, no matter how many numbers are added and/or subtracted. When 10 or more numbers are added and/or subtracted, the rule can predict too many digits.

Example: Standard Rule for Multiplication and Division

    The application of the standard rule for multiplication and division can yield only three possible results: it can predict the correct number of significant digits, 1 digit too many, or 1 digit too few. It predicts the correct number of digits less than half of the time.  On very rare occasions it predicts 1 digit too many, overstating the precision.  Most of the time it predicts 1 digit too few, causing valuable information to be lost.

Example: Alternate Rule for Multiplication and Division

    The application of the alternate rule for multiplication and division can yield only three possible results: it can predict the correct number of significant digits, 1 digit too many, or 2 digits too many.  It predicts the correct number of digits over half of the time.  It predicts 1 digit too many less than half of the time, overstating the precision.  On very rare occasions it can predict 2 digits too many, overstating the precision.

    So, which rounding rule is best for multiplication and division?  The standard rule is less accurate and allows valuable information to be discarded over half of the time.  The alternate rule is more accurate and perfectly safe, but can overstate precision by having 1 or (on very rare occasions) 2 un-needed digits.  Which is the lesser evil, discarding valuable information most of the time or overstating precision less than half of the time?

    Another compelling argument for the alternate rule comes, ironically, from a detailed study of the standard rule for addition and subtraction. By relating the operations of addition and multiplication, the standard rule for addition (which everyone accepts) can be related to the alternate rule for multiplication. The fact that the standard addition rule preserves information content directly implies that the alternate rule for multiplication must preserve information content as well. Thus the standard rule for addition & subtraction and the alternate rule for multiplication & division form a self-consistent set of rounding rules that serve to preserve information content.

    While the alternate rule can allow precision to be overstated, it still achieves the original purpose - preventing students from grossly overstating the result of a calculation by writing down the full calculator result.  Because the alternate rule achieves its purpose, is more accurate that the standard rule, is consistent with the accepted rounding rule for addition & subtraction, and prevents the loss of valuable information, it is my personal choice.