This particular example shows the problem that students do not have either proportional or compensation reasoning. The two go hand in hand. Once they have this type of reasoning, then they will automatically do this transformation, and even consider the relevant algebraic rules to be obvious.
So before teaching the algebraic rules they should have activities that build both proportional and compensation reasoning. Any one who is interested might investigate "Thinking Math" developed at King's College.
John M. Clement Houston, TX
> > With my seventh-grade students, I would do transformations, > but, depending on context, > I would not necessarily expect them to be able to do it. > > One time, in particular, that I recall using it is with the distance > formula. > > For whatever reason, there were many students who knew that r = D/t, > and when we started out with problems that required solving for D > there would immediately be questions, because their recollection of the > formula > "didn't solve for" what they needed. > > Other students could tell them that D = rt, > and I would always do the transformation for them, > to show them that the two formulas are "equivalent" > before I would show them how to solve the problem both ways. > > I always found it fascinating that the students often had no problem > understanding how to solve the equation (once the values were plugged in), > but they were totally confused by the transformation, using the variables. >