Maria, your exposition is excellent, and I strongly agree. In my own experience as an engineer, I am nearly always happy to picture in my head multiplying by the reciprocal, rather than create any mental picture of dividing by a fraction. On reflection I probably think about it as the inverse of a scaling operation: the quotient is what I must have started with if I scaled the divisor by the dividend.
However, I wonder if you would allow me to clarify the following point, as it seems to me that division involving non-natural numbers is important outside pure math. > In practice, nurses, pizza cooks, carpenters and so > on don't "really" divide by fractions - they work with numerators and > denominators separately. This may not be the case when dealing with decimal fractions or percentages, when they might wish to crank a formula, or hit the division key on a calculator. Therefore I think it is worthwhile to give students a chance to develop a mental picture of division by a fraction, even if they choose to forget it later, and rely on the algorithm. I see from your wiki page that you might not agree with me that percentages and decimals are special cases of fractions. However, at least in some curricula, fraction arithmetic is taught first, and the others follow. A couple of examples: (1) if a toy car completes a five foot track in three-quarters of a second, what is its speed? (2) if you want to take 20% sales tax, or value added tax, off an invoice to find the pre-tax price, then you divide by 1.2 (3) devise a formula to convert lap times measured in problem (1) into speeds in miles per hour, or kilometres if you prefer. I think we will find numerous other examples in science, engineering and finance. (I think we could describe some of these calculations as non-integer ratios, or as the need to reverse or invert a multiplication.) I think authentic and concrete concepts of division of all real numbers can be useful to help us to use calculators confidently, and to use, develop, and adapt formulae that include division. (Aspiring electrical technicians will probably also want to extend the concept to complex numbers.) Thanks for your patience. David _______________________________________________ IAEP -- It's An Education Project (not a laptop project!) IAEP@lists.sugarlabs.org http://lists.sugarlabs.org/listinfo/iaep