>> I only have a few moments before I need to leave to tutor >> someone, but here's a somewhat silly one off the top of >> my head: >> >> Determine the exact value of a^2 - b^2 if >> >> a = 7.0241132301442003123012230341430201 >> b = 6.0241132301442003123012230341430201 >> >> Calculators are allowed.
> I get > 65241132301442003123012230341430201/5000000000000000000000000000000000. > (Yes---it's in lowest terms.) > > Though I would accept your decimal answer---which you arrived at the > semi-hard way. (You even made the semi-hard way harder than it needed to > be: 2 a - 1 = a + (a - 1) = a + b.) > > It's easier to write a^2 - b^2 = (a - b)(a + b). In this case, a - b = 1, > so the answer is simply the sum of the two numbers given. Your restrictions > are therefore unnecessary: Kids should be able to do multi-digit addition > where it's sometimes necessary to carry a one. Even in their heads, given > that they can write down the digits of the answer as they produce them.
About 3 or 4 minutes after I sent this post in, as I was driving out of the parking lot where I work, I realized that one could correctly answer the question asked by simply writing down a^2 - b^2 with the values of a and b filled in. Better instructions are to say to give the exact value as a single decmial numerial, or something like this.
For what it's worth, I didn't have Robert Hansen's method in mind (or even consider the possibility of that approach). For me, seeing a^2 - b^2 immediately brings to my mind (a - b)(a + b), so I just made it so a - b was as easy as possible (without being 0) and a + b doable with minimal effort (for messy looking values of a and b).
I don't like these kinds of questions for math tests (contests and certain timed standardized tests that measure different things being another matter), since the problem doesn't test much of the processes and procedures and skills that one wants to be testing for, at least not in a robust way. You either see how to do it (easily) or not, and in either case it's difficult to pin down how much the student knows from this one problem. Of course, paired up with several problems having the same type of difficulties would be better, but I'd prefer to see something requiring several steps (somewhat independent if possible, so that it's easier for the grader to "read with the student" to see what they know and don't know) and more "algebra work" in evidence.