Excellent Exercises − Completing the Square

The is the first of an occasional series that I'd like to post about intelligent exercise design for use in a math class (whether as part of a presentation, homework, or test). My primary point is that if someone just thinks that they can solve problems, and walks into a classroom and starts making up random problems to work on -- disaster is sure to strike. There are so many possible pitfalls and complications in problems, and such a limited time in class to build specific skills, that you really have to know absolutely every detail of how your exercises are going to work before you get in the classroom. Not expecting to do that is basically BS'ing the discipline.

So in this series I'd like to show my work process and objectives for specific sets of exercises that I've designed for my in-class presentations. Are my final products "excellent"? Maybe yes or no, but certainly that's the end-goal. The critical observation is that a great deal of attention needs to be paid, and the precise details of every exercise have to be investigated before using them in class. And that some subject areas are surprisingly hard to design non-degenerate problems for.

For this first post, I'll revisit my College Algebra class from last week, where I lectured on the method of "completing the square" (finding a quantity c to add to x^2+bx such that it factors to a binomial square, i.e.: x^2+bx+c = (x+m)^2... which of course is solved by adding c=(b/2)^2.). As per my usual rhythm, I had four exercises prepared: two for me and two for the students. Each pair had one that would be worked entirely with integers, and a second that required work with fractions. The first three went as expected, but the fourth one (worked on by the students) turned out much harder, such that only 3 students in the class were able to complete it (which sucks, because it failed to give the rest of the class confidence in the procedure). Why was that, and how can I fix it next time?

First thing I did at home was turn to our textbook and work out every problem in the book to see the scope of how they all worked. Here I'm looking at Michael Sullivan's College Algebra, 8th Edition (Section 1.2):

What we find here is that all of the problems in the book share a few key features. One is that after completing the square, when the square-root is applied to both sides of the equation, the right-side numerator never requires reduction (it's either a perfect square or it's prime).  Second and perhaps more important is that the denominator is in every case a perfect square -- so the square-root is trivial, and we never need to deal with reducing or rationalizing the denominators. Third is that with one exception, in the last step the denominators of the added fraction are always the same and need no adjustment (the exception is in #43, where we adjust 1/2 = 2/4; noting that even when combining fractions on the complete-the-square step, I had a few students flat-out not understand how to do that). That does simplify things quite a bit.

Now let's look at my fraction-inclusive exercises from class:

As you can see, item (b) (the one I did on the board) works out the same way, featuring a right-side fraction  with a prime numerator and a perfect-square denominator. But item (d) (that the students worked on) doesn't work that way. The denominator of 18, after the square root, needs to be reduced, then rationalized, and that causes another multiplication of radicals on the top; and then to finish things off we need to create common denominators to combine the fractions. That extends the formerly 8-step problem to about 14-steps, depending on how you're counting things.

You can see on the side of that work paper that I'm trying to figure out what parameters cause those problems to work out differently. One is that if there's any GCD between the first coefficient and any of the others, then some fraction will reduce and produce non-like denominators in the last step. And that in turn will result in a non-perfect-square denominator on the right after you complete the square (because of adding fractions with initially different denominators). So my primary problem in item (d) was using the coefficients 6, 4, and 9, which have GCDs between the first and each of the others.

Finally, here's me trying to find a reasonable replacement exercise, which is harder than it first sounds (of course, trying to avoid all the combinations previously used in the book or classroom);

It took me 4 tries before I was satisfied. The first attempt had a GCD in the coefficients (and thus a denominator radical needing reduction/rationalizing), before I figured that part out. The second attempt fixed that, but accidentally had a reducible numerator radical, which makes it unlike all the stuff before that (√44 = √4*11 = 2√11). The third worked out okay, but I was unhappy with the abnormally large numerator radical of √149, which is a little hard to confirm that it's not reducible (the "100" and "49" kind of deceptively suggest that it is). So on the fourth attempt I cut the coefficients down some more, so the final radical is √129, which I'm more comfortable with.

Now we could ask: shouldn't students be able to deal with those reducible and rationalizable denominators when they pop up? In theory, of course yes, but in this context I think it distracts from the primary subject of how completing-the-square works. More specifically, the primary (but not sole) reason we want completing the square is to use in the proof of the quadratic formula -- and coincidentally, neither of those complications appear if you work the proof out in detail (the numerator radical is irreducible, the denominator is a perfect square, and like denominators appear automatically). So as a first-time scaffolding exercise these are really the parts we need. If students were to encounter more complications in book homework on their own time, then that's great, too (although as we've seen in the Sullivan textbook, that doesn't happen).

In summary: Completing the square exercises can get extremely bogged down with lots of radical and fraction work if you're not careful about how they're structured in the first place, losing the thread of the presentation when that happens. More generally: It may be necessary to work out every exercise in a textbook, as well as all your in-class presentations, beforehand in order to scope out expectations and challenge level. Hopefully more examples of this on a later date.


Show Work vs. Justify Answers

My current testing protocol is that all of my remedial math classes have multiple-choice tests, but all of my college credit-bearing classes have open-response tests (i.e., not multiple-choice). This is a minor change this year, as previously I felt completely constrained by the various department-level final exams in our system, which are multiple-choice for most everything up through calculus (so as to make it easy for the department staff to score them). 

Anyway, for the in-class tests that I personally give, I recently grappled a bit with exactly what direction I should give in this regard. Of course many instructors use the phrase "Show your work", so much so that students frequently anticipate that as the direction. But does that address a real issue? Some people's work process is just undeniably crappy: scattered, jumbled, incoherent. While that might indeed be their work process, does it really do them or anyone else any good?

What I've recently settled on is this direction: "Justify your answers with well-written math." This gets more to the heart of the matter, that one is using mathematical language to explain why something is to another person. There's a specific syntax and grammar to this (just like French or Russian or anything): any arbitrary "this is the way I do things" doesn't cut it, because we need a shared language to be understood. And it prepares students to read a math book on their own. And help other students in need, and be helped by them. And it allows the instructor to give useful feedback, by identifying a specific logical gap. And probably some other stuff that I'm overlooking right now.

So at the level of College Algebra and above, I've started to grade half-credit on this basis as of this semester (for full credit, students need both the correct answer and properly-written math statements showing small-scale steps). Later in Trigonometry they can deal with more formal identity-proofs, etc., but I think this frames the expectations for students properly at an early point.

Do you agree that this is a much better directive than "show your work"? Can you think of a better phrasing for the requirement?