Excellent Exercises − Simplifying Radicals

Exploration of exercise construction, i.e., casting a net to catch as many mistakes as possible: See also the previous "Excellent Exercises: Completing the Square".

Below you'll see me updating my in-class exercises introducing simplification of radicals for my remedial algebra course a while back. (This occurred between one class on Monday, and different group on Tuesday, when I had the opportunity to spot and correct some shortcomings.) My customary process is to introduce a new concept, then give support for it (theorem-proof style), then do some exercises jointly with the class, and then have students do exercises themselves (from 1-3 depending on problem length) -- hopefully each cycle in a 30 minute block of time. In total, this snippet represents 1 hour of class time (actually the 2nd half of a 2-hour class session); the definitions and text shown is written verbatim on the board, while I'll be expanding or answering questions verbally. As I said before, I'm trying to bake as many iterations and tricky "stumbling blocks" into this limited exercise time as possible, so that I can catch and correct it for students as soon as we can.

Now, you can see in my cross-outs the simplifying exercises I was using at the start of the week, which had already gone through maybe two semesters of iteration. Obviously for each triad (instructor a, b, c versus students' d, e, f) I start small and present sequentially larger values. Also, for the third of the group I throw in a fraction (division) to demonstrate the similarity in how it distributes.

Not bad, but here some weaknesses I spotted this session that aren't immediately apparent from the raw exercises, and these are: (1) There are quite a few duplicates between the (now crossed-out) simplifying and later add/subtract exercises, which reduces real practice opportunities in this hour. (2) Is that I'm not happy about starting off with √8, which reduces to 2√2 -- this might cause confusion in a discussion for some students who don't see where the different "2"'s come from, something I try to avoid for initial examples. (3) Is that student exercises (c) and (d) both involve factoring out the perfect square "4", when I should have them getting experience with a wider array of possible factoring values. (4) Is that item (f) is √32, which raises the possibility of again factoring out either 4 or 16 -- but none of the instructor exercises demonstrated the need to look for the "greatest" perfect square, so the students weren't fairly for that case.

Okay, so at this point I realized that I had at least 4 things to fix in this slice of class, and so I was committed to rewriting the entirety of both blocks of exercises (ultimately you can see the revisions handwritten in pencil on my notes). The problem is that simplifying-radical problems are actually among the harder problems to construct, because there's a fairly limited range of values which are the product of a perfect square for which my students will be able to actually revert the multiplication (keeping in mind a significant subset of my college students don't actually know their times tables, so have to make guesses and sequentially add on fingers many times before they can get started).

So at this point I sat down and made a comprehensive list of all the smallest perfect square products that I could possibly ask for in-class exercises. I made the decision to use each one at most a single time, to get as much distinct practice in as possible. First, of course, I had to synch up like remainders to make like terms in the four "add/subtract" exercises -- these are indicated below by square boxes linking the like terms for those exercises. Then I circled another 6 examples, for use in the lead-in "simplifying" exercises, trying to find the greatest variety of perfect squares possible, not sequentially duplicating the same twice, and making sure that I had multiple of the "greatest perfect square" (i.e., involving 16 or 36) issue in both the instructor and student exercises. These, then, became my revised exercises for the two half-hour blocks, and I do think they worked noticeably better when I used them with the Tuesday class this week.

Some other stuff: The fact that add/subtract exercise (c) came out to √5 was kind of a happy accident -- I didn't plan on it, but I'm happy to have students re-encounter the fact that they shouldn't write a coefficient of "1" (many will forget that, and you need to have built-in reviews over and over again). Also, one might argue that I should have an addition exercise where you don't get like terms to make sure they're looking for that, but my judgement was that in our limited time I wanted them doing the whole process as much as possible (I'll leave non-like terms cases for book homework exercises).

Anyway, that's a little example of the many of issues involved, and the care and consideration, that it takes to construct really quality exercises for even (or especially) the most basic math class. As I said, this is about the third iteration of these exercises for me in the last year -- we'll see if I catch any more obscure problems the next time.

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