Tuesday, March 25, 2008

Completing the Square

Toward the goal of sharing more of the humdrum, everyday business of teaching math, here are a few notes about how we do completing the square in my classes. I have no cool tricks or creative activies for this, and I would very much like to see more of yours. Nevertheless, completing the square is, inexplicably, a favorite topic of mine.

The first time I taught it I relied somewhat on the formulaic addition and subtraction of the square of half the middle coefficient, but that left the class utterly confused and frustrated. Now I rely less on this rule and more on pattern recognition and intuition. This appears to make for better retention, though the students do have trouble applying it to numerically messy cases, where a formulaic approach - if ever mastered, that is - would be safer.

We start by reviewing how to square a binomial, because while we have of course worked on multiplying binomials and using standard factoring patterns earlier, the error of squaring a binomial by squaring each term is remarkably resistant to instruction. We always need to refresh that by writing out the factors and multiplying, carefully, term by term. Indeed, any time that a review of completing the square is called for later, squaring binomials from scratch is the point I will return to, and invariably it will turn out that many students have forgotten what the squared binomial looks like. After we've done the multiplication from first principles for a handful of examples, I point out the pattern in the middle and last terms of the product and ask the students to pick up speed, which they do. I write up a few binomials where the second term is a fraction and remind the students that one advantage of fraction form over decimal form is that fractions are really easy to square.

I'll call on individual students or have the class shout out answers, and will alternate between having students suggesting problems and solving them ("J., will you give us a binomial?" "S., will you square that for us?") and I have repeatedly been surprised at how engaged the students tend to become during this exchange, since the topic, after all, isn't that inherently exciting, and we aren't doing anything particularly nifty. Part of the reason may be that it is easier than for many other topics to sense just where the students are and to tailor the next example so that it matches their readiness.

When I notice that the class is beginning to get that "now what...?" feeling, we reverse the process: I write up a perfect square trinomial and have students factor it. We keep doing this until the students again reach the point where this is too easy, and then start looking at cases where only the quadratic and linear terms are given and the students need to figure out what numbers would fit in the blank spaces in a form such as this one: Later, writing this form on the board will be sufficient to cue a large fraction of the students to what they are trying to do.

We move on to rewriting simple quadratics (where a=1 and b is an integer) in vertex form. Later I will show them that adding and subtracting the square of half of the coefficient of the linear term will give us just what we want, but at this stage we simply identify the squared binomial, multiply it out, and compare this with the original quadratic to see what we need to add or subtract. For example, to write in vertex form we will recognize that (x-4)^2 is the square term, and since expanding this gives a constant term of 16 we'll need to subtract 13 in order to ensure that we have the same quadratic that we started with: This approach seems to stick fairly well in students' memories. Many students who do not correctly add and subtract the half of the middle coefficient later (they'll insert an x in there, or halve it incorrectly, or something) will be able to rewrite simple quadratics in vertex form, and I can see from their scratch work in the margin that they're just comparing the expanded square with the original quadratic. I'm pleased with that, because the equivalence of the quadratic in its two forms is one of the big ideas I want them to take away, and the fact that we aren't dealing with different quadratics even though they do look different isn't nearly as self-evident to the young ones as it is to us.

So, that was not terribly exciting or innovative, I concede. But how do you teach completing the square?


Anonymous said...

I draw a square, eg a + 5 by a + 5, divided into a big square, two rectangles (the 'wings') and a little square (25). We talk a bunch about the parts. I eventually get something like x^2 + 20x + 97 into the discussion, digress to a story***

when we were little, my sister liked snacking, but couldn't always make up her mind. Sometimes she would try grown up crackers, triscuits, but didn't like them, and put them back after nibbling them***

and draw the perfect square with my sister's nibble gone from the little square. (squeals and "oh gross"). And then we undo my sister's damage (in this case, but putting 3 back to fix the triscuit (or, as the mathematicians say, complete the square)

Not wonderful, but it seems to hold.


druin said...

Excellent :) I love how you kept them engaged and worked with pattern recognition rather than just "here's how you complete the square".

By the way, in an earlier post you mentioned about posting to ILoveMath. You can actually link your blog posts in the lesson plans section so that you retain the ability to update it easily. I tried it out with my own blog under the Statistics section.

Keep up the good work!!!

Anonymous said...

i like your way of building their intuition. i try to do that in general, but i totally failed when teaching them how to "complete the square."

i thought it would be easy (silly me), but there are so many steps, and the book presents it in such a procedural manner, that my students lost exactly what they were doing. they saw it as a set of steps to memorize, which is exactly what i want to avoid.

next year, i'm going to do it like you.

i think that the big problem in presenting this thing is that the big picture gets lost... why are they doing it? at least that's what my problem was. so i'm going to try doing it backwards first. show them an equation like (x-2)^2-3 and tell them why this form is so useful (can be used to quickly find the vertex, graph, and find the zeros). THEN i'll go through the rest.

i did like teaching completing the square for the "a ha, math be cool" moment. students had learned or heard about the quadratic formula earlier, but it came from nowhere. i motivated the class by trying to excite them to show that math wasn't magic and that we were going to come up with this awesome thing ourselves.

i'm always amazed at how much i can get away with by saying it's "awesome."

H. said...

Jonathan, I'll probably try that story on one of the kids who needs to make up the quadratics test - I'll see how it goes.

Mrs Temple - you're actually Druin?! I'm honored to see you here! I love I Love Math, and now posting to the site moved a few notches further up on my todo list.

I've long been hoping that one day I'll get to teach a stats course so that I can learn the stuff myself - just took a tiny course on it in college, not nearly enough, and have forgotten most of even that.

Sam - working backwards from vertex form, again and again, is worthwhile. And I decided that for Intermediate Algebra sacrificing some of the complicated cases (where you have to find half of a fraction, etc) was worthwhile in order to keep focus on the big picture. I also agree that deriving the quadratic formula by completing the square is very satisfying - hoping to post some notes on that soon.

druin said...

Well shucks, thanks for the warm welcome :) I actually post around the net as several names... Druin most of the time, mrs. temple is from my gmail account (hence blogger), and Shelli/HS Math on teachers.net...

I've enjoyed reading your posts and check in on your blog a few times a week to see what you've written. I am really into pattern recognition/discovery methods, so I love your method of teaching quadratics.

Keep up the great work! :)

Unknown said...

The method that I have used for the last two years seems to work well for me.
I sort of "scaffold" their understanding from factoring a perfect trinomial square to completing the square.
I might begin with a perfect trinomial such as y = x^2 + 6X +9 and have them factor it. Then I write y = x^2 + 6X +9 + 7 and talk about writing it in vertex-graphing form (since this is when I introduce completing the square). Building on this, I may use y = x^2 + 6X + 16 and have them recognize that this is the same as the previous question, they just need to break the 16 up first. Then, after this type becomes comfortable, I use something like y = x^2 + 6X + 4 and wait until they realize that they need to make the 4 into a 9 so they add 5 to both sides of the equation, or they add then subtract 5 from the same side. I continue this way until they are comfortable with completing the spuare with y = ax^2 + bx + c where a = 1.
Then I use questions that have a common factor for a and b, then I move into ones that do not have a common factor.
I find that students following this process understand what they are doing and we don't need to "memorize steps".

Rebecka said...

I just taught completing the square how you suggested today and it went sooooo well! Thank you! They saw and recognized the patterns on their own. So when I taught completing the square...I didn't have to teach it using a procedure, they already knew what to do. Awesome!

H. said...

Thanks for telling how that went, Rebecka - it made my day:) I haven't taught completing the square in more than two years, and it's so much fun to think that someone found this useful now!