Today I dove into a lesson that was entirely new to me. I remember having to "complete the square" in Calculus but I don't think I was ever quite sure of the algorithm and I spent some time reteaching myself over the weekend in preparation for this week's lessons. I even derived the quadratic formula from a trinomial in standard form- it was a thrill for this math nerd!
I tried to look for a way to make completing the square easy, fun, engaging- anything other than a long algorithm to memorize.
Here's how my lesson went today:
I began by giving the students a "puzzle" to solve with Algebra tiles. Could they make a square with x^2 + 6x + 9 while following the rules we'd already established for how Algebra tiles can touch each other? The kids made the square easily and were unimpressed; after all, we'd been through this weeks ago when we started factoring!
Next, I gave the students x^2 + 4x + 6 and asked them to make a second square. They worked at it for a while and then complained that I had given them an impossible task. I was pleased to note that most kids started with the x^2 tile in the upper left and 2x to the left and bottom of this tile. Instinctively, they realized that they would need to evenly split the linear term to form a square. They tried other combinations after this and then tried to get creative- adding tiles, hiding tiles, overlapping tiles, and breaking all sorts of Algebra tile rules.
I asked kids how many "1" squares would be necessary to make a square in the previous problem. They knew it was 4.
We moved from these concrete examples to guided notes that worked through the algorithm. In each step, I referred back to the models we'd made to reinforce how the algorithm matched what they had done on instinct. And thus began the whining, complaining, whimpering, and near mutiny! They wanted nothing to do with this lesson, don't see the point, think it's too hard, are scared of the quiz, etc, etc.
I took as a sign of divine providence the fact that I had a meeting scheduled with my supervisor just after school. We're writing curriculum for Common Core implementation. Since she is just recently out of the classroom, and had recommended that I include completing the square in the course, I decided to ask what she would have done. Her answer: she just taught the algorithm, plain and simple- not the answer I was looking for. She was impressed by the work I'd had the kids do with the Algebra tiles but just kept explaining how important completing the square is in later math courses. Unfortunately, I didn't get any suggestions to improve my lesson which was what I was hoping for. I can't see completing the square being useful until at least Pre-Calc (and my students are in middle school, so that's at least three courses away) so I'm returning to the question I've been wrestling with since the weekend: is completing the square necessary at this point and are my students developmentally ready for it? I question the wisdom of doing some problems just to say it's done. I want to improve my lesson, so if you have any magic answers, please share!
Tomorrow, we'll finish our guided notes, complete a foldable to summarize the process, and try several example problems. Wish me luck!
I usually start with having them square a few binomials and look at the relationship between the constant term in the binomial and the constant term and coefficient of x in the trinomial. My kids usually like the "short cut" feel of this. THen we have some incomplete trinomials where the y try to guess what "magic number" will make it also be a perfect square. Then we move on to ones where the there is a constant that is not the magic number and discuss what we might be able to do to make it be the magic number...hope that gives you a little help.
ReplyDeleteI found your blog through pinterest because of your foldables. :) I am a high school math teacher, from algebra 1 to advance algebra 2 to 2nd year calc. Your question on completing the square. On one hand: GOOD FOR YOU! While you are instilling a great deal of problem solving and getting to the why of the mathematics, this skill at alg 1 seems very...uninteresting.
ReplyDeleteUntil you get to quadratics. Completing the square is necessary to convert standard form to vertex form. (for most trinomials-it won't work that well for irrational numbers) You already know we use it to derive the quadratic formula. Well enough. But on the other hand, it becomes so useful for when they deal with conic sections (Algebra 2). You'll need completing the square so much for dealing with solving, graphing, and manipulating conic sections.
On the gripping hand, you're going to have to come clean with the kids-this is a tool to solve quadratic equations when factoring isn't so simple. This is another way to get the vertex without having to memorize a formula for the line of symmetry. (or recognize what part of the quadratic formula is the line of symmetry!) It is simplying another tool. A tool they should practice now, practice later, and the mastery and use of it will be clearer later. (God, HALF of the battle of calculus is just the algebra part. Recognizing that completng the square can simplify so many calculus problems...)
I hear ya on your troubles of making it interesting or in context of a problem. But this is true math, the manipulation of expressions and equations with a creative twist.