Some advice that was given to me when I first started teaching regular PreCalculus was to skip or deemphasize completing the square. However, I didn’t shy away from the topic because I felt it could be approached in a way that appealed to the fact that the average person enjoys solving puzzles. I plant the seeds by introducing “Avoid the Freshman Mistake Puzzles” as warmups or puzzles for those last few minutes of class, for about a week before we dive into the topic. This handout that I created is the most formal thing I have written up (we normally do this more informally in class).
I’ve also covered this topic in my Intermediate Algebra class (a post-Algebra 2 or Pre-PreCalculus class) with a population of students that have generally struggled with math in previous years. One of the struggles that we often have in teaching high school math is the need to teach or review fractions and how to do that with a population that is very fraction-phobic. This year, “completing the square” turned into an accidental ah-ha moment for many in terms of understanding fractions. (Working on my National Boards this year, I wish I would have had the camera rolling for this whole-group discussion…) Using words here, I know I’m not doing a good job of capturing the discussion, but basically the problem the class was wrestling with was how to find out what half of 3/8 was. Various students described their thinking and we ended up looking at a visual representation of 3/8 in a rectangle and cutting each shaded part in half and seeing that half of 3/8 was 3/16. Then the excellent generalization question came from the audience: “So can we always just double the denominator to cut a number in half?” Those are the types of questions that make a teacher proud of their students :). From that came the observation that that’s what happens when we multiply horizontally using fractions. Other ideas that came up in the discussion included the way we found half of 4/9 in a previous problem and the connection between thinking of it as four “ninths” and then half of it was just two “ninths,” versus going to 4/18 and then reducing, and why the denominator changes when multiplying but not adding fractions. And all this on a 5th period on a Friday afternoon, right after lunch and about an hour before the weekend.
Just thinking about how far this class has come from September to now makes me so proud of them! This is precisely the sort of thing I was refering to in my previous post about beginning-of-the-year review. Review in context, in this case not even a “real-world” context but a context of an “advanced” algebra move, is much more meaningful rather than just drilling a bunch of fraction addition and multiplication problems.