Tuesday, 25 October 2016

Rethinking Factoring Special Quadratics

Have you ever had a day when a lesson you took the time to rethink actually worked noticeably better? Let's be honest - I don't have the time (and sometimes not the motivation either) to rethink all my lessons. "What worked well enough last year is good enough for this year" happens far more frequently than I'd like to admit. I try to make notes if something really doesn't work or if I have a brilliant idea after the fact. And I do my best to act on those notes to my future self. Occasionally, if I teach more than one section of a course, I will make changes on the fly as I teach the second class. But the reality of teaching full-time and raising a family is that every lesson may not be as good as it could be. This is a difficult reality for me.

The change I made to yesterday's lesson was a simple one - I did the opposite of what has been done in the past. Let me back up for a minute (and I apologize if you've heard this all before)... The math teachers at my school all share lessons for all courses. I am the renegade who sometimes does things differently. I have been spiralling my grade 10 applied classes for several years and I spiralled my grade 10 academic class for the first time last year. I put a lot of thought into the order of topics and how each would be approached and blogged daily. This year I am tweaking what I did last year - the biggest change being that I am introducing more quadratics concepts earlier in the course. I am trying to be intentional when I look at past lessons and ask myself whether this is the best way to approach the topic. I looked at the "department lesson" on factoring special quadratics (at this point I have no idea who created it - it could have been me???) and just wasn't happy with it.

The old:

... followed by exclusively difference of squares practice questions. Then:

... followed by exclusively perfect square trinomial practice questions.

The new:

As I wrote above, I got students to do the opposite of the old lesson. Instead of expanding, they factored. This was good practice for them and they could see that there was a shortcut within the patterns. We had a whole-class discussion, talk with the people at your table, test your conjecture(s), come back to whole-class discussion kind of thing going, but we got there. They came up with the patterns (I didn't tell them) and they saw the value in what we were doing (I think). I think the old lesson tended to fall flat because they didn't see a need for more ways to factor - it was just confusing. They didn't see these special cases as being helpful. I hope this year's students do. They also know that they can also successfully factor them as they would any other trinomial if they don't notice that they are dealing with a special case. (Confession: Until I started teaching grade 10 applied, I did not think of a difference of squares being a trinomial where the x-term has a coefficient of 0. Factoring these with algebra tiles was a revelation!)

One of the things I love about spiralling is that it freed me from common test days. When my students need more time on a topic, I give them that time. So tomorrow we are factoring a little more. A few students are really solid with all types of factoring, but most have a more tenuous grasp of what to do when. My room is currently all set up for some factoring speed dating. Tomorrow should be a fun-filled day of factoring!

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