My 7th graders have been going over quadratics for the past couple of weeks. I have been out quite a bit on school business, so the progress has been slow, but very rewarding. Last week, students discovered that if you change the value of "a", it has an effect on how fast the parabola grows. I then had them graph a bunch of parabolas whose line of symmetry was the y-axis only to have a student ask,
"Can we move the parabola left or right?"
"Well since you asked..." So I did what any responsible teacher would...I had them graph a bunch of parabolas whose vertex sat on the x-axis which led to the next question,
"Can we move them up/down and left/right?"
So Friday, we are graphing parabolas in the form y =a(x-h)^2 +k and they are getting it. This is stuff I couldn't do until Algebra II with my high school students and these 7th graders are just crushing everything I throw at them. I would even give them a vertex and a second point and they were giving me an equation because they figured the stretch factor using the second point. Things are looking good and I am thinking:
Man this is just toooo easy...
Yeah, I know, pride comes before the fall. Which is what started to happen today. I have had a planning block. Now that we have graphed a bunch of parabolas in vertex form, where do I go from there? Do I start dealing with standard form? Do I show them how to expand (x-h)^2 in order to arrive at standard form? I am still not sure what the ideal path would be. But being the "try anything once" kind of guy I am, I figured that since I have already had them:
- Graph quadratics organically (area vs. radius; area vs. side length)
- Graph quadratics with a not equal to 1.
- Graph quadratics with vertex on y axis.
- Graph quadratics with vertex on x axis.
- Graph quadratics in the form y = a(x-h)^2 +k
...then I would focus in on what was necessary to graph a parabola: vertex and stretch factor. If they could identify the vertex and a stretch factor, they can graph anything, right? So today I wanted them to graph a bunch of parabolas in standard form, look for the line of symmetry and recognize the relationship between a,b and the line of symmetry. I didn't expect them to necessarily "discover" that the line of symmetry is x = -b/(2a), but I figured that if we graphed enough of them, we might start to notice some patterns. Once we have the line of symmetry down, then we could start looking at x intercepts which would lead us to factoring and completing the square as well as quadratic formula. (If my sequencing on this is bad, please save me from myself. )
This is where it started to go bad. GeoGebra is a great program, but it doesn't save a weak lesson. Kids were all over the place with their parabolas and we were getting lines of symmetry like x=.7923496, which wasn't going to help at all. I was about to put us all out of our misery and jump ship when Chandler says, "Mr. Cox come here, I think I found something."
She had about 10 parabolas that all had the same line of symmetry. She was making adjustments to the a and b values and recognized that the c value had no effect on the symmetry. So rather than aborting the mission, we just changed course.
"How about choosing a line of symmetry and keeping the vertex on that line?"
They got right to it. Tomorrow they are going to come to class with five different a,b,c values and the corresponding line of symmetry. We will see where it goes.
Note to the reader: Quadratics are not a 7th grade standard and these kids will go through it on a deeper level as 8th graders. So, I am not worried about "finishing" this with them. I have the flexibility to let concepts marinade for a while. Usually, I would have just followed the pacing of the book, but I have become very dissatisfied with that. I am pretty sure that I want to continue from linear relations right into quadratic relations and that graphing is a good gateway to all the other skills that go along with quadratics. I am just not sure how one skill will best lead into another. Any suggestions?
Note to self: Quit gettin' ahead of yourself and be sure to see the lesson through the eyes of a student rather than your own.
Oh yeah, and thank Chandler.