By request from @jreulbach at http://ispeakmath.wordpress.com/ – this is how I teach graphing. On twitter she mentioned that her students struggle with graphing, but could solve a system using substitution. Very ironic because mine get graphing but took forever to get substitution!
A lot of how I teach graphing came from the curriculum my school uses for 9th grade. Also, with any subject I teach I try to make as many connections to anything else they know as I can. For my ninth grade class very first we started of with what an intercept is. I introduced that with football. When someone intercepts the ball, their path and the path of the ball cross.
To talk about slope I showed them this cartoon. They get confused when graphing because for slope you go up first, then over next. For graphing a coordinate you go over first, then up/down. We had a graphing competition that seemed to resolve that. They were in teams and had to find the correct point if I gave a point, and the correct slope if I gave a line. First team to get it won that round.
Those two things didn’t take very long, about one day each. It was stuff they’ve learned before, just needed reminded of.
As we went through the rest of the unit we used the organizer found here. but instead of printing this out to give to them, I had a blank grid we filled in as we went along. Only the honors students did the matrix, everyone did graphing, elimination, and substitution. We learned graphing first, then the other two. I also used this method for notes. We only filled in the organizer once they had it down.
I also had them write slope-intercept, standard, point-slope, and recursive formulas for lines we were working on. and practice going back and forth between the different forms, and graphing from the different forms.
Once we know how to graph one line very well, we get into two lines. I start by introducing it as another kind of intercept. But instead of looking at the x or y axis, we are going to use another line. I go back to football on this and explain it like someone tackling another player. The two have to cross paths for that to happen.
After we graph the two lines I have them pick the point where it looks like the lines cross, plug that into both equations, and see if they are both true at that point. If they are both true, that one point works for both lines. So that has to be where they cross. If it doesn’t work, that’s not the right point. I’m REALLY emphasizing checking work with all my students and that seems to help also. It helps them get better at catching their own mistakes before they practice too much the wrong way.
For the younger grades, that aren’t into systems yet – they are just learning graphing – I still use the same slope and intercept instructions, and we still practice the confusing slope vs. graphing a point problem with the game. That seems like the biggest confusion for them. You start vertical for finding slope, and horizontal for graphing.
With the younger ones I also have them create pictures with graphs, then give someone else the instructions for making their picture. Here’s an example of how to do pictures with graphing, but instead of giving my students one I have them create their own. Depending on how long we have to work on it I’ll have them do a picture with anywhere from 10-20 points as the minimum.