It's the end of Full Week #3, and I am in the midst of linear functions with my Precalc class. I decided to try something a little different with graphing lines this year. I'm tired of teaching every kid to re-arrange terms into y=mx+b form only to have them then not remember how to graph a line of that form. This year, I've been telling kids to just come up with TWO points that fit an equation, and then to connect them. This allows them to find the slope immediately via looking at the graph, and then if they want to, they can either extend the line (if it's a simple case) or plug in a point to find the y-intercept, as per usual. I like this "graphical" approach for two reasons: 1. It re-inforces, at every step, the idea that a line is the locus of solutions to that equation. 2. Most kids with a decent number sense can look at a linear equation (of any form) and think up in their heads at least one (x, y) ordered pair that satisfies that equation. If a kid can't think of the ordered pairs intuitively, then at least the idea of plugging in an x and solving for its related y value makes SENSE to the kids. As opposed to the whole re-arranging terms thing, which they only half-understand, at best.
So, for example, today we saw in the textbook a problem that said, "Find the equation of the line through (2, 3) and parallel to 3x - 2y = 5." The way we approached it was we first took a look at the line 3x - 2y = 5. Some kids figured out that (3, 2) and (1, -1) were two points on that line. We graphed and connected the two points to graphically find the slope of that line = rise/run = 3/2. Then, for its parallel line that we're looking for, we graphed (2, 3) then moved towards the y-axis via the same rise/run until we graphically found its y-intercept, which happens to be zero. So the equation we're looking for is y = 3x/2 + 0.
Obviously, out of habit, some kids want to immediately re-arrange the given equations into y=mx+b form and to do the whole problem via algebraic manipulation, which is all good with me (as long as they can do it correctly). At this point in their junior year, I feel like I want to just try different strategies to fill in the holes of their understanding. Maybe some kids are already good at the traditional methods, and that's great. For the rest, I want to offer them a little bit of a different strategy that might make more sense to them.
Sprinkled into this week was Sweeney's Slope Rida sing-along (yes, I sang with my kids to the instrumental version... I also showed them Sweeney's rapping video, even though I couldn't do the rap myself... and I even got another teacher's econ class to come in, be their audience, and to sing along!!), as well as his trick for remembering the difference between zero and undefined slopes and the first of the graphing stories from Dan. (I plan on doing a graphing story per week. Keeping it light, maybe on Fridays!)
Overall, it has been a good week. :) There's a lot of stuff I'm throwing at my kids this week, so I won't really know until next week how well they are doing, exactly. But, I think it'll be OK!
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