We've been hard at it.....mathematical discourse can exhaust kids and their teacher. I'm worn out from a pretty excellent week.
I've been helping students examine their conjectures....and track their mathematical thinking. We had an absolute blast thinking about what different lines that had a slope=3 might look like. OK OK OK....I caught you yawning. I've written about this already since school started, but it was such a good lesson activity I have to write about it again.
But what made it great fun was using the modeling software, Geometer's Sketchpad. We diligently plotted our first line, making sure it went through the origin. Then students tried to find any point on the coordinate plane from which they couldn't draw a line that had a slope of 3. Sure enough they'd suggest point after point....yet were still able to draw a line.
Finally....someone asked or observed.....aren't these all parallel? which we discussed at some length. And then one of my sweet little cherubs asked if we couldn't have eliminated the whole line drawing exercise by realizing that if the line has a coefficient of 3, it will be parallel.
I sent them off to discover if there were other patterns assoicated with coefficients and perpendicular lines....they'll report back next week.
This is where math gets fun for them and why the discourse is so essential to an engaging math classroom. Students get to ask the questions that they believe...and then we test them out. Pencil and paper are wonderful....but really pale in comparison to the power of the Sketchpad and SmartBoard combo. (I'll have to write my good friend, Bill Ferritier, about this dynamic duo.) As someone thought they had a way to defeat the slope=3 problem, they come to the SmartBoard add their point, trace the rise over run to the next point and then draw the line. Using the Measure slope function, each student could quickly prove/disprove their idea.
Soon the class found out that there was also a Measure function that told the equation for the line. While I hadn't planned on using it, I thought why not...they're interested. Boy am I glad I did. By this time we had about 13 or 14 lines on drawn and it wasn't a minute until the student had highlighted and "measured" the equation for all the lines.
From that data....they were also to conjecture that the "+" or "-" was the y-intercept. We had discussed the y-intercept, but somehow this discovery made them feel like they owned it now. And then we had to test out their theories that where the line crossed the y-axis was the "b" in everyone's favorite linear form y=mx + b . Sure enough. They'd add a point on the y-axis, draw their next point, draw & measure the line....and the equation was still 3x and their point on the y-axis was listed in the equation.
All in all, it probably took me about 20 minutes to do all this experimentation. It has probably been the most valuable 20 minutes I've spent all year. My students and I were real co-learners today as they developed ideas, they ran the tests and then ran another and another and another...until we were prety sure that our idea was solid.
On a related note: we're teaching students how to use Geometer's Sketchpad in their Study Skills class. I'd never heard of the Vesica Piscis before I started using Sketchpad. It's a powerful tool and the Sketchpad tutorials quickly showed this technique and how it could be used to construct equilaterial triangles. The challenge was to extend that technique and draw a regular hexagon.
I love my digital natives. They thought it wasn't going to be a big deal until we made them measure their hand-drawn line segments and they realized that what they drew wasn't congruent. Their bravado suddently vaporized and using this medium really challenged their thinking. In my class of about 30 students only a couple were able to figure it out pretty quickly....and so it will extend to another day of learning.
And because they are digital natives....as soon as they learn this, they'll be fearless in applying it to all sorts of things I'd never dream of doing. It will be a beautiful thing to watch.