## Wednesday, February 20, 2013

### Visualizing Concepts

Here is an MS update. I feel pretty productive lately, as I always do during the second semester. I also feel quite productive with my Grade 11s, and as a result I'm taking on three new kids potentially, at least for a while. Grade 12's are doing OK, but the pressure is sure ramping up for their IB exams, so there's not a whole lot of "cool" instructional things that I can be doing with them...

Referring to the system {5u + v = 21, 10u + 3v = 48}, one middle-of-the-road kid explained excitedly to her friend, "Oh! I got it! If 5u + v = 21, then 10u + 2v must equal 42. If you put that here into the second equation, then you will still have 1v on the left side. Which means that 42 + v = 48, and v must equal 6!"

Not bad for being day 1 of systems algebra. (Of course, they've been doing the same reasoning using shapes and visual substitution for several days now, so the transition to symbols was seamless. I didn't bother with even any examples on the board this year, and it worked out just fine without one.)

I went up to a boy who was one worksheet ahead, and pointed at the equation {y + 6x = 10,
y + 2x = 2} to ask him what immediate conclusion he could draw based on inspection. He immediately said, "X is 10 minus 2, divided by 4." I had to backtrack to ask him, "How do you know? Because 8 is equal to....?"

He finished my sentence, "4x." Great, now we're talking about the process.

I showed him the traditional notation for showing that work / reasoning via elimination, even though I know that in his head he's doing substitution from one (smaller) equation into the other.

y + 6x = 10
-(y + 2x = 2)
4x = 8

I am very pleased with how this unit is going. I think I've gotten it down pretty pat; if I remember correctly, last year I didn't have to change any worksheets at all, and so far I haven't had to alter any worksheet either. This is one unit where I can just sit back and watch the kids' thinking unfold, more or less, and I can be very hands-off in their own building of the concepts of algebraic substitution and elimination. At some point, I looked around the room when I noticed the noise level rise, and saw that it was because literally every pair of kids is engaged in some sort of intense mathematical discussion with their neighbor. Awesome-o!

In Grade 9, we've been working on learning / understanding / memorizing geometry area formulas via cutting and pasting. We each re-arranged, glued down a parallelogram to form a rectangle, to help them understand why A=bh for a parallelogram. Then, we each re-arranged a trapezoid into a thin parallelogram, to help them understand why A=(b1+b2)(h/2) for a trapezoid.  This year, I went a step further and highlighted the base 1 and base 2 sides using a green marker, so that the kids can see that they line up within the parallelogram when we do a "move and flip" of the top half of the trapezoid. This simple highlighting technique is superbly visual for my visual learners to see why the new parallelogram base MUST be (b1+b2).

Then, I proceeded to give them two practice trapezoid problems. In both cases, I had them draw out the parallelogram that it becomes, labeling the side lengths to emphasize the numerical connection between the trapezoid and its resulting parallelogram. It worked great! My concrete thinkers really latched on after the first example. One of the weakest kids in my class went up to the board and bravely (and correctly) drew out the trapezoid, its new parallelogram dimensions, and the calculated area, without any previous verification that he was on track. I was proud!!

I just love geometry lessons like this, because they involve both tactile and visual learners and help the concept "stick" in their heads. Slowly, I'm getting their geometry concepts up and running in order to do the 3-D project this year.

Addendum March 4, 2013: Here is a visual that Lara has made on Pinterest. Thanks, Lara!