I wrote earlier about a quiz – a single system of linear equations to be solved graphically, then solved again algebraically. I figured it was fifteen minutes, max. It took over forty.

Here’s the system:  $3x-4y = 8$  and $y - 4 = \frac{6}{5}(x-3)$

I took to heart comments some of you left here, and others made on facebook. But most of all, I had long conversations with the kids (two full classes) themselves. And then follow-up conversations.

Some of the more interesting points:

• kids really liked the post-quiz analysis sessions. And honestly, they were good discussions. I think me walking in saying “I’m not sure what happened; I need your help” made it different than a regular class discussion.
• some kids have become more comfortable using point-slope to graph directly. However, not all of them have. And none of them had attempted any more than the most rudimentary manipulations of an equation in point-slope form – and then only to mindlessly move it into slope/y-intercept form. The cleanest solutions required thoughtful algebraic manipulation of the equation.
• the demo of getting rid of fractions, and how it makes life easier, that was well-received. A week later and they were still ready to talk about that part.
• one kid wants to write up a discussion of this system on a poster/project.
• some kids created a fraction in the first equation (by solving for y) for graphing, then substituted into the second equation. There were a good number of kids who worked with denominators of 20…

Some things I take away:

• I really want to teach flexibility. It means that kids need to see awkward problems. And they need to be faced with questions where the wrong choice is available. And they need opportunities to examine right choices and wrong choices, or better choices and not as good choices. And talk about them.
• I won’t use this as a quiz again.
• I will use this as a warm-up, and stop the class after letting a little (emphasis on “little”) frustration build. We need a little frustration to make the debrief engaging, but the best part was the debrief.
• I need to remember to occasionally work in discussions where I don’t know the answer, and share with students that this is the case. It’s good for them to see, and they react favorably.
• I like the feedback I get here. Thank you. And I will ask for more.