March 13, 2015

Systems of Equations

My Algebra classes having been playing with solving systems for over a month. A very interrupted month, but a month nonetheless. I decided to go all puzzles at the beginning of the semester to re-engage my classes. And it was awesome. I didn't even realize the puzzles were systems when I first assigned them! Then we did some algebraic manipulation. And we ended with more context. I had no continuity (7 snow days and a week's vacation happened during this unit) so I want to figure out how to run this unit in the future.

We started with Noah's Ark. It was hard. I tried "convince your classmates, then come to me when you have a consensus" but they got mad. So instead I did a few cycles of "work for 10 minutes, then I'll have students share who made progress, then you can work for 10 more minutes" That worked.

Next we did Mimi's shape puzzles (only the first 5 pages because we'll only be solving systems in 2 variables in this course). This went amazingly. They worked at the boards and not only solved them, but explained their work!

Tape diagrams. I like this sheet (except number 5, I meant that to be something trickier than 60*3=men but I don't remember what). It doesn't need to happen in this unit though. It would probably be good to move to September as a way to use diagrams effectively.

We've been solving balance puzzles for months. I thought it would be great to segue into equations/graphing by writing equations from balance puzzles. (See twitter conversation.) Turns out they're too good at solving them to make two variable puzzles challenging enough that they see any need for equations/graphing. It was cool to see how all their equations intersected at the same point when I graphed everything they came up with in Desmos. Next time I'd do this as a whole class activity only. Challenge: write as many equations as you can for this (two shape) balance. Solve it. Check out what happens when you graph all those equations on Desmos. Yay!

Then we spent a while solving algebraic systems: Graphing without context (intro above). Substitution without context (sticky notes intro). Elimination without context (intro with - you can add 5 to both expressions, you can add y to both expressions, if y=5 you can add y to one expression and 5 to the other expression. Okay? Hey look! If I add these two equations ones of my variables disappears. Yay!).

Most kids liked elimination best, which is fine by me. But I did want them to realize that if a problem is set up for substitution, that might go faster. So we did Sam's efficiency rating scale. Then we did a scavenger hunt (solve one system, the solution is on another board, do that problem next) where each problem might be best suited for a different method.

So far I'd only given kids systems with nice solutions. So I had them solve three systems. The point was supposed to be to discover that lines can coincide or be parallel instead of always intersecting. But despite the fact that all three systems included the line y=2x+4, they struggled because they weren't set up nicely for elimination (issues like lining up the = signs arose for the first time). Next time I will have better systems for this task and more challenging systems for previous tasks. Solving three systems takes forever even if they're nice ones... But we finally learned the word coincide and reviewed parallel and practiced those a bit.

We ended with systems in context again, but words rather than puzzles this time. First we practiced interpreting equations with some awesome examples of potential misconceptions (just student pages 1 and 3) then we did the Mathalicious lesson Flicks.

Next time I think I'll do tape diagrams and Noah's ark whenever we need puzzles, not necessarily this unit. Shape puzzles systems is a good starting place. We could even do systems by elimination from there because the method comes up and it is a better transition than solving equations was this year. Then a brief intro to graphing using balance puzzles rather than that whole activity. Hopefully they won't need so much practice graphing so we can use Desmos more. Finally, substitution. With stickies. The context problems don't have to wait until the end but it's not a bad place for them. I think this seems like a reasonable plan (though I have no compelling reason for the order of the methods). It would be awesome to run this unit without all the interruptions. We'll see what crazy weather we end up with next year!

I haven't really been writing much about what I'm doing in Algebra this year since nearly everything I do is from somewhere else in the MTBoS or a cut up Kuta sheet. But maybe it's helpful to see how I put together the pieces to build a lesson? And to remember what resources exist? I'd like to get back to sharing (and recording for myself) so feel free to comment with what you'd like to hear about!

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