Three typical lessons

Systems of equations are something that I had trouble teaching last year, so when I rebuilt the unit I had some concerns. I think that most of my goals have been met. Also, things were moving pretty quickly, which is a major concern for me this time of year.

Day 1 was pretty much solving "catching up" problems and then giving those a name. Day 2 was solving systems graphically, and tying that into the "catch up" problems. Day 3 was convincing kids that there's a limitation on the graphical technique, and then providing them with a technique for "merging" equations by substitution.

I think that there are a couple of good things about these p-sets/worksheets. First, there are some good questions interspersed throughout. I also like the way that the abstractions are latched onto some concrete structures. These are examples (plumbers, walkers and music subscriptions) that students are familiar with from the linear equation unit earlier in the year. I also like the way, in Day 3, I had the students see the limitations of solving by graphing.

Here are my docs. Feedback, please? Help me make these better?

Systems Day 1 Worksheet

Systems Day 2 Worksheet

Systems Day 3 Worksheet

Here's what makes it fairly typical of my work right now:

1. It's a problem set. Problem sets and problem solving are more or less at the center of my classroom, right now.
2. It's made from scratch. My school doesn't have materials for me. I wasn't happy/didn't understand the stuff that Park Math, Exeter, Arlington Algebra or Sam's VFC had made available. I have no idea how I taught this last year, because I have nothing saved. My lesson plan was probably scribbled on the back of an envelope and I probably printed off a worksheet from the interwebs. Yeah, first year was awesome.
3. I wrote it on the day of the lesson after reflecting in a lesson plan document where I thought about "hard parts" and some "good questions" to ask in class.*
4. I had very, very little time to teach this. I know we all have coverage issues, but I REALLY have coverage issues. 

* It's like a diary. It'll be great fun to read next year when I'm planning.

"Your class is the proof that there does not need to be a daily structured schedule for students to accomplish and to want to accomplish."

A student writes:

I will give you my honest perspective on the class, and I believe everyone in the class feels the same way. Your class is immensely enjoyable, and the one class I look forward to throughout the day. The class and the projects are interesting and engaging. The class is extremely chill and open-ended; we can go at whatever pace we like, as long as we can still finish on time. Your class is the proof that there does not need to be a daily structured schedule for students to accomplish and to want to accomplish – We all of our own will and decision come to class each day, mainly because we enjoy it. It is a stress free work environment, as everyone just hangs out, has fun, helps each other, and in general just enjoys themselves. You should mention how nearly every day we are baffled as to how time flies by so fast in that class, and how we feel there is never enough time for the class. Working on the projects is actually fun to do even outside of class on our own free time. Your class helped teach me how to troubleshoot issues more effectively and is the only class in the school that brings ideas from inception to reality so effectively and in such a great way. The feelings of accomplishment from completing specific goals, learning concepts, and completing projects, is unfound in any other class. Plus, I get to have a website with a portfolio of programs and games I MYSELF have created, and now have gained (at least what is to me) actually useful and practical knowledge, which I can't say for many of the things I learn in school.

That's my programming class. Now, how do I get my math classes closer to that?

A quick one about problem-solving in class

I think that I just noticed another piece of the problem-solving puzzle. I've just finished a really good unit on factoring with my ninth graders, and a lot of the frustrations that I've had with this group have been absent for the unit. I think I know why.

Before, I would have anywhere from a half to two-thirds of the class working hard on problems, and the remainder goofing off. They would either be stuck, not interested in doing any work on the day or just punching each other in the head. (I teach high school boys. They do this sort of thing.)

Anyway, that didn't happen with this unit. They self-organized into helpful little groups. They started saying mature things like "I need more practice on multiplying binomials" or "What's the next step after this?" They taught each other. They shared tips and techniques.

Why? Here's my quick analysis:

1) The unit was set up well. I taught them the concept of factoring when we were doing the distributive property in October. I taught them about multiplying exponents in November. They were all comfortable with the prerequisites to this unit.

That means that they could self-assess. They knew that if they got no x^2 term, that something was up. There was very little of that x * x = 2x sort of thing.

2) More importantly, I think, is that I was clearer than I had ever been about what they needed to learn. After the first day or so, I told them that there were four levels of sophistication that they needed to hit: Multiplying polynomials, Factoring the Diff of Two Squares, Factoring Trinomials, Factoring Stuff Completely. I told them that their job each day was to move themselves up a level. I started class with a "Reality Check" that helped them (and me) assess what they needed to work on. I had problems and activities available for every level. Students got to choose what to work on, so if a kid was feeling frustrated he knew exactly where to head back to.

The lesson for me is this: always share the map with the students. If they know where they're going, they'll feel more empowered to make sure that they get themselves there.