Well, it looks like we're going to block schedule next year. Blach for 3 reasons.
1. "Higher Ups" kept up the pretense of asking for our opinions, but those never seemed to be addressed, and then suddenly the decision seemed to be made.
2. I'd love it if someone would prove me wrong, but now I teach 5 classes of students, and so have approximately 25 students per class (and more often more than that). With block, I'll have 6 classes of students, and so will have 20% more students/papers-to-grade in the same amount of time for the same amount of money. Texas is going towards (to) 4 years of math per student, and now POOF they will have more teacher hours in essence for free. I do NOT mind hard work. I WORK hard to make sure I'm doing a good/great/acceptable-to-high-standards job. I DO mind being taken advantage of.
3. I think students need daily math practice. I think they need time to absorb material, so I can't necessarily cover 2 topics in one day and expect it to be successful.
I just introduced piecewise functions to my precalculus classes on Friday. The first year I taught it, I was surprised it wasn't a "gimme" topic. Students were confused by the notation and couldn't always successfully graph/analyze/use such functions. The second year, I tried a different way of teaching it, and still I had more confusion than I was comfortable with. This year, I think I nailed it. (famous last words??)
I started with an electric company example where they charge 10 cents per ___ for the first 500 ___ of electricity per month and 15 cents per ___ for anything more. We talk about why it may be structured that way. I stress the company charges per partial ____ too (continuous). They get to the point where they see the shape of the graph. Then I keep alluding back to this example later to make the connection.
Then I make sure they have colored pencils and a fresh sheet of paper. On the top (as I do on the overhead), I make them draw a number line across the page from -5 to 4, say, (spanning the whole page). I break it into 3 regions and 3 colors and talk about neighborhoods and if x is in one neighborhood, f(x) is ___. Then in the appropriate colors RIGHT under the number line in the correct neighborhoods, we define f, and make a table (all in the right color). Then on the bottom 3rd of the page, we make a coordinate plane, where the x-axis lines up directly with the one at the top of the page and graph the pieces in the right colors.
Then I say, that's too much work to explain this way every time, and we're "lazy/efficient", so here's the shorthand way of writing a piecewise function, and with the same pieces, write it as normal and make the connection with what each piece means. Hopefully, this year, it will be a "gimme" topic.