Must be doing something right?

Highlight of the Day:

Context: AP Calc class, talking through some multiple choice (no calculator) problems from a practice test.

what is  ;

Me (typical first question): So....what should we do?
Student: Substitute the 2 in for t to find the answer
Me: Why would we want to do that?
Student: Well, f(x) is the integral there, which means it's really the anti-derivative of the function. We want the derivative of that anti-derivative, which basically means we want the stuff inside the integral. So if you just substitute 2 in, you'll get the answer.
Me: You basically just explained the first part of the Fundamental Theorem of Calculus that you struggled with when we first talked about it. Woohoo!

Granted, I know his explanation skips the step where t in the integrand becomes x after you take the derivative, but the conceptual understanding of the relationship between derivatives and integrals is there. Made my day.


Help and Critique needed: GSP file

Hello friends.

This is what I've been doing this afternoon. It's a Geometer's Sketchpad file that I hope to use with my calc class when we talk about volumes of revolution. I tried to upload it using Javasketch for those of you that don't have GSP, but found out that JavaSketch doesn't support function plots, or pretty much anything I used to create the sketch. Typical.

Anyway, I'm looking for feedback. Play with the sketch, press the buttons, see what you think. Ideally, I'd like this to be along the lines of a Dan Meyer "What Can You Do With This" type activity, but I don't think I'm there yet.

It's not finished, mainly because I don't quite know how to create functions for the last 1.5 pictures. Any help is appreciated! Thanks, friends.

A new version is posted here. Thanks to iTeach in the comments section (@PersidaB on twitter) for coming up with cubic regressions for the lasat 2 pictures. I didn't think to use my graphing calculator...wanted to do it all in GSP and couldn't figure it out. Thanks again! More feedback is always welcome, too. :-)


Budget Cuts and Future Plans

Thursday we had a staff meeting led by our school Superintendent and the Human Resources director. The topic? Next year's budget cuts. As of the last governor's proposed budget, our school was scheduled to have $1.5 million cut from our budget for next year. The new governor has not approved a budget yet, so we're not sure if that number will remain, or increase.

Currently we have about 50 students in the deaf department high school. Twelve or thirteen will be graduating. Currently we have five 8th grade students. Our numbers in the high school are going to be dropping. Our numbers in general are dropping. Next year we will have 66 students in the deaf department. That is, 66 students from preschool through 12th grade. Not very many.

After telling us these numbers, HR discussed how the administration has been planning to deal with the budget cuts. They will be leaving vacant positions open, trimming utility and technology expenditures as much as possible, and decreasing hours for wage employees (teacher's assistants, bus aides, interpreters). They will also decrease some admin positions from 12 month positions to 10 or 11 months.

After all that, they still need to cut over $430,000. Layoffs are coming. It makes me sad to think of people that I have worked with for the past two years being laid off. At this point I feel some degree of security in my job because there are only 2 of us math teachers in the deaf department, teaching middle and high school. The administration has also informed me that I might be the person teaching the newly required Personal Finance and Economics class that all students will need to take (eek!). Looking around, though, I see people who have worked for more years than I, who have more education that I have, that are in jeopardy of losing their jobs because they are the most junior member of their "team." I don't know what will happen in the next few weeks, nor am I 100% certain that I won't be writing to say I'm out of a job for now. I am one of the most recent hires...low (wo)man on the totem pole, so to speak. I'll keep you posted.

One more thing...this information about the budget cuts and the future of our school further justifies my thought that I should broaden my scope for a master's program. Maybe Secondary Ed: Math? Or math in general? ...but I don't want to be proving deltas and epsilons and doing way far out math...I don't quite know yet. Any suggestions?


Graphing Sine and Cosine

Recently in Trig, we've embarked on the task of graphing sine and cosine curves. Before this, we constructed the unit circle and students became familiar with the exact values of sin and cos at the special angles. To introduce the parent graphs, I used an activity that I found at Kate's blog last year (who I'm sure got it from somewhere else, but I don't know where).

To first see the "unwrapped" unit circle that is the sine (and later, cosine) function, students used yarn to mark intervals along the circumference of a unit circle, then used spaghetti to measure the y (later, x) value at each place. They then transfered these lengths of spaghetti to an x-y plane with x intervals of length matching the circumference intervals. This was a great picture for them of where the shape of the sine function comes from. One student even explained it (works much better in ASL), as though you've taken the bottom half of the unit circle and spun it around to make up the second half of the sine function. (my adaptation of worksheet)

Graphing sine and cosine with changes in period and amplitude came easily for my students, but when we started translating with horizontal and vertical shifts, the students were lost. The first day was a major failure on my part and I knew that I needed to have a new approach when I started the next day. I placed an open call for help on Twitter and was forwarded two GeoGebra applets (thanks Dave!). I quickly came up with a guided investigation to go along with the first applet.

We went to the computer lab, and had some success! Normally, when I've taken classes to the computer lab to use Geometer's Sketchpad or GeoGebra (or other programs), the students end up playing with the program, going through the motions, and they leave with little to no connection back to the paper and pencil world of the classroom. It's probably the way I present it, but they may understand the concept more indepth while using the computer program, without any transference back to the original idea or any application to the next topic.

This time, I required the students to answer the questions in a word doc, and I taught them how to use Print Screen to capture the image on screen and paste it into their document in order to later compare it to another graph. The next day, we continued by using the second applet (with some of the parameters changed) to walk us through the process of graphing sine functions with translations *and* changes to period/amplitude. We closed the day with an application of what the applet taught us about sine to help us graph cosine functions with translations. As a bonus, my boss happened to come and observe me explaining the second applet to the students. He just loves it when teachers use technology (and happens to observe me on days when I'm using it well...lucky me!)

Results: I know at least one of my (two) students benefitted from this process. We are still not at a point where they can graph the functions independently, but they have some strategies to help them, and a deeper understanding of what the numbers in the ugly looking equations mean and how changing them, changes the function. As their Algebra II teacher, I know I could have done a better job when we did function transformations. That probably adds to their confusion and weakness in this area. Something to think about for next time.

**update** I know I haven't even posted yet, so I feel like I shouldn't update, but I did see some good progress in my students today. Now we have a shared experience that I can link back to as they continue to develop their graphing/graph analysis skills.