Tag Archives: group work

Power and Maclaurin Series

23 Mar

I know my SIFE class is probably what more people want to hear about, but hey, it’s AP crunch time and Calc BC has been getting FUN. Finally, after 3 years of teaching it (and one year of suffering through it 17 years ago)!

Today’s share: I finally feel semi-decent about how I’m introducing the power and Maclaurin series. The last couple of years, my students have been full of complete panic when I introduced the first free response series examples because they looked so entirely different from the way the book introduced it. I tried looking at other books to try to supplement to see if they do a better job, but none were very satisfactory.

So I finally re-wrote it as group work for discovery. Let’s be real, after going through painstaking tests for series convergence and divergence for 8 sections straight, the power series should be an easy transition, but the fact is that it is NOT.

I think I’ve finally figured out why, or at least partly why. Firstly, the books kind of sneak in a second component to what an “interval of convergence” means for Maclaurin series. They quickly transition from using the “interval of convergence” to represent all values of x that make the series converge (like we’ve been doing for 8 sections) to suddenly using the phrase to also represent how two completely different functions are now graphically the same. Secondly, the books also fail to explicitly equate how the geometric series now uses x to represent the common ratio and now calls it a Maclaurin series. It is all very fuzzy and seemingly brand new, as opposed to being a continuation of the exact same concept we learned way back in the beginning of series.

So here is how I have introduced these two concepts, the power series to the Maclaurin series. Each of these worksheets are done in groups. I have different accountability measures for making sure each are doing the work together, but that is for a different post.

DAY 1 (95 min block): Worksheet A. This is to introduce the power series via the geometric series but by making the students explicitly point out the connection. I also explicitly point out the connection on which test to use using the same flow chart on the power series as well. Practice problems on using the Ratio Test on power series is given as classwork –> homework.

DAY 2 (55 min class): Worksheet B. This is where I try to redefine “interval of convergence” to mean something extra and introduce the Maclaurin series without actually introducing it. At the end of this day, w/ the catalyst of my cheesy/nerdy enthusiasm for how cool this is, I got quite a few “ohhhh. coooool!” from the kids. (FYI, this has never happened before in prior years.) No homework this day.

DAY 3 (95 min block): Worksheet C, pgs 1-2 first (and separately from the rest of the packet). I had them find the Maclaurin series for sine and cosine. At the end of these pages, have them MEMORIZE these function=series equivalents because otherwise, they keep wanting to re-derive every time!

While I am walking around checking, stamping, and collecting their pgs 1-2, they are memorizing. By the time I get to their desks to collect their pgs 1-2 I also have them clear their desks before handing them pgs 3-5. They now have to pull those series equivalence they just crammed out of their brains and on to paper.

I stop the class when about half the class has started the second part to go over an example of how to use our given four series to do their first “manipulating” of the four series together. Hw: finish finding the intervals of convergence and do the first three AP free response problems on pg 4. I encourage/ beg the students to please not Google the answers. They have a terrible “study habit” of having the answer too close and relying on it too much to find the answer.

DAY 4 (95 min block): Worksheet C, pgs 4-5. This day is yet to come for me (it is this Friday). I have plans to go over the answers in detail and the point distribution for the first three, then give group work time to go over the next three.

Taylor series and LaGrange Error Bound is yet to come. I have no clue how to introduce those yet. =/

 

The New Classroom

8 Sep

This past summer, I was in an program run by the Exploratorium up in San Francisco. It was an amazing month of the best professional development I have ever had in all my 5 years of teaching. I had taken the Physics class because for the first time this past year I taught AP Calculus and the application/ Physics portion was where I was weakest in.

I learned a million new things that will hopefully start creeping out as these blogs keep rollin, but one of the things that the program helped to inspire was the setup of my new classroom. In the first week of attendance I had attended a random seminar on how hands-on science exhibits are created. One of the hurdles that these professionals had come upon was that students were hard to engage! One thing that they discovered that did a beautiful job of engaging all the students was to have certain exhibits around a round table with the teacher amidst the children. They found that in this environment, the students couldn’t keep their hands off!

They also found that having screens mobile also helped avoid the classroom student mindset where they only listen and don’t engage. The change of somehow taking their focus off the “front” of the classroom and being able to move their focus to different places seemed like a brilliant idea in giving them more ownership of the classroom. Thanks to working with ProofBlocks for so many years now, I was very akin to creating huge white boards of my own ($12 for a 4′ by 8′ piece at Lowes or Home Depot).

So as soon as I came back into the classroom, I decided a good classroom setup was more important than I had given it credit for. I lucked out this year on a bungalow where all four walls were corkboard so I got to work.

It took me 5 hours just to clean up the dump, take everything off the walls, and put up the black butcher paper (something else I had learned from this one seminar was the importance of focusing the kids by using black as backdrops). I then went to Lowes and cut up a bunch of white boards to hang all around my classroom, one for each of the 6 groups I had in the class (**w/ the increase in class size, I use the two ends of my own white board to make 8 groups now.)

This is what the back of my classroom ended up looking like:

Without the white boards

Without the white boards

With the white boards

With the white boards

The black around my front white board was added to make it seem more like a curtain show when I’m up there so these kids can focus on the board and not be distracted by the clock, any cute posters, etc etc. Plus, it just looks cleaner.

white board

Lastly, inspired by the Exploratorium’s King of Rock, and having visited Nic Hess at the Hammer Museum just that week, I decided to give my own tribute to the King of Pop.

Exploratorium's Elvis

Exploratorium's Elvis

My Michael

My Michael

I wanted it to be in the corner and on the ceiling too, but MJ was pointing perfectly back to my white board in this position. =P

So far, this set up has been amazing for more reasons than one.

  1. I no longer bother with posting student work. Student’s work are automatically changing with the work on the white boards. Everyone only looks at student work the first time it goes up anyways.
  2. It saves paper. I used up a lot of butcher paper setting up my classroom. At least I won’t be using anymore for group posters. =P
  3. Changes are easy to make on white boards. And we all know how often students are hindered in putting something down on paper because they’re afraid it’s wrong.
  4. Each white board is mobile. I can have them take the white boards off the wall and work on a task around their desk (like at the museum!) and when they’re finished, they can put it up.
  5. My personal favorite, I can make up group presentations on the fly whenever the class leads that way! I was never able to do that before because either I would have to have the poster paper ready in advance OR they would get way too shy to come up on my white board to show their work! Just this last reason alone was worth all the trouble.