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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. =/

 

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Calculus BC Flashcards

15 Mar

I’m not a proponent of having kids just memorize. In fact, I don’t pass these out until after all my (pre-series) chapter tests and before their AP test. I pass out the last page on series tests separately after the series chapter (but right before the power series).

Made to be printed back to back and cut out (flashcards!).

Also, there were a few edits on the last series flow chart. Link edited in previous post.

 

Series Convergence/ Divergence

7 Mar

Super nerd alert. I have been working the last 3 hours on perfecting this flow chart that took me 3 years to figure out how best to go about proving series for convergence or divergence. If you have any idea what I’m talking about, I hope you find it useful. =)

(Please be aware that these are not perfect descriptions of the tests, only summaries. The second page is a fill in the blank copy. I think it would be fun to have a group competition to try to figure out every blank, then review/grade together as a class.)

screencapture

Play-Doh Solids

28 Apr

(The lessons from this chapter are not coming out in the proper order, but hopefully they can still be useful independently…)

20130409_161828What better way to gauge how well your students understand solids than by PLAY-DOH! This was tons of fun.

Each team got one tub of Play-Doh, some fishing string (to cut), and a plastic knife. The Play-Doh had to rotate hands every challenge. The challenges were on Powerpoint, one challenge per slide.

First slide, person #1: in 3 minutes, build your best… tetrahedron. GO!

At the end of the 3 minutes, everyone has to throw their hands in the air and wave ’em around like you just don’t care!… or just hold them there. I collect all the CORRECT tetrahedrons and then judge the BEST ones for 1st, 2nd, and 3rd.

The challenges/slides go something like this:

  • build your best tetrahedron
  • octahedron
  • dodecahedron
  • model of what the following net will build:
    tetrahedron-net
  • Regular pentagonal prism
  • Polyhedron whose cross-section is a triangle
  • model of what the following net will build:
    net_of_pyramid

You can add or subtract what you like to the slides and to the time allotted for building.

To consider for next year: Add a drawing section and have them next DRAW THE NETS to a given solid.

Research on Motivation

21 Nov

A good Monday morning to all!

I saw this falling asleep last night and I had to share as soon as I had the chance. It’s a TED talk by Daniel Pink on motivation.

Basically, providing incentives to complete a certain task, like rewards and punishments, actually only get better results if the task is rote, mechanical, and very focused. 

If ever you need creativity or critical thinking, on the other hand, incentives actually produce significantly poorer results than not providing any incentive at all.

Interesting, isn’t it?! I’m a huge stickler for creating accountability in all my activities, ideally by motivation, but often by incentives. Even if part of me understood some of the ideas here, it really makes  motivation just that much more tangible to reproduce.

For example, activities that might be productive to use incentives for are times table drills, identifying a series of triangles as SSS, SAS, ASA, AAS, or HL, practicing solving equations, etc. I’d say most other activities should stray from using incentives.

So how do we build intrinsic motivation? That IS the big question, isn’t it. Well according to Dan Pink, there’s 3 main elements:

  1. autonomy
  2. mastery
  3. purpose

Perhaps what we do most days is helping students build the first two. Purpose is the big challenge though. Helping students understand WHY these concepts are interesting and worthwhile. I think I’m going to try to focus on that once again. Maybe I’ll try an open-ended project in Geo.

Anyways, on a side-note, when people ask me what made the biggest difference for my Calculus Camp and those scores, it was definitely the motivation factor that camp gave them to pass. They had the know-how, the capability, just like all the classes before them, but why would they sweat over passing an AP Calc exam, when all of them only paid $10 for it and about half of them were thinking about community college anyways? Taking them up to camp changed that, but more importantly, it changed that for them together as a class. Those who never had competitive, college/univ. bound friends were not only mixed in w/ those who were, but they were all becoming friends.

If you teach/taught in a low-performing school (not that my LA school is low-performing anymore!! =)), one of things you would’ve quickly realized is that “they just don’t care.” Even the brilliant ones seem to take on this “C is still passing” attitude (not all, but most). What else can you expect when no one else, neither family or friends, expect any more of them?

Well, up at camp, we got away together as a class,  did team builders, played ridiculous games, got delirious at night together, ate Fear Factor-worthy things for punishment, and endured 15 hours of Calculus. When they got back, they were suddenly so bonded that they weren’t just trying to pass the test alone, but w/ their entire class. It was AMAZING. They suddenly wanted to pass this test because that was what we had spent the entire weekend focused on and even the weeks prior in getting it funded.

I only wish I had done all that on purpose. I had only taken them up b/c I remember having gone to Calc Camp when I was a student and finding it helpful. I also remember it being a bit boring so I added some flair. =) I guess I just need to figure out how to add motivational flair to math on a daily basis now.

Movies in Math. Math in Movies.

18 Nov

Here is a website with a compilation of short movie clips on math and physics that you can freely download.

http://www.math.harvard.edu/~knill/mathmovies/index.html

I use the Shrek clip as the introduction to inverses and contrapositives in Geometry.(This one was also on YouTube.) The PowerPoint I used is in the Box widget.

M&M Catapult

2 Jun

I’ve been eye-ing this for awhile now so I was so excited to run this!! SO excited! I first read about this from Sweeney Math and seriously fell in love with it because the math in it is not forced. I really dislike those forced math projects, though I must admit, I use them sometimes anyways for engagement purposes.

Student’s Goal: To build a consistently-shooting M&M Catapult and find the equation of the M&M’s trajectory when shooting from the floor. Using that equation, calculate where the M&M will land when shooting from on top of a table.

Teacher’s Goal: Again, this was for my post-Calculus students so I really wanted them to understand some of the physics behind what we were doing. I created a few pre-questions to prime them a bit. I really wanted them to be able to see the integration between all the different maths over the past several years. (Pictures of problems below)

Copy of my modified lesson plan is to the right in the Box.net widget. As a note to that copy, I gave them one front-back page at a time. In other words, they had to get pgs 1-2 completed before they getting pgs 3-4 and they had to finish pgs 3-4 before getting pgs 5-6.

Materials Needed:

  • box of fat popsicle sticks
  • one clothespin per group
  • multiple glue guns
  • a wooden board to attach the catapult to (I asked the woodshop teacher who had a bunch of cut up desks available)
  • 2-3 bags of M&Ms (they crack with every launch so students kept wanting new ones)
  • tape measure
  • masking tape to hold M&M. I didn’t have any so we cut small Dixie cups I had laying around for geometry.
  • timer(s). I used my Flip because each launch lasted around one second and I can be accurate to within 1/30th of a second with it.

Overview of Project

  1. Build a catapult that shoots consistently to the exact same spot.
  2. Time how long the M&M is in the air for and measure the horizontal distance the M&M shoots
  3. Use a bit of physics and the vertex form to figure out the equation of the M&M’s trajectory
  4. Give students the height of the desk they will launch from. They calculate where they need to place the bull’s eye if they were to shoot if off that desk!

Tips and Hints

  • I gave my students freedom to build the catapults as they saw fit. I just gave them the hint that it should launch to the exact same place every time AND that they should know at which angle it launches if their M&Ms were flying crooked.
  • Doubling up the “diving board” makes each launch a bit more consistent b/c it gets rid of the bend.
  • Try to run this outside or in the gym. It was raining and windy in sunny So-Cal the days we were doing the trials so we had to limit the launches to within the classroom. This makes the launches limited to the height of the room, making the times shorter than I had wanted.
  • Try to avoid using the stopwatch the shorter the launches are.
  • Make the target fit into about 1 square feet. I’d say the bull’s eye was about 2 in radius. I had several groups get a bull’s eye on their first shot!

… the rest is in the file “MM catapult” to the right.