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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

Amusing

20 Aug

The new school year has begun!!

While Common Core Geometry is consuming my time, a quick share:

In creating a floor plan for their “dream home,” not a single student included a laundry room. 

I found that amusing.

Length, Area, and Volume

17 May

Them-PosterDid you know… that if you DOUBLE up the size of a chair by doubling up its dimensions in every direction, you will need FOUR times as much paint to paint that chair and that it will weight EIGHT times as much?

Did you know… that if a gorilla was to really proportionally increase to be the size of King Kong, that his guts would just explode and gush out of him from the sheer weight of itself? (The same concept prevents an elephant from ever getting a c-section!)

Did you know… that if an ant was really the size of the ants in “Them!”, not only would they not be able to stand up on their legs, they would also suffocate immediately?

Did you know… that if you proportionally double up in size, you’ll only be able to jump the same height as you did when you were normal size? Likewise, if you got 100 times smaller, you will also be able to jump the same height as you did when you were normal size! (So how long should it have taken for the kids in “Honey, I Shrank the Kids” to get home?) 

tumblr_m78krxad8d1qiceiuo1_500Interesting, right?  All these facts  take into consideration that very first “Did you know…,” that changing a linear dimension changes the area and volume differently. Intuitively we all have some understanding of this. I mean, imagine making an itty-bitty model of a wee little ant out of Play-Doh. How easily will the ant stand on its tiny legs? Now can you imagine building a 2-foot model of the same ant? You’re going to have some difficulty making it stand on the same proportional legs.

So I found it all the more interesting when I went online to search for great lessons on this concept and found NONE. I had never been able to teach this topic to very much detail due to our crazy pacing plans and, frankly, due to my hazy knowledge of the science behind it. However, this year, I had a couple of advantages that allowed me to actually try to plan a lesson- time, of course, and an awesome student teacher who is ridiculously skilled in the sciences!

Anyways, Day 1, I started with a lesson on the basic concept* and followed it up with a follow-up to an exit ticket from the area chapter. It was pretty interesting to see how many students didn’t make the connection to the worksheet that was previously given. I’d say it was about 50/50. Half who figured it out and half who simply multiplied the 52 square foot tiles by 12 inches per foot… (I posted this student’s work because she had asked me if she could simply multiply by 12 or if she had to convert all the numbers on the page first and re-work it. Like a good teacher should, I said, “Exactly! Test it!”)

tiles

The next day was the fun day though. I had almost an entire block period to now talk about the implications of same concept in reality.


 

Again, looking online, I found almost nothing fun, nothing very deep, and nothing very useful to me, except these notes… which I devoured. And from these, we came up with this lesson

__________________________________________________________________________________________

FACT: If you double up the dimensions in every direction, then the area multiplies by ____four____________ and the volume multiplies by ______eight________.

QUESTION: If you have a chandelier barely hanging from a rope and you doubled up the size, including the rope that it hangs on, will the chandelier still hang or will the rope snap?

Why or why not? (Make your argument here. You will not be graded based upon whether your answer is right or wrong, but on how well you make your case.)

 By this point, most students understand that although the dimensions are doubled, the weight of the chandelier has now become 8 times heavier. However, italicizing the part about the rope has them also consider the fact that the rope has also gotten 8 times heavier proportionally. A very true fact. I have them in groups and ask them to talk about it and then write out their argument, emphasizing that they will NOT be graded on correctness. 

I then take a vote. The vote for all 3 of my classes was close to 50/50. I say awesome, and before I give away anything else, I have one person from each camp make their case. I then take out a piece of rope to begin my “Consider…” lecture:

Consider… a really heavy object hanging from this piece of rope. I want to make it stronger and so I double up the length, like so, because I know that this will increase the VOLUME by twice as much. Will that make the rope stronger? [Class agrees, no way!] So then, how do I make the rope stronger?? … exactly! I need to double up the rope by making it thicker. So what does that mean? What does the strength of the rope depend on? The volume? (no) The thickness (yes!). And how do we measure thickness?? … the cross-sectional AREA!

So will the rope snap of the chandelier snap or not?! Discuss…

So yes, the rope definitely snaps. The cross-sectional AREA only increased four-fold (which is what the strength of the rope depends on) while the weight increased eight-fold. In this case, two chords of the thicker rope will hold the larger chandelier.

__________________________________________________________________________________________

FACT: The height that a body can jump vertically is directly proportional to the ratio of muscle mass to the total mass of the body.

jump height

, where C is a constant (enough).

QUESTION: If a human being who can jump 1.5 feet is doubled up in size (pretending like it’s actually possible), how high will the new larger human be able to jump? Explain.

 This was the section that my student teacher had helped me out the most in. He came back w/ an equation that we boiled down and simplified to the one above (apologies to any physicist who might get offended by this). I helped to explain what this equation meant and how it made sense. I made sure they all knew that working out actually made your muscles heavier, which makes total sense in this equation. I had them discuss and then take a vote. 

Their logic behind this one was so adorable. Most of them completely ignored the equation and went straight to guesses. “If twice as tall, muscles get bigger so they’re stronger… they can jump higher.” But the winner of all arguments…

He jumps shorter! Like Mario! When Mario grows, he can’t jump as high! 

The group cracked up and then voted, yes, he jumps less than 1.5 feet… like Mario. 

Final answer? This person will jump the same height of 1.5 feet. His/her muscle weight will definitely increase eight-fold, but so will the person’s total weight, canceling each other out. 

QUESTION: Now reverse the idea. If the same human being is shrunk to half the size, how high can the new small human jump?

Likewise, the smaller human can jump 1.5 feet.

(So how long would it have taken the kids in “Honey, I Shrunk the Kids” to go home?!)

It would’ve taken the same amount of jumps as when they were full size! This was an awesome discovery after reading that one professor’s notes.

__________________________________________________________________________________________

FACT: Ants don’t have lungs. Air is received through “blind tubes” that cover the surface of the ants’ bodies where air enters through.

QUESTION: If an ant was to be blown up as big as the ants in the movie, “Them!”
(pretending the physical structure is possible), what can you conclude about the
air intake of the giant ant?

 We didn’t have time in class to go over these, but these should be easier after all that we’ve learned. I should’ve given it to them for homework… 

 FACT: With larger animals came the development of blood circulation and lungs as a more efficient way of oxygen distribution. Lungs increase the surface area available for oxygen absorption by the blood.

QUESTION: If a human was blown up twice as big (again pretending like the physical structure is possible), how do the lungs have to increase along with the body in order to support the oxygen level needed to survive?

__________________________________________________________________________________________

Anyways, if there are any other lessons/facts out there that you think are good for this concept, let me know! I didn’t have this much fun lecturing in awhile!

*Some of the problems from that worksheet were either directly taken or modified from the NCTM Illuminations site.

Toblerone Challenge –> Can Top Challenge

28 Apr

Here was another fun one.

The Word file here: Toblerone Challenge

Screen Shot 2013-04-28 at 11.20.36 PM

A few logistics:

  • They got to work either alone or in partners. Winner gets ONE Toblerone bar, so if partnering up, they must share.
  • 20 minutes was all that they had. After the Play-doh activity the previous day, they understood that the winner would be the BEST net for this Toblerone box in those 20 minutes. No extension of time.
  • Only one sheet of cardstock was actually passed out to the groups. All other materials were available to them for pickup, if they needed- rulers, scissors, compasses, protractors. They had to build everything until they needed the gluing, pretending like I was the one who will be gluing (there was no glue involved).
  • Collect all nets immediately and put them aside to determine winner when they are doing classwork.
  • At the end of class, when announcing the winner, I took apart the real Toblerone box under the doc cam to see how the actual engineers did it. I then wrapped the chocolate bar w/ the winning net and awarded the winners their chocolate.
  • Hw was to find the lateral area, base area, and total surface area of the box.

Immediately after they turned in their net and before any kind of clean-up, there was a 5-minute bonus challenge.

Screen Shot 2013-04-28 at 11.36.11 PM

I had the actual can in my hands w/ the actual label on it. The label was only taped down on one side so I could show how it wraps around the can and opens up to be a rectangle. I pretty much laid it all out save handing the can to them. I think I still only had about 2 groups per class understand how to find the perfect circle to go on top of the can. Most of them just eyeballed it. Ha!

Winners of that challenge got a mystery can that I had pulled the label off of last year for last year’s lesson. At least the expiration date was on there to prove it wasn’t expired yet! They just wouldn’t know what it contained until they actually opened it. Heh.

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.