Power Series on Desmos

This is a crosspost from my Photo 180 blog.

Power Series work in AP Calculus BC.

\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-...

Process: Since it’s a infinite series, look at partial sums to get an idea what this graph looks like.
So look at
y_1=\frac{x}{1!}
y_2=\frac{x}{1!}-\frac{x^3}{3!}
y_2=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}

Perfect time to use technology.

Texas Instruments Method

Go to y1. Enter in y=x.
Graph.
Wait 3-5 seconds.
Go to y1. Subtract a \frac{x^3}{3!}.
Graph.
Wait 3-5 seconds.
Go to y1. Add on a \frac{x^5}{5!}.
Graph.
Wait 3-5 seconds.
Go to y1. Subtract a \frac{x^7}{7!}.

2014-03-05_13h53_31 2014-03-05_13h53_44

REALLY crappy resolution. Awful zoom system. Where’s the factorial sign? Hopefully you remember what the previous graphs looked like. Lots of waiting. Ugh.

Desmos: version 1

Graph y=x
(NO WAIT STEP)
Subtract a \frac{x^3}{3!}.
(STILL NO WAIT STEP)
Add on a \frac{x^5}{5!}.
Subtract a \frac{x^7}{7!}.
2014-03-05_13h55_46
Great!

Desmos: version 2

Students teaching teachers: Have one your students find this out for himself, and remark that they can enter in the entire series, but he’s having trouble finding the infinity sign. SLIDERS!
Link.
Whoa. (That seemed like it should be much harder to type in. Took me 5 minutes to type in the original latex code at the beginning of this post. Took me < 1 minute to actually enter the series into desmos. I had to help 1 student out of 15. That’s it. Wow easy.)
2014-03-05_13h58_56

Watch the power series create sin(x) step by step by moving a slider. LIVE!

We live in good times.

 

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Two James Tanton Questions

It’s midterm week at school, and James Tanton threw out two interesting questions in two days. I spent a little time programming “solutions” to these problems (not solutions, just verifications for an infinitesimally small portion of the natural numbers).

Problem One:

Here’s my processing.org code for this problem. And here’s the output of the code, each time the sequence gets longer, it prints out the new “max” sequence length. 2014-01-28_08h29_27

I didn’t use any of processing.org’s graphics but I had the prime function optimized, so it was quick work.

Problem Two:

Here’s the python code for the “solution”. And here’s the last six lines of the output of the code. 2014-01-28_08h31_32 I checked all numbers under 1,000,000, and all the sequences were finite (they stopped at a multiple of 13). The starting number whose sequence ended in the largest multiple of 13 was 964,665, and the multiple of 13 had 384 digits (BIG NUMBER! There are only ~10^80 particles in the entire universe). Fun stuff.

[Edit: 1/28/2014 9:08am] Ok, ok. James asked for a proof for the second problem. Here you go :-) 2014-01-28 09-05-38

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Prime Matrix – Processing

Saw this tweet yesterday:

Pretty cool. Josh Giesbrecht also did some great work with Mathematica to replicate the image.

I also replicated the image with processing.org. The (live) code can be found here.

screen-0000

(update 1-16-2014 11:38, image above is now lossless and will stand up to zooming.)

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Four and Five – Cartalk Puzzler

From cartalk.

Ray: Get a piece of paper and write the number four, leave a little space, and write the number five. What common mathematical symbol, when placed between the numbers four and five, will result in a number that is greater than four but less than six?

Tom: It has to be a mathematical symbol? It can’t be, like, the word “or”?

Ray: No, it’s got to be something that’s commonly used in mathematics. You’ve used it many, many times – maybe even today.

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Pebbling a Checkerboard Game (Or Chessboard)

I happened upon this tweet when I got to work this morning:

So I watched the fantastic numberphile video Pebbling a Chessboard. I wanted to get the kids to play the game and I didn’t have any checkers in the classroom, so I decided to program the game in Processing. Thankfully the programming went smoothly, and I finished it in time to have the Pre-Calculus class try to beat the game (it’s the day before winter break, and we just finished sequences and series). Fantastic timing.

Rules:

  • Goal: remove all checkers from the green prison.
  • If you click on a checker, and there is space to the right and above, then that checker will disappear, and two clones will appear to the right and above.
  • Theoretically this game board extends to infinity to the right and above.

The game is hosted by openprocessing.org or hosted by recursiveprocess.com. It should work in any web browser, including smartphones. If you’d like to increase the checkerboard size, just go to the openprocessing.org site and “tweak” the code. The size of the board is set on the first two lines.

Enjoy!

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Why I Blog (Nowak requested post)

Fellow Empire State rep Kate Nowak requested answers to the following in her post: Tell Me Why You Blog.

1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?

I got hooked on reading math blogs because I was interested in becoming a better teacher and I wasn’t finding resources in other places to satisfy this desire. Math blogs put me in touch with teachers willing to try new things out and succeed, or more importantly, fail in public.

2. What keeps you coming back? What’s the biggest thing you get out of reading and/or commenting?

Confession: I don’t comment much on blogs. Maybe I use twitter as a sounding board for my opinions? Not sure why.

I keep coming back because I keep getting good stuff. The economics of this system are tough, if you people in the #mtbos stop producing good stuff, then I’m out. Thankfully, the stuff coming out has only been getting better and more interesting.

3. If you write, why do you write? What’s the biggest thing you get out of it?

Mostly I write for myself. I process things better after writing about them. This is from a guy who hates to write (or reflect). I record things on my blog first for myself; second for others. If other people find use in my blog, fantastic. But I’d still have some version of it if it were only my Mom reading it (hi Mom!).
It’s great that people find use for things I’ve posted. Just this morning I woke up to find 15+ responses on twitter talking about a post that I put up 3 years ago. Soooo cool. Just awesome. And that there’s that stupid Oreo thing. That must count for something.

4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to’s? Stories?

Dunno. Give me personal stories. I’m no help here.

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How Harvey Mudd Transformed Its Computer Science Program — And Nearly Closed Its Gender Gap

From Tech Crunch. Must watch video.

Harvey Mudd is known as one of the most elite science, engineering and mathematics colleges in the world. But historically, its computer science department hasn’t exactly been known as a paragon of gender diversity — in 2006, only 10 percent of Harvey Mudd’s computer science majors were female.

But under Dr. Maria Klawe, the renowned computer scientist (and Microsoft board member) who joined Harvey Mudd as president in 2006, the gender ratio at Mudd’s CS department has changed dramatically. This year, fully 48 percent of the CS majors in Harvey Mudd’s junior class are female.

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