A Recursive Process http://blog.recursiveprocess.com Math teacher seeking patterns. Sat, 25 Feb 2012 01:42:54 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 Light Limaçons and Cardioids http://blog.recursiveprocess.com/2012/02/24/light-limacons-and-cardioids/?utm_source=rss&utm_medium=rss&utm_campaign=light-limacons-and-cardioids http://blog.recursiveprocess.com/2012/02/24/light-limacons-and-cardioids/#comments Sat, 25 Feb 2012 01:42:54 +0000 Dan http://blog.recursiveprocess.com/?p=1170 While putting away the bowls, I noticed some nice limaçons and cardioids made from the refraction of light.


Video of the changing shape - Limaçons

 

For more info:

]]> http://blog.recursiveprocess.com/2012/02/24/light-limacons-and-cardioids/feed/ 0 25 Billion (From Dan Meyer) #anyqs http://blog.recursiveprocess.com/2012/02/24/25-billion-from-dan-meyer-anyqs/?utm_source=rss&utm_medium=rss&utm_campaign=25-billion-from-dan-meyer-anyqs http://blog.recursiveprocess.com/2012/02/24/25-billion-from-dan-meyer-anyqs/#comments Fri, 24 Feb 2012 13:27:35 +0000 Dan http://blog.recursiveprocess.com/?p=1156 Well? Live link: http://www.apple.com/itunes/25-billion-app-countdown/


For those (geeks) looking for more detail on how the rate is determined, you might be able to find the rate in the somewhat obfusticated javascript file found here: http://images.apple.com/global/scripts/downloadcounter.js

]]> http://blog.recursiveprocess.com/2012/02/24/25-billion-from-dan-meyer-anyqs/feed/ 3 Obsessions #anyqs http://blog.recursiveprocess.com/2012/01/29/obsessions-anyqs/?utm_source=rss&utm_medium=rss&utm_campaign=obsessions-anyqs http://blog.recursiveprocess.com/2012/01/29/obsessions-anyqs/#comments Sun, 29 Jan 2012 13:59:24 +0000 Dan http://blog.recursiveprocess.com/?p=1146

Some more information can be found here (look in the description of the video). A second spoiler video can be found here.

]]> http://blog.recursiveprocess.com/2012/01/29/obsessions-anyqs/feed/ 0 Pole to Pole Run http://blog.recursiveprocess.com/2012/01/24/pole-to-pole-run/?utm_source=rss&utm_medium=rss&utm_campaign=pole-to-pole-run http://blog.recursiveprocess.com/2012/01/24/pole-to-pole-run/#comments Tue, 24 Jan 2012 16:21:20 +0000 Dan http://blog.recursiveprocess.com/?p=1139 Continue reading →]]>

Pat Farmer, an endurance runner, set out to run from the North Pole to the South Pole to raise money for the Red Cross.

Utterly awesome.

So if he sets off on April 2nd, 2011 and he runs 16 miles a day on the snow, and 50 miles a day on roads and trails, then when will he reach the South Pole?

]]> http://blog.recursiveprocess.com/2012/01/24/pole-to-pole-run/feed/ 0 Skallops – Geometry Heaven http://blog.recursiveprocess.com/2012/01/12/skallops-geometry-heaven/?utm_source=rss&utm_medium=rss&utm_campaign=skallops-geometry-heaven http://blog.recursiveprocess.com/2012/01/12/skallops-geometry-heaven/#comments Thu, 12 Jan 2012 15:29:32 +0000 Dan http://blog.recursiveprocess.com/?p=1117

Skallops are a construction toy that uses laser (no sharks) cut wood clips to put together playing cards. They are up on kickstarter now and they can make some incredible shapes. I just ordered a set myself.

]]> http://blog.recursiveprocess.com/2012/01/12/skallops-geometry-heaven/feed/ 1 Paper #wcydwt http://blog.recursiveprocess.com/2012/01/06/paper-wcydwt/?utm_source=rss&utm_medium=rss&utm_campaign=paper-wcydwt http://blog.recursiveprocess.com/2012/01/06/paper-wcydwt/#comments Fri, 06 Jan 2012 20:38:00 +0000 Dan http://blog.recursiveprocess.com/?p=1102 Neat video:

Note: All photos were printed on recycled paper.  Wasn’t that nice of them. 

So how many photos? Answer found here.

]]> http://blog.recursiveprocess.com/2012/01/06/paper-wcydwt/feed/ 0 IB Thoughts? http://blog.recursiveprocess.com/2011/11/29/ib-thoughts/?utm_source=rss&utm_medium=rss&utm_campaign=ib-thoughts http://blog.recursiveprocess.com/2011/11/29/ib-thoughts/#comments Wed, 30 Nov 2011 01:19:45 +0000 Dan http://blog.recursiveprocess.com/2011/11/29/ib-thoughts/ Continue reading →]]> So our school is looking into implementing the International Baccalaureate program at our school in addition (?) to our AP programs. I know little about IB, and I’m interested in your knowledge and opinions on the IB program. Lemme have it. Good stuff? Things to watch out for?

]]> http://blog.recursiveprocess.com/2011/11/29/ib-thoughts/feed/ 6 Fibonacci and Recursion http://blog.recursiveprocess.com/2011/11/18/fibonacci-and-recursion/?utm_source=rss&utm_medium=rss&utm_campaign=fibonacci-and-recursion http://blog.recursiveprocess.com/2011/11/18/fibonacci-and-recursion/#comments Fri, 18 Nov 2011 15:27:28 +0000 Dan http://blog.recursiveprocess.com/?p=1065 Continue reading →]]> To introduce the fibonacci sequence using the recursive series to my Pre-Calc H class (although most of them know it), I came up with something on the fly that worked nicely. To set up the fibonacci sequence, I showed them a recursive program that calcuates the sequence in a very clear (but inefficient) way.

def fibonacci(n):
    if n==1:
        return 1
    elif n==2:
        return 1
    else:
        return fibonacci(n-1)+fibonacci(n-2)

for number in range(1,10):
    print "fibonacci number " + str(number) + " is ",
    print fibonacci(number)

This is probably the first time they’ve seen programming in person, so to make it real I gave a post-it to each student and then had them stand up in a line. I had the person at the start of the line write n=1 on their post-it, and then they continued to the end of the line, n=19. I gave the first two students in the line their fibonacci numbers f_1 and f_2 (1 and 1 respectively -haha).

Then I asked the person at the end of the line what his fibonacci number was (f_{19}). After a bit of questioning back and forth he agreed that he needed to know f_{18} and f_{17}.

Well this continued all the way back to the start, and then after f_{3} got her fibonacci number, the information went down the line and finally:

…

f_{19} got 4,181, and I was desperately hoping that no-one made a mistake in their number, and went to check the result on wolfram alpha. Success! Before having the students sit down, have them calculate the ratio of the two fibonacci numbers that are under their number. So f_{17} calculates the ratio: f_{16} / f_{15} and then have them write their decimal on the board in increasing order. Save this for later.

To show the class how inefficient the above algorithm is, I calculated f_{19} and every time f_{1} and f_{2} were “asked” what their number was, it would print their names.
The Code:

def fibonacci(n):
    if n==1:
        print "Sage"
        return 1
    elif n==2:
        print "Jessica"
        return 1
    else:
        return fibonacci(n-1)+fibonacci(n-2)

print fibonacci(19)

The Output (Click to expand, but it’s very long. An interesting number of lines *hint*):

Jessica
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4181
>>>

So that algorithm is very inefficient for calculating fibonacci numbers because it has no memory of the previous fibonacci numbers. When f_{19} asks f_{17} for her number, she doesn’t remember it, and she needs to calculate it again when f_{18} asks for it.

Finally to cap it off, talk about the ratio of the fibonacci numbers that you’ve written on the board. These numbers tend towards the golden ratio (\phi).
Awesome.
Continue deep into math teacher land talking about phi and fibonacci numbers. Possibly share an applet like the following which shows that phi is present in seed growth in plants like sunflowers.
Seedhead.
Or show (or assign) a video about the Golden Ratio and Fibonacci like Keith Devlin’s talk where he discusses what is true and what is false about the golden ratio.

Cheers.
Dan

]]> http://blog.recursiveprocess.com/2011/11/18/fibonacci-and-recursion/feed/ 1 Dr. Square http://blog.recursiveprocess.com/2011/11/11/dr-square/?utm_source=rss&utm_medium=rss&utm_campaign=dr-square http://blog.recursiveprocess.com/2011/11/11/dr-square/#comments Fri, 11 Nov 2011 12:18:09 +0000 Dan http://blog.recursiveprocess.com/?p=1067 Continue reading →]]> Much like the Collatz conjecture, the Dr. Square Puzzle (from the blog mathforlove) is an intriguing problem.

With some playing around, we came up with what I think is an excellent (and solvable) puzzle. He dubbed it the Dr Square puzzle, because it involves one of the steps in taking the digital root (dr) and squaring numbers. Here’s how it goes.

Step 1: Choose a starting number.

Step 2: Square the number.

Step 3: Sum up the digits of that number.

Step 4: Repeat steps 2 and 3 until you understand what’s going on.

Example. Let’s take the number 26. Squaring it gives 676. The digital sum of 676 is 19. Squaring gives 361. Digital sum of 361 is 10. Squaring 10 gives 100. Digital sum of 100 is 1. Squaring gives 1. Digital sum gives 1. So we stay at 1 forever once we get there.

More briefly, we could write 26 –> 676 –> 19 –> 10 –> 100 –> 1 –> 1 –> 1 –> etc.

Naturally, I wrote a python program to check the first million number and so far it looks like they are right. If you take off the comment symbol (#) from lines 6, 20, 23, and 24 then it will print the path to each number.

]]> http://blog.recursiveprocess.com/2011/11/11/dr-square/feed/ 1 Geogebra Dynamic Conics for PreCalc http://blog.recursiveprocess.com/2011/11/02/geogebra-dynamic-conics-for-precalc/?utm_source=rss&utm_medium=rss&utm_campaign=geogebra-dynamic-conics-for-precalc http://blog.recursiveprocess.com/2011/11/02/geogebra-dynamic-conics-for-precalc/#comments Wed, 02 Nov 2011 19:45:57 +0000 Dan http://blog.recursiveprocess.com/?p=1054 Continue reading →]]> For Precalc H, I created a series of dynamic Geogebra files, where the teacher or student uses sliders to control the center and the a (and b) values of the conic. The file calculates and graphs the appropriate conic, foci, directrices, etc.

All the files can be downloaded here.

Sample:

Or you can go to geogebratube to run them in your web browser.

Circle

Ellipse (wide) and Ellipse (narrow)

Hyperbola (Left and Right) and Hyperbola (Up and Down)

Parabola (Function) and Parabola (Not a function)

 

Cheers.

]]> http://blog.recursiveprocess.com/2011/11/02/geogebra-dynamic-conics-for-precalc/feed/ 0