New blog series -- 'Little known free math gems"

 "Little known free math gems"

These are tiny, free, and surprisingly powerful programs for mathematics students from algebra through college-level which can be used on virtually any laptop or desktop computer, even old ones.   

All of my screenshots will be from a laptop running Windows Vista Pro with 1.88 GHz on 1GB of RAM.

I'll be reviewing the following: I hope to do one review each week. 

1. DPGraph
2. GrafEq
3. MVT
4. Winplot
5. dfield & pplane
6. Linear

Also I plan to include:

1. screenshot techniques (cropper, mspaint)
2. useful & interesting wolframalpha commands
3. desmos
4. Geogebra resources
5. Touch Trigonometry

If you have any specific requests or know of a program I have not mentioned, please feel free to contact me. 

The many ways to plot implicit functions

There are many different ways to plot accurate and complicated implicit functions, for free and without eating up your processor.

Let's look at this implicit curve, for example:

ax^2 + bxy^3 = c  

where a, b, and c are real numbers.

In Geogebra, you can click the sliders button to create a, b, and c then type the function exactly as it appears  above in the input bar. Drag a, b, and c (or animate them if you wish by right clicking) to see the function change.

Click images to enlarge.

Create the sliders first then type the equation.

Drag the sliders to see the changes in the graph.

We can also use http://www.desmos.com to do the same thing. Desmos has the nice feature of highlighting "points of interest" (in this case, the x-intercepts are highlighted). You can also share your graph easily by clicking the blue icon at the top right and choosing your share-tool of choice.


Click images to enlarge.

Click "all" to add sliders for each variable.

Drag the sliders to see the graph change.

Share your work in a variety of ways.

And for just about any calculus course, we are done.  Geogebra and Desmos are great for polynomials but if you want to graph something like cos(xy) = c  where c is a constant you will need a different set of tools.

Gnuplot is a free, command line based, graphing program. It will not plot implicit functions by default.  It's possible to hack it by plotting a 3d function with level curves projected in the xy plane. Here is the code needed to plot cos(xy) = 1. The line "set cntrparam levels discrete 0" means "plot one contour line at z = 0".

set view 0,0
set isosamples 50,50
set contour base
set cntrparam levels discrete 0
unset surface 
set grid
unset key 
unset ztics
set xlabel "x"
set ylabel "y"
set xrange [-pi:pi]
set yrange [-pi:pi]
f(x,y) = cos(x*y)
splot f(x,y)


Click image to enlarge.

The output of the above code.

There is an easier way of doing this and here is where GrafEq, a tiny, free windows program, really shines. Just type in your function and go.


Click image to enlarge.

Here is a sampling of some of the beautiful implicit graphs you can quickly, easily, and accurately plot with this tiny little-known program.

One can also type

plot cos(x*y)=0 for x=-pi...pi, y = -pi...pi

into http://www.wolframalpha.com and obtain the following output:


Click image to enlarge.

Communicating mathematics online

How does one quickly and easily communicate upper-level mathematics entirely online?

I never interact with my students in a physical classroom. All communication happens via email, virtual meeting room (Adobe Connect), Skype, and/or phone. All of the course materials are hosted on Moodle, but I do not feel that having my students use multiple choice (or free response limited to a few characters) for all assessments is sufficient to gauge their understanding and it's certainly not sufficient to engage them in a mathematical dialogue aimed at sharpening their exploration and written communication of mathematics.  

All of the graded assessments in the AP & College math courses I supervise and instruct are free response so I can clearly see a student's thought process and offer feedback as needed. There is a great deal of material per student per course and the logistics of communicating all of this marvelous math can become overwhelming. 

Fortunately, there are two main options for students and instructors which work very well, both are illustrated below. 

1. Write work by hand, use a free app called CamScanner to send to instructor. 

• Pros
• Quick and easy -- write, scan, email. 
• Automatically color-balances & arranges images into albums (for sending an entire test at once)
• Saves to pdf, jpg, tiff, or other formats, each page ~ 200 kb 
• Very high quality image output
• Cons
• Requires an Android or iOS device, so accessibility may be an issue.
• Can be difficult or slow to edit without special software (Foxit reader is free and has nice PDF editing features) 

Example output:

CamScanner (Android | iOS ) (click to enlarge)

Detail of above (click to enlarge)

2. Use free & open-source LaTeX to typeset work. 

• Pros
• Forces students to articulate their reasoning clearly (crossing out, drawing arrows, and writing all over the page simply won't transfer.)
• Output is beautiful
• Graphics can be integrated very easily
• File size is very tiny (5 kb + images)
• Easy to edit
• Typing math becomes as natural as typing English
• Professional documents can be typed for any course, not just math!
• Learn it now instead of later!
• Cons
• Heavy up-front time investment to learn the syntax. I have a continually updated page devoted to getting students started featuring: videos, templates, and additional resources. 
• Installation of a LaTeX distribution can be tricky, but fortunately there is http://www.sharelatex.com
• Errors can be difficult to spot at first and frustrating to correct without help from an experienced user. 

Example Output: 

LaTeX (click to enlarge) download source here.

Detail of above. (click to enlarge)

Professional goals for 2013

• Improve my tweet/blog ratio. 

I have roughly ~4,600 tweets but only 23 blog posts (counting this one).  I started this year knowing that I wanted to have a professionally oriented twitter and blog and I've managed to do both.  Upon reflection it seems that I can still tweet daily with content being mostly info that comes across my reader feed and hashtag chats but should designate at least one day per week to blog.

The focus of my "reactionary" posts should be the topic du jour of the hashtag chat, my reaction to an interesting blog post, article or paper, etc.

I should also seek to create original content.  I am endlessly creating and tweaking content for my students and staff and much of what I do could likely benefit others in my field.  This dovetails into my next resolution...

• Participate more in "real world" professional development

One of the downsides of doing one's work entirely online is how easy it is to fall out of real-world communication with... everyone and everything. Looking back on 2013 I am thrilled that there is such an active education and technology community in Baltimore, MD but thoroughly disappointed in how few events I attended in person (none!). This has to change in 2013. 

• Be more of a leader

My professional position seems to be rather uncommon. I work with highly gifted math students in rolling-admissions online courses that are based on an individual mathematical dialog between student and instructor.  I also develop curriculum for these courses and supervise instructors. After four years of this (as of Nov 2008) I feel like I have a very good idea for what works and what does not.  It's time to articulate these ideas. 

I look forward to a year of learning and professional growth via engaging with educators in person and online!

landscape panoramas before & after snowfall

I have the pleasure of spending my holiday in rural Indiana, specifically Boonville. Below are some photos I took before and after some snow (with permission of the landowners, of course). If there is any beauty contained within these photos, it is because the landscapes and camera did all of the work. I hope you and your family are having a safe and happy holidays.

Sincerely,
Joe

Before: 2012 12 25 
After:  2012 12 26

Click to enlarge each set of images.

Here are some others I took on the 26th which don't have a "before" shot. 

Click to enlarge.

A lonely barn on the way to a quiet wooded path.

The road goes ever on...

The land is so flat in Indiana.

A spectacular half-frozen pond in the woods.

Another view of the pond.

 My next post will be math related, I promise.

Volunteer Juggling Lessons

Juggling has been my passion since I was 11 years old and every time I go out and give a workshop or lesson, planned or impromptu, I can't help but wonder if I am unlocking the love for juggling in people who didn't even know they had it. After nearly 18 years of juggling, I figure the law of large numbers is on my side and there is at least one such case out there somewhere, even if I don't know about it.


If you are interested in having me give a juggling workshop at your school or for your event, please feel free to contact me.

mathteacher1729 at yahoo dot com

At a recent hike + cookout I gave an impromptu juggling workshop. (meetup.com group)

Here are some of my adventures from 2012.

Sat 11 Nov 2012 -- Univ. of Maryland College park's Kappa Delta sorority hosted the Girl Scouts for International Girl's Day.  One of the activities emphasized physical activity and fitness and I worked with a few of the UMD Juggling club members to lead a juggling workshop.

Thurs 25 Oct 2012 and Tues 5 June 2012 --  Maryland School for the Deaf (Columbia campus).

In June I was invited to give a short juggling workshop to the 4th and 5th graders of MSD.  A member of the UMD juggling club joined me and it was a great day.  The students were very receptive to it all and it was surprisingly easy to demo and teach juggling without saying a word. One of the teachers acted as a translator but it was hardly needed, as juggling is very visual. The message I had was very simple

• Juggling is fun!
• The more you practice, the better you get. 
• Juggling is for boys and girls. 
• You can juggle on your own or you can juggle with friends. 

I explained that when I started I dropped all the time (still do) and that I'm still learning new moves even though it's been about 18 years since I started.

The students were very eager to get started and there were a few who really caught on quickly! I was invited back again because it was a great physical activity with a positive message and it did not require any sound.

In October there was a large haunted house, the only one in Columbia, from what I understand due to various fire and safety restrictions.  The school ends up making lots of money which goes to help the students and nearly 600 people, hearing and deaf, show up. There is no way for them all to go through the haunted house at once, so they go in small groups of 5 or 10 while everyone waits in the gym for their turn.  There are lots of activities in the gym, and one of them was -- you guessed it, juggling lessons. Several students remembered me from my last visit and were eager to juggle again.

Possibly my favorite moment was a student who really got a kick out of throwing a ball up and then I would catch it while juggling. I would end the pattern by throwing the same ball back to him and he would catch it.  The first time we did this he turned heel and ran to the opposite side of the gym to show his mom.  One of the teachers at the school said this student was having self-confidence problems and the juggling made his evening and helped him socialize the next day with other students because he had a happy story to tell. There were several other students who spent most of their time juggling in the gym while juggling and another teacher said that they are ordinarily rambunctious and disruptive, but they seemed to be more or less under control while juggling.

16 Sept 2012 -- MD Renaissance Festival. I always bring my props with me wherever I go, and this was no exception.  There was as "juggling school" set up and I asked if it was ok if I helped.  The workers gladly said yes, and I gave them my card for next year then set up shop.  At the Ren Fest there was an interesting mix of excited young kids who were eager to juggle and ... their slightly drunken parents. ;-p All in all it was a fun and kid-friendly event.

Pre-June 2012 -- Mostly at the Inner Harbor, Baltimore MD.  During the summer I would juggle at the inner harbor on a nice weekend.  People would stop and watch and I would happily show anyone interested some basics.  Every now and then I would meet someone who already knew how to juggle and I would show them some more advanced moves based on what they already knew.

The Electoral College

As a mathematically literate citizen, my brain hurts every time I contemplate the electoral college method of selecting a president in the USA. Below are some quick facts and graphs which communicate how ineffective and inappropriate it is for a modern functioning democracy.

First of all, major props to CGPGrey for this YouTube playlist on the electoral college featuring the following three videos:

How the electoral college works. 

The trouble with the electoral college -- all votes are not equal. 

and

What if the electoral college is tied?

Watch them all. 

CGPGrey's videos are the best. They are succinct, well-researched, entertaining, and thorough summaries of the absurdity which is the electoral college. Votes are literally unequal across states. Here is an interactive map showing the ratio between population and electoral votes per state.  

Finally, have a look at these maps which distorts states by the amount of money spent for their votes.  

The USA needs election reform if is to reach anywhere near its full potential as a democracy. 

I haven't even addressed gerrymandering or Citizen's United...