Tuesday, January 20, 2009

Graphing a Picture

Beauty, creativity, mathematics.

I would say that beauty is part of what we were created to display.
"Your people shall all be righteous ... the branch of my planting, the work of my hands, that I might display my beauty." (Isaiah 60:21 ESV)

Several years ago I attended a T-cubed Regional Conference in St. Louis where a teacher, Pam Burke, gave a presentation called "Picture This!" (ppt) about a 'Graphing Art' project. She talked about how she would only give an assignment or project that had maximum educational value for the time spent. Her handout contained impressive examples of pictures students created with their TI-83 or TI-84. I modified this 'Math Art with TI' activity to serve as our final activity in my Exploring Math & Science with TI-Nspire CAS J-term.
The freshmen in the class probably hadn't done much beyond graphing linear functions, but I introduced, briefly explained and required at least 5 of the following different types of graphs:
  • linear (even inequalities)
  • quadratic
  • absolute value
  • square root
  • cubic
  • cube root
  • reciprocal
  • greatest integer
  • exponential
  • logarithmic
  • sine
  • cosine
  • tangent
  • semicircle
  • polar
Using transformations of the functions and equations, many of them created something that reflected their personality. The above picture was done by our school's top golf star. (Speaking of golf and the TI-Nspire, there is a mini golf course algebra activity at this link.)

Students learned how to limit the domain. They asked great questions and had an exciting time exploring. I had the students write out all the functions they had on their screen and I quickly collected their Nspire file using connectivity software.

The TI-Nspire was such a superior learning tool for this activity compared to the TI-84 for several reasons
  1. The TI-Nspire has better resolution - more pixels
  2. The TI-Nspire is much faster. You don't have to wait anywhere near as long for the device to generate the picture. Along with this, I remember students doing this project with the 84 and by the time a modification was graphed they forget what change they made. With the TI_Nspire, the graph and function entry is on the same screen. When you scroll through the function, the corresponding graph is highlighted. (See the above video.)
  3. Teachers who don't have their students use a program on the TI-84 are limited to around 10 functions. There really isn't a limit on the TI-Nspire. It goes from f1(x) to f99(x). You can also use g(x), h(x), etc. You can backspace over the = and replace it with a greater than or less than symbol to do an inequality.
  4. With the TI-Nspire you can even graph polar on the same screen as functions.
  5. Again, unless doing this project as a program on the 83/84, you are locking up your resources. Students can't use their graphing calculator for anything else while working on the project. With the TI-Nspire you save the file and open up a new document if, for example, the science teacher wants students to do something with their handheld.
  6. Limiting the domain is much more intuitive with the TI-Nspire. Using a piecewise function template looks like the mathematics in textbooks. Whereas, multiplying or dividing the expression by a quantity including Boolean operators, may be mathematically educational, but fairly novel.
  7. Collecting their 'program' or screen shots is so much easier with the TI-Nspire. Any of the connectivity options work faster and better than what was used with the TI-84. If you want to get it from their handheld to yours, you don't need to go through 2nd Link-Receive-Waiting-Send. (I made up a handout called Managing your TI-83/84 to help students remember this process.) On the TI-Nspire, just connect the two handhelds and the one who wants to send something clicks Tools and Send and you have it. The Computer Link Software is quick and intuitive. The TI Connect-to-Class Teacher Software steps this up to enable mass distributing and retrieving files from students' handheld. Additionally, at this link it is announced that coming soon is a means of sending, collecting, and automatically grading wirelessly. It is called the TI-Nspire Navigator. This TI-Nspire TI-Navigator will be available Spring/Summer of 2009 in the UK according to this site.
  8. Since the purpose is investigating transformation, the TI-Nspire is impressively designed to dynamically facilitate this. E.g. graph f1(x)=x^2. Press ESC to escape off the function entry line so you can move the mouse to the graph. You can grab the graph in two ways: one will change the spread and the other will adjust the vertex, all while showing the effects of the equation.
  9. TI-Nspire CAS came in handy to at least one student. He wanted to use a hyperbola for the golf tee. He used CAS to confirm his algebraic manipulation in solving for y.
  10. Animation. (See the above video or another experiment in transformations - the US Flag.)
FYI, information about the TI-Nspire Summer Workshop that will be held out our school has been posted. I'm looking forward to that fun-filled week.


Anonymous said...

An example may help to see the difference on your point #5.
On the TI-83/84 you can graph a limited domain by entering y1=-0.5x/(x<-6) or y2=cos(x)+10/((x>-5)(x<5)).
The Boolean operation causes it to divide by 0 when the inequality is not true. Interesting, but an odd concept for students.
This can be done on a TI-Nspire with f1(x)={-0.5x, x<-6 or

darrint said...

That reminds me of something completely unrelated. There's a software game called Perl Golf. In it you take a working program and try to express it more and more tersely until it's a one liner and uses the fewest possible characters. Of course it works in languages other than Perl. Wonder if it would work on a TI?