(skip to content)

• Homepage
• Publications
  • Math-related
  • TeX-related
  • Software
• Dissertation
• Curriculum Vitae

• Jacobians of Curves
• Computation
• ABC Conjecture
• Core Courses
• Qualifying Exams
• Preliminary Exam
• Mechanical cryptography
• Fourier Series
• Mentoring
• Neural networks

• Quotations
• Genealogy
• Bookmarks

• Elliptic Curve Seminar Website
• Computing Website

Talks | Fourier Series for High School Students

Short talk (20 minutes) given at San Antonio meeting January 1999.

Abstract: As part of the activities of the Southwest Regional Institute in the Mathematical Sciences, graduate students at The University of Arizona organized special workshops for high school students and their teachers. This presentation will focus on the workshops on Fourier Series. We show how one might geometrically motivate the well-known formulas for Fourier Series coefficients, and we demonstrate software that we developed for the hands-on aspects of the workshops. In one of the activities of the workshops, the high school students used this software to identify telephone touchtones by their corresponding fourier spectrum.

• Slide 1: SWRIMS High School Workshops
  Over the last few years, the SouthWest Regional Institute in the Mathematical Sciences has been organizing workshops for high school students on a variety of topics. I'll give a quick overview of SWRIMS activities, hint at why we run the workshops, outline the format of the workshops in general; but mostly I intend to give all of that short shrift, so that we can focus specifically on our two workshops on Fourier Series.
• Slide 2: SWRIMS Activities (handwritten)
  I should type this slide up!
• Slide 3: High School Workshops (handwritten)
  I should type this slide up! The time discrepancy comes from the hour for lunch. Also, we have run some workshops without paying anyone, so those are close to free.
• Slide 4: Touchtone patterns
  Here are the wave forms from telephone touchtones. Question: if you are given only the wave form (you hear it, for example), how do you determine which button was pressed? [Maybe compare to the denser images on subsequent two slides.]
• Slide 5: Periodic functions (handwritten)
  What are waves? Mathematically, they are periodic functions. So let's study those for a while, then come back to the question of identifying touchtones. Here are some examples of simple waves, and a more complicated wave obtained by adding them together. Moral: we can build complicated waves from simple waves. Question: do we obtain all complicated waves by this process?
• Slide 6: Amazing Fact (Fourier) (handwritten)
  
• Slide 7: Picture of complicated wave (from Aaron)
  Here's a complicated wave. According to Fourier, it must be some additive combination of simple waves. We played a game, in which teams tried to figure out the coefficients, simply by guessing around.
• Slide 8: Lots of waves (guess the Fourier coefficients!)
  
• Slides 9-12: Finding coefficients geometrically
  Of course, after obtaining the formulas, we also go back and use trig identities to prove that the integrals are zero.
• Slide 13 (same as slide 8): Lots of waves
  Now the game can be played. But we discovered that most students were still just guessing around.
• Slide 14: Welcome to the Fourier Workshop
  
• Slide 15: Touchtone 1
  
• Slide 16: Touchtone 1, one coefficient enabled
  
• Slide 17: Touchtone 1, other coeff enabled
  
• Slide 18: Touchtone 1, both coeffs enabled
  
• Slide 19: Touchtone 5, both coeffs enabled
  Compare the peaks in the spectrum; these indicate which touchtone is pressed.
• Slide 20: Touchtone grid (handwritten, not yet made)
  Show how each row and column has a frequency, so each touchtone consists of two overlayed pure tones.
• Slide 21: Dirty satellite image (from Andre)
  We were using canned software, students suggested ideas, but Andre controlled the software.
• Slide 22: Satellite spectrum with obnoxious line
  
• Slide 23: Spectrum with line removed
  
• Slide 24: Cleaned-up satellite image
  
• Slide 24: Spectrum of sound recording
  Student made secret recording in corner, then we added noise, gave it out to everyone.
• Slide 25: Spectrum of sound with noise
  They then had a huge data array in Matlab, and we showed them various commands for graphing, finding large data values, changing values, multiplying through with filter functions. Their goal was to clean up the sound.
• Slide 26: Bart Simpson (Damned if you do, damned if you don't) with noise and filter
  Don't show this one, unless people ask about the filter.