Park City Mathematics Institute
Visualizing Functions
Project Abstracts

Drafts of Project Files (password required)

Exploring Newton's Method through Geometer's Sketchpad®
Gary Meinhardt and Stephanie Ruprecht

To explore and observe the recursive nature and iterative tendencies of Newton's Method through numerical and graphical processes in order to determine zeroes of various functions. Through this process, the students will create recursive formulas that will enable an easier method for getting infinitely close to the actual value of any zeroes on a given curve. Also, drawbacks of Newton's Method will be explored and discussed.

Hooked on Conics!
Peter Kaczmar and Lars Nordfelt

Hooked on Conics! is an activity to help precalculus students discover conic sections and their properties. Most of the activity uses files that require the software application The Geometer's Sketchpad®. Usually students would work as pairs (2 students to each computer) and enter their responses on the computer. Students will investigate the geometric definitions, the rectangular and polar equations, and the effect of parameter changes on the graphs. They will study the ratio (the eccentricity) of distance from any given point to a focus to the distance from the given point to the corresponding directrix and will then be able to define all conics in terms of this ratio.

An introduction to periodic functions using a monochord
James Stallworth and Susan Antonsen

Using common household materials, students will construct a monochord, a one-stringed instrument, which they will then use to investigate the relationship between string length and musical octaves. We will include a video demonstrating the construction of the monochord along with step by step instructions. After the construction and initial exploration, they will discover the relationship between ratios of string length to the notes on Pythagoras' and Ptolemy's musical scales. The students will then use their musical instrument and the calculator based laboratory to see what the sounds look like in terms of period, amplitude, and frequency.

The Mathematics of Infectious Diseases
Calvin Armstrong, Jerry Neidenbach, Jeff Reinhardt, and Alieze Stallworth

Students explore a recursive SIR model that is based on the differential equations model developed in 1927 by W.O. Kermack (public health physician), and A.G. McKendrick (biochemist) for predicting the behavior of various epidemic diseases. Students engage in several activities that use the SIR model including a spreadsheet simulation. At the conclusion of the activities students are able to determine the smallest percentage of the population that should be immunized in order to prevent the spread of a disease.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.