Functions Summary

Monday - Friday, July 19-23, 2004

Monday

Jim brought in some graph paper (from Staples): www.meadweb.com It has space for graphs as well as tables and writing. Jim led us through some examples of parametric functions using graph paper, acting out, physical models, and technology. Connie and Joyce helped with the graphing calculator part. Connie recently attended a one week TI-84 workshop. Jim asked us to describe if and how parametric equations were used in our classroom. Marta uses graphing calculators and parametrics with her Integrated III class. Kelley, Sean, and Joyce use parametrics some in Pre-Calculus mostly at the end of the year. Joyce and Connie have also worked with parametric equations in AP Calculus.

  • Seth is using Sketchpad to produce parabolas by using parametrics and an elbow triangle, moving a fixed ratio along each of the two connected segments and then connecting them.
  • Jim mentioned that Post script language uses parametrics.
  • Connie's calculator tip: If you want to see how your two paths are moving when graphing using parametrics, make the t-step really small.
  • Jim mentioned a way to graph regular polygons on the calculator using parametrics. Connie found this exploration on page 302 of Discovering Geometry: Graphing Calculator Investigation - Drawing Regular Polygons.
  • Jim also mentioned that if you take a parabola and multiply it by any invertible matrix (has to have an inverse - determinant isn't 0), then you still have a parabola.

Tuesday

Functions Group Project Specifications

Parabolas: From introduction to sophistication or the basics and beyond.

Celeste, Marta, Lynda and Donna plan to develop activities and lessons to foster understanding of parabolas from physical models, tables, graphs, and equations. They also plan to work with transformations of equations for parabolas from tables, graphs, pictures, equations, and use of technology. Rani will examine geometric transformations using matrices.

Seth and Kelley are creating a lesson to generate parabolas in an unusual way using three points, ratios, and tangent lines. The lesson will be developed using Geometer's Sketchpad software.

Connie and Joyce are working on preparing introductory parametric activities using parabolas generated from physical models. Graphing calculators and Sketchpad will also be used to further develop the understanding of parametrics.

Thursday and Friday

Group members have been working of their projects and learning a lot about Sketchpad in the process. As of Friday, here is the status of the projects:

Project 1: Lynda, Kelley, Seth, and Connie are generating parabolas using various physical models and generating the equation for the function using technology.

Project 2: Donna, Marta, and Celeste are working on a lesson plan with a new twist on the well used and loved chess board problem. This will be a lesson involving generating ordered pairs, graphing the pairs, and deriving equations from the data. They will be focusing on various payment schemes (traditional doubling scheme, a linear sequence such as 1, 3, 5, 7, 9, ... which would be added up for a running total, squaring each entry such as 1, 4, 9, 16, 25, etc.). Students will be comparing the various schemes and learning about various kinds of functions, their similarities and differences. Sketchpad was helpful in creating a chess game board for the lesson.

Project 3: Rani is creating a unit lesson examining transformations of graphs (rotations and reflections) from patty paper physical models to sketchpad to multiplying using matrices. Jim has had some suggestions for Rani and connections between her project and parabolas and functions.

Joyce and Jim are providing resources and help where needed.

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IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540
Send questions or comments to: Suzanne Alejandre and Jim King

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. 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.