## Functions Summary## Monday - Friday, July 19-23, 2004
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.
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.
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:
- Lynda is taking pictures of physical models of parabolas and will be writing up a lesson plan. The lesson will have students importing photos into Sketchpad, using transformational techniques to find an equation to fit the functions, and answering questions. Lynda has been getting some ideas from the pictures Joyce posted to the PCMI discussion group last fall, sites recommended by Suzanne, and a site found by Connie.
http://www.drizzle.com/~frostj http://www.joma.org/vol2/articles/wattenberg/JOMA_article/forms_javascript.html http://interactive-mathvision.com/mck/MAAPrep2003/SionFountain/SionFountain.html http://interactive-mathvision.com/mck/MAAPrep2003/MAAPREP03A.html http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/calc2004/appparab.html - Connie (with some help from Joyce) is preparing a lesson to generate parabolas by rolling a ball up, across, and down a grid (cardboard) that is on a slant. She has discovered that her digital camera will take video clips that can be examined frame by frame and that her camera takes 15 frames per second. She was able to estimate the coordinates and find a quadratic regression for the equation. Since time as well as x and y coordinates are involved, parametric equations could be used to model this data.
- Kelley is using Geometer's Sketchpad to produce a dynamic lesson generating parabolas using the focus and directrix (a line and a point not on the line).
- Seth is creating a Sketchpad lesson which uses three initial points to create a simple Bezier curve in the form of a parabola.
Joyce and Jim are providing resources and help where needed. PCMI@MathForum Home || IAS/PCMI Home
With program support provided by Math for America 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. |