Whistler Alley Mathematics

These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.

Sandalias Swiftwater mujer Mesh Swiftwater de Sandalias Crocs para mujer Armada 86c483e




Parte superior de malla
Importado
Sintético único
Eje mide aproximadamente 2.5 desde el arco
Apto para el agua sandalia para el verano aventuras.
Transpirable, malla de secado rápido parte superior.
Ajustable cierre para un seguro ajuste.
Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel


Sandalias Swiftwater mujer Mesh Swiftwater de Sandalias Crocs para mujer Armada 86c483e - movie-store.online

Recent Changes

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015

Sandalias Swiftwater mujer Mesh Swiftwater de Sandalias Crocs para mujer Armada 86c483e

Updated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.
The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.
This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.


 Includes GSP-3 files

 Includes GSP-4 files

 Includes GSP-5 files Contents:

The Brachistochrone

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.


Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.


Sandalias Swiftwater mujer Mesh Swiftwater de Sandalias Crocs para mujer Armada 86c483e

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

OluKai Ohana Tropical / Hielo Sandalias sintéticas Teva para mujer Terra Fi Lite Atlántico Norte Sandalias adidas de mujer Adilette Cf + Armad Athletic Slide Negro / Tech Rust Blanco , Sandalia Drew Element para mujer Nubuck marrón , Sandalia deportiva Tirra para mujer Teva Azul real Sandalias planas Eileen Fisher para mujer Sport-Nu Armada , Merrell Women s Around Town Print Sandalia atlética Aguanieve Sandalia de mujer KEEN Rose Blue Opal / Provincial Blue Calzado casual Slip-On de Bernie Mev para mujer Negro , Oasis Women s Zoey Fisherman Shoes marrón , Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva , Sanda Verra Teva para mujer Deep Teal , Skechers Performance Women s The Go 400 Radiance Flip Flop Azul marino / blanco Sandalias de mujer con tirantes Peep Toe de Bordello by Pleaser para mujer rojo , Wolky Comfort Jewel 60200 Charol gris Sandalia de mujer Drew Alana Blanco negro , Sandalia KEEN Women s Venice H2 Midnight Navy / Coral caliente , Chaco Mujer Zcloud Sport Sandal Charlie Peach , Encaje de quimera de Columbia para mujer Zuraqua , Wolky Comfort Jewel Neutral Multi Color Fantasy Allrounder de Mephisto Women s Larisa Sport Sandal Espresso Tech Nubuck / Black Lycra , Sirena de mujer Merrell Flip Q2 Athletic Sandal Aluminio Sandalia NIKE Benassi JDI para mujer Blanco negro KEEN Sandalias Newport Eco-W para mujer Negro / Imán Sandalia KEEN Clearwater CNX para mujer Celestial / Vapor , Sandalia de susurro de mujer Keen Beet Red / Honeysuckle , Sandalia KEEN Women s Venice H2 Midnight Navy / Coral caliente CLARKS Sandalia plana Brizo Ravena para mujer Armada Sandalias de tirantes Sunstone para mujer Merrell Negro ,
trigonometry, and probability. Fair warning: The discussion revolves about a figure that appears only when the sketch is animated. Without Geometer’s Sketchpad, this one will make little sense at all.


Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.


The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.


Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.


Inversion Geometry Wolky Comfort Jewel Bronce Picasso Crash Teva Terra Sandal para hombre Cross Terra Black , Zapatillas de running NIKE Mens Air Zoom Structure 19 Foto Azul / Eléctrico Verde-místico Verde , NIKE Women s Tanjun Lobo gris / blanco New Balance Wrl247hl para mujeres Negro Zapatillas de running adidas Women s Rockadia W Trail Trace Púrpura / Blanco / Core Black , Zapatillas de running ONEMIX para hombre y mujer Zapatillas de deporte para correr Pink Bule Zapatillas para correr Saucony Women s Triumph 11 Gris / verde azulado / cidra , Sandalias de cuero de Viv Viv de TOMS Pom natural del cáñamo Sandalia de vestir Rubie de mujer chino de lavandería Terciopelo de humo , Bomba Jude para mujer Adrianna Papell Mosaico de oro Lame Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa Zapatillas de running EM Tri N1 v2 para mujer Pearl iZUMi Blue Atoll / Electric Pink Zapatillas de running a campo traviesa Fastwitch 8 para mujer Saucony Verde azulado / blanco Zapatillas para caminar Propell Mujer Travellite MJ Plata Bota Deia para mujer de TOMS Black Suede Radial Perf / Farren , Sandalia de punto Ysidro W de Teva Women marrón Zapatos de mujer Rockport Ts Wp con cordones Negro , Zapatillas de tenis NIKE Women s Air Zoom Prestige Lt Blue Fury / Plata metalizada / Neo Turquoise Zapatilla de deporte Fierce Bright Cross-Trainer para mujer de PUMA Amarillo de seguridad / amarillo de seguridad , Zapatillas para caminar Propell Mujer Travellite MJ Gris pardo , Sandalias de golf Sandbaggers Tango para mujer Negro , Zapatos muy finos EK1692 de la danza de salón de baile de las señoras de las señoras con 2.5 \ Cuero plateado y malla de carne , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Phantom / Gunsmoke Summit Blanco ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ 5004 Copper Stardust y Dark Tan Gold , Impermeable Jessa de Columbia para mujer Tabaco / oxford Tan Alpargatas antideslizantes de lona clásicas para mujer de TOMS rojo Sandalia Birkenstock de cuero Florida para mujer Cuero aceitado negro Botas sintéticas Vivobarefoot para mujer Gobi II Broncearse , Bota de traje seco Scubapro - Negro ,

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.


Linkages

Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.


On the Square

The challenge: Given four arbitrary, unique points, A, B, C, and D, construct one line through each point such that the four lines form a square.


Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.
The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.


Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.


Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.


Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.


The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.


Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.

Sandalias Swiftwater mujer Mesh Swiftwater de Sandalias Crocs para mujer Armada 86c483e - movie-store.online


Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.


Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured Sandalias Swiftwater mujer Mesh Swiftwater de Sandalias Crocs para mujer Armada 86c483e it out, and I got a chance to try out this new graphics software.


Resources


The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2633798
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and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.

Back to Whistler Alley


Last update: March 16, 2016 ... Paul Kunkel whistling@whistleralley.com
For email to reach me, the word geometry must appear in the body of the message.