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.

Charles David Jane by Charles David Mujeres inquietantes Mary inquietantes Jane Microsuede negro 1e60f89

Cork ^ Suede
Heel mide aproximadamente 3.75 pulgadas
Indoctrinate un nuevo esencial en la venida sazonar con el simplemente elegante Charles por Charles David y el comercio; deslizamiento en cuña.
Microsuede, entrecruzado superior.
Deslizamiento diseño.
Round-toe silueta.
sintético forro y plantilla. Ligeramente acolchado plantilla. cuña de corcho y tacón de plataforma. Sintético suela.Importado.Medidas: Altura del tacón: 3 3 & frasl; 4 in Peso: 10 Plataforma oz Altura: 3 y frasl; 4 en mediciones de productos fueron tomadas usando tamaño 7.5, ancho M. Tenga en cuenta que las medidas puede variar según el tamaño. Peso de calzado es basado en un solo artículo, no un par
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

Charles David Jane by Charles David Mujeres inquietantes Mary inquietantes Jane Microsuede negro 1e60f89 - 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

Charles David Jane by Charles David Mujeres inquietantes Mary inquietantes Jane Microsuede negro 1e60f89

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.

Charles David Jane by Charles David Mujeres inquietantes Mary inquietantes Jane Microsuede negro 1e60f89

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

Zapato de malla Nohea de OLUKAI - Hombre Serpiente natural , Sandalia Marcy de mujer de caja amarilla Negro , Sandalia de Adinis de la mujer de Lucky Brand Palo de rosa , Sandalia Knoll para mujer Marc Fisher melocotón J.Renee Womens Soncino Cuero metalizado Taupe , L Artiste de Spring Step Women s Style Tapestry Sandalia de cuero Blanco Marrón Multi Revere Miami sandalia de confort para la mujer w. Rojo / Croc , NAOT Kayla Elegant Sandalias de mujer Charol negro L & rsquo; Artiste de Spring Step Cloe Sandal para mujer Negro Sandalia de vestir Townsend de Eastland para mujer marrón , Zapato de malla Nohea de OLUKAI - Hombre Dark Java / Rojo Sandalia Pamela Mujer Mephisto Scarlet Bucksoft , Sam Edelman Womens Barry Piel de serpiente Royal Natural Print La sandalia de gladiador FLEXX Women s Band Together Denim nubuck Sandalia Teva Encanta para mujer Gris pardo , Melissa Women s Diane + Jason Wu Slides Marrón / negro , Birkenstock Betula Licensed Womens Leo Birko-Flor Basic Navy , Comfortiva Mujer, Faye Sandalia de tacón medio gris Sandalia Retana Wedge de Vince Camuto para mujer Azul nocturno , OTBT Nomadic Peltre claro , Easy Street Adagio Wedge Slide para mujeres Patente / estiramiento blanco Sandalia casual destello de las mujeres de ECCO Negro , Entrenador de Mujeres Cassidy Cuero azul acero , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Quilate , Drew Womens Element Nubuck negro , Sandalia de cuero Showrunner para mujer de Nine West Cuero negro , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro Good-memories 2018 verano mujer zapatillas plano antideslizante interior informal peludo playa sandalias Gris claro , Sandalias de cuero de Viv Viv de TOMS Chambray negro texturizado ,
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 Zapatillas de running Saucony Women s Triumph ISO 2 Coral / Plata Zapatillas de running Nike Kids Air Max Zero Essential GS River Rock / Black-CobbleStone , Zapatillas de deporte casual Gazelle de adidas para hombre Colegiado Real / Blanco / Oro Met. , Converse Chuck Taylor All Star Seasonal Color Hola Papel rosa , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Azul La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Esmeralda / menta , Calzado de trail running Inov-8 para mujer Terraclaw 250 Negro / Berry / Azul , Finn Comfort Women s Ikebukuro Oxford Patagonia atlántica , Skechers GOrun 5 Zapatos de mujer Morado Negro / verde azulado Zapato para caminar WW411v2 New Balance para mujer Negro Sandalias de mujer Havaianas Slim Sandalia Tropical Blanco negro Sandalias de mujer de Britton Leather Open Toe con punta abierta Negro Sandalia de vestir de cuero Nine West para mujer Light Gold Synthetic , Ryka Womens Divya Azul / rojo / blanco Zapato de skate Diadora N-92 Verde blanco Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Black Silver , Zapato de caminata Skechers Performance para mujer Go Walk 4 Pursuit Negro , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Blanco Sandalia de cuña OAHU de China Laundry Women Basil Suede Sandalias planas para piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural , Carlos by Carlos Santana Sandalia de mujer Zoey Slide Negro Zapatos de voleibol Mizuno Women s Wave Tornado X. Negro / Plata , Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra 02 Zapatillas de ciclismo de carretera Fizik R5 UOMO BOA Negro / gris oscuro , LifeStride Women s Lively 2 Pump Sonrojo , Zapatilla deslizante Lilah de Daniel Green para mujer Armada Botín Hayes Gore de Cole Haan para mujer Ante de piedra de hierro , ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Pisos tachonado Zapatillas Zapatos Azul claro , Bota Darcy para mujer Easy Street Negro / Negro Cocodrilo / Gore Hush Puppies para mujer, Adee Chardon Botines Cuero rojo oscuro

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.


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.

Charles David Jane by Charles David Mujeres inquietantes Mary inquietantes Jane Microsuede negro 1e60f89 - 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 Charles David Jane by Charles David Mujeres inquietantes Mary inquietantes Jane Microsuede negro 1e60f89 it out, and I got a chance to try out this new graphics software.


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.

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

Geometry Construction Reference

Thirteen elementary straightedge 5471305
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.