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.

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Cuero
Neutro y muy acolchado.
Perfecto para caminar y realizar actividades físicas livianas.
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


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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

Zapatillas 19359 deportivas para Spira caminar de Spira CloudWalker para mujer con resortes Nimbus/ Carbón de leña/ Menta 73b6c1e

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.


Zapatillas 19359 deportivas para Spira caminar de Spira CloudWalker para mujer con resortes Nimbus/ Carbón de leña/ Menta 73b6c1e

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 Nohea Nubuck - Mujeres Basalto / Tapa Trotters Mujer Simmy Mary Jane Flat Moho Trotters Mujer Simmy Mary Jane Flat Ante azul marino Drew Shoe Women s Tulip Comfort Negro Skechers Go Walk Evolution Ultra Rapids Sneaker para mujer Blanco negro Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Gris oscuro Zapatillas Isla de las mujeres Softinos Lila Zapatillas de deporte Romy de Easy Spirit para mujer Cuero gris oscuro / negro , Sandalia Juno Slide de Vionic Women s Rest Broncearse Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Negro / Microdot , Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Hyper Pink , Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Gris / blanco Drew Shoe La alegría de las mujeres Suede de Oliva / Stretch Ballet de niña de flores Aerosoles femenino Negro exótico , Calzado de senderismo Go Joy 15603 de Skechers para mujer Negro / rosa caliente Zapatillas para caminar antideslizantes Spira WaveWalker para hombre Naval Blanca Botas de mujer Naturalizer Wanya Cuero Saddle Tan , Zapatillas clásicas Vionic para mujer Walker Blanco / rosa Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Gris SAS Womens Sunny Azul profundo Zapatillas Skechers Performance para mujer You-BCA 2017 Azul marino / blanco , Zapatillas deportivas para caminar MBT Jambo (Le) para mujer Denim Blue / Stone Washed Navy , Zapatos atléticos diabéticos Katy Turquoise de Dr. Comfort para mujer Turquesa Zapatillas deportivas para caminar Spira CloudWalker para mujer con resortes Blanco / gris / menta , Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Grey Knit Sandalia Vionic Shelley Slide para mujer Caramelo , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Piso de mujeres de Coral Pies extra anchos Diabético ortopédico Zapatos atléticos Sneakers para caminar gris Zapatillas para caminar Cinch de mujeres Devotion Plus de Ryka Vino / 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 Zapatillas deportivas 574S Sport New Balance para hombre Varsity naranja / blanco NIKE Men s Flex 2017 RN Blanco / Pure Platinum / Cool Gray , Sandalia deportiva Coleman 2.Zerogrand Slide de Cole Haan, cuero negro Cuero negro , Mujer adidas Cloudfoam Qt Racer Negro / Blanco / Carbono , Correpasillos Merrell Women s All Crush Shield Trail Runner Negro / púrpura , Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco negro Salomon Women s Speedcross 4 W Trail Runner Surf hawaiano Kenneth Cole New York Mujer Kingcro Triple Hook y Loop Sneaker Negro , Zapatillas de running PUMA para mujer Ignite Negro / Plata / Turbulencia / Patina Verde , Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Pale Gray / Pale Gray-dark Gray , Sandalia de vestir FitFlop KYS de cuero para mujer Blanco negro Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Juanetes caminando Whitney Womens Sneakers Athletic Zapatos Negro , Sandalia de tira de Ghillie Bre-Italy Bella Vita para mujer Cuero negro , Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Rojo-b (estilete) , La sandalia Beglad de mujer FLEXX Cognac Vacchetta SAS Mujeres, Sanibel Thong Sandal Oro Metálico Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco , Zapatillas de tenis Prince Advantage Lite para mujer blancas / rosas Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Obsidiana / obsidiana-blanca-pista roja , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Negro , Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Plata / Metálico , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 1700- Beige Nubuck / Gold Trim Zapatos muy finos EKSA7001 de la danza del salón de baile de las señoras con el talón de 3 \ Light Tan Velvet Reebok Zapato de baile Pilox 1.0 para mujer Negro / Vitamina C / Gris Ceniza / Blanco , Bota de montar Xandy para mujer LifeStride Poliuretano negro , 3N2 Mofo Turf Trainer Negro , J.Renee Womens Bilboa Plata blanca , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5004- Gold Sparkle , Botín tobillo para mujer de Seychelles Estaño TOMS - Slip-On de lienzo clásico para hombre Lienzo lavado con óxido

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.

Zapatillas 19359 deportivas para Spira caminar de Spira CloudWalker para mujer con resortes Nimbus/ Carbón de leña/ Menta 73b6c1e - 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 Zapatillas 19359 deportivas para Spira caminar de Spira CloudWalker para mujer con resortes Nimbus/ Carbón de leña/ Menta 73b6c1e 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 770932
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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.