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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Sintético
Importado
Caucho único
Rearfoot GEL acolchado Sistema: Atenúa choque durante el impacto fase y permite para una transición sin problemas a medio cuerpo.
SpEVA Entresuela: material Mejora el rebote características y disminuye la entresuela avería.
DuoMax Sistema de apoyo: Una doble densidad sistema de la suela intermedia posicionado para mejorar soporte y estabilidad.
Trusstic Sistema: Reduce el peso de la unidad única mientras reteniendo el estructural integridad de la zapato.
Trail Suela específica: Las orejetas invertidas proporcionan cuesta arriba y cuesta abajo tracción en todos tipos de terreno
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

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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.


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

Vionic Womens Cypress Trail Walker Gris / verde azulado , Calzado de running Columbia Montrail Caldorado III para mujer Earl Grey, azul costero Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker GTX W para mujer Negro / carbono / verde ceniza Zapatilla de deporte de velocidad vertical para mujer Vasque Gárgola / verde , Timberland Flyroam Go Lace Chukka para mujer Negro azabache , Zapatillas de trail running New Balance WT00 Minimus Zero v2 para mujer New Balance Púrpura / rosa , Salomon Women s X-Mission 3 W-w Imán / Negro / Rosa Violeta Zapatillas de running Salomon Women s Speedcross 4 GTX W Trail Cereza de Barbados Merrell Siren Edge Hiker para mujer Negro / púrpura IEKLFKFVF Popular Yellow Checkered Scrap Sneakers para mujer Nmd Soft Spot Black 2018 Zapatillas Merrell Women s Vapor Glove 2 Barefoot Trail Running Libertad , Correpasillos Merrell Women s All Crush Shield Trail Runner Negro / púrpura Zapatillas de trail running Merrell para mujer Pace Glove 3 Negro , Zapato ATA 3 Challenger de mujer ONE ONE de HOKA Joya persa Skechers Women s Go Trail 2 - Zapatillas de senderismo con agarre Gris / rosa Zapatillas de trail WT10V4 para mujer New Balance Negro Correpasillos Merrell para mujer 3 Trail Runner báltico Zapatillas sin cordones Flex Merit para mujer Merrell Armada Corredor de trail Speed-Wak W-W de mujeres Salomon Coral Punch / Negro / Infrarrojo Zapatos de entrenamiento cruzado Inov-8 Unisex F-Lite (TM) 240 Gris / Verde azulado / Berry Zapatillas impermeables Columbia para mujeres Trail Redmond Guijarro / iluminado por el sol , Zapato casual bajo NIKE Lupinek Flyknit para hombre Cargo Khaki / Mica Green , Columbia Montrail Women s Variant X.S.R. Gray Ash, Phantom Purple Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Saúco , Zapatillas de mujer Salomon X Ultra 3 W para excursionismo Dirigir Zapatillas para correr Saucony Women s Cohesion 10 Gris / verde azulado / cidra , New Balance Women s Summit - Desconocida Trail Running Shoe Verde / negro , Zapatillas de trail running Ryka Women s Intent XT 2 SR Negro / plata cromo Zapatos Salomon Sense Pro Max Running Trail para mujer Virtual Pink, Black, Enamel Blue
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 Sandalias OLKAKAI Ohana - Hombre Dark Shadow / Dark Shadow Nike Men s Air Presto Essential Negro / Verde pino / Gris neutro Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Semi Congelado Amarillo / Noble Tinta Blanca , Calzado de running NIKE RN 2017 2017 para mujer Anthracite / Pink Blast / Black / Cool Gray , Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral , Zapatillas de running adidas NMD_CS2 PK Originals para mujer Lino / Correr Blanco , Trotters Mujer Simmy Mary Jane Flat Grafito , Zapatillas de viaje Proather para mujer II Teal, gris Brooks Womens Ravenna 9 Ebony / Divapink / Blanco Zapatilla de pies Cush N de Prop & eacute; t para mujeres Arena , Zapatillas para caminar Nell Ryka para mujer Taupe / Blanco Sandalia Kristina Fisherman del Dr. Martens para mujeres Broncearse Zapatillas de running NIKE Womens Huarache Run Ultra PRM para mujer Negro Negro Zapatillas de running Duramo 6 W adidas Performance para mujer Core Black / Tegrme / Ftwwht , Sandalia Minnie Trotters para mujer Nubuck negro Sandalia de tacón Edline de mujer Ivanka Trump Coñac Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Mayor ventilador gris / blanco , Zapato Cross-Trainer X Conviction Women de ASICS Aqua Splash / Diva Blue / White Zapatillas de golf adidas W Adipure DC de adidas para mujer Core Black / Gold Met./Core Black Koolaburra de UGG Women s W raychel Slide Sandal Sonrojo Sandalia Ryana Marilyn Slide para mujer Blanco negro , Sandalia plana Sofie de Badgley Mischka para mujer Marfil , Sandalia Heather Slide de mujer Vía Spiga Midnight Suede Sandalia de gamuza de gamuza violeta Taryn Rose para mujer Negro 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) 2828- Satén dorado satinado Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Bronceado oscuro Bota occidental con flecos y tachuelas Roper para mujer Black / Crystal Stud , Funtasma de Pleaser Men s Jazz-02 Platform Oxford Negro CLARKS Mujer Ashland Bubble Slip-On Metálico

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.

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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 ASICS Gel-Kahana 8 B077XNY8QN Trail Runner Runner femenino Dark Steel Glow/ Pink Glow/ Mint 1f00abe 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3591155
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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.