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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Textil y sintético
Importado
Caucho único
multicolor Zapato de running con cuello acolchado y Sistema Quicklace con bolsillo de encaje para facilitar el encendido / apagado
Moldeado Entresuela de EVA para amortiguación en urbano o entornos de senderos
3D Agarre para la tracción en varias superficies
Sensifit envuelve de forma segura pie
Sensiflex tecnología para ajuste perfecto
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

Salomon Women s X-Mission 3 W-w 3 Menta Women Athletic/ Deep Dahlia/ Athletic Green X 95e0229

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.


Salomon Women s X-Mission 3 W-w 3 Menta Women Athletic/ Deep Dahlia/ Athletic Green X 95e0229

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

Arc teryx Womens Norvan VT GTX Liberty / Arcturus , Zapatillas deportivas de malla al aire libre de malla transpirable de Onemix para hombre rojo La Sportiva Akyra GTX Running Shoe - Mujer Pizarra / Azufre Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Negro / Blanco / Stealth / Cool Gray Zapatillas de trail WT610V4 para mujer New Balance Gris / rosa Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Negro, negro, blanco , Zapatillas de senderismo V753WW085 para mujer Apex Gris Correpasillos Merrell para mujer 3 Trail Runner Blue Sport , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral , Zapatillas para correr Saucony Koa St para mujer Verde | , Zapatillas sin cordones Flex Merit para mujer Merrell Armada Zapatillas para correr Inov8 Women s Parkclaw 275 GTX Negro / Coral Entrenador ECCO Biom Fjuel Racer Cross para mujeres Petal Trim / Rose Dust / Coral Blush , Zapatillas de running adidas Galaxy Trail W para mujer Gris oscuro / gris Tech / naranja brillante Zapatillas de running NIKE LD Runner LW para mujer Negro / Negro-cool Gris-blanco , Corredor de Trail ventilado Biom Venture para mujeres de ECCO Dark Shadow / Dark Shadow , Zapatillas de running Trail Trail Nomad para mujer Saucony Azul marino , Zapatillas de running NIKE Zoom All Out Low para hombre Crimson total / Crimson total , Inov-8 Roclite 280 Trail Runner para mujeres Verde oscuro / verde azulado / rosa El corredor de mujeres Xa Pro 3D W de Salomon Deep Peacock Blue / Black / Aruba Blue Zapatillas de caminar de yoga minimalista para mujer de ala Vivobarefoot Guijarro , Zapatillas de running Gore-TEX 910v4 New Balance para mujer Pigmento / Porcelana azul , Calzado de running NIKE Zoom All Out Low para hombre NIKE Cargo Khaki / Negro , Zapatillas de running Skechers GOtrail para mujer Taupe / rosa Zapatillas de trail running Inov-8 Trailtalon 235 (W) para mujer Teal , Entrenador ECCO Biom Fjuel Racer Cross para mujeres Coral Blush , Zapatillas de trail running Salomon Speedcross 4 para hombre rojo Salming Trail T3 Zapatos de mujer Orquídea / Rosa Orquídea oscura / rosa , New Balance Women s Summit - Desconocida Trail Running Shoe Verde / 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 Under Armour Women s HOVR Phantom NC Marfil (105) / Acero Zapatillas antideslizantes Skechers for Work para mujer Eldred Blanco Zapato de running ASICS GT-1000 6 para mujer Mid Gray / Carbon / Flash Coral Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel , Sandalias KEEN Clearwater CNX-W para mujer Vino de uva / beso de uva , Under Armour Women s RailFit 1 Negro (002) / Negro , Zapatillas de running New Balance 520v3 para mujer Artic Fox / Black , Skechers Performance On The Go 400 Vivacity Flip Flop Negro Zapatilla Slip-On Hydro-Life W de Teva para mujer Granito New Balance Women s Summit - Desconocida Trail Running Shoe Rosa / negro Drew Shoe La alegría de las mujeres Negro / Estiramiento Zapatos de mujer RYKA Sky Walking Azul marino / Lila , Zapato para caminar de trabajo Flua MBT para mujer Negro , UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Azul verde , Hush Puppies Walker clásico de las mujeres Blanco , ASICS Tiger Unisex Gel-PTG MT Tan / Tan , BareTraps Women s Fergy Slide Sandal Natural , Sandalia Aerosoles Midday Women Slide Tela roja 50 sombras de blanco 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) 2824- Ivory Satin & White Glitter, tacón de aguja , 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 , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo Sansha Salsette 1 Zapatillas de baile unisex rojo Zapatillas de beisbol de softbol de lanzamiento rapido FUSEV1 metal New Balance para mujeres Blanco Mocasín de gamuza Kilty de Minnetonka para mujer Tormenta azul Zapatillas de ciclismo de triatlón Tri X-Speed ​​II para mujer de Louis Garneau 2016/17 - 1487227-096 Blanco Zapatillas Ivana de Spring Step para mujer Beige Bomba de Angel II para mujeres estilo suave Brocado , Cobb Hill Rockport Women s Parker CH Flat Lino , Hush Puppies Chowchow Chukka para mujeres Gamuza añil vintage Zapatillas Pretty You London Azafrán Borgoña

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 Salomon Women s X-Mission 3 W-w 3 Menta Women Athletic/ Deep Dahlia/ Athletic Green X 95e0229 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 3522155
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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.


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