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
Importado
Cuero y suela de goma
Eje mide aproximadamente no aplica del arco
Zapatos, Zapatillas, Bombas etc.
Mula
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

Estuardo Mira Cush Mule Mule de Stuart Stuart Weitzman Off White Cush Nappa a25a9df

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.


Estuardo Mira Cush Mule Mule de Stuart Stuart Weitzman Off White Cush Nappa a25a9df

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

YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Amarillo Birkenstock Betula Licensed Womens Leo Birko-Flor Basic Black Sandalias FitFlop Bumble de mujer con diapositivas en cristal Negro Easy Spirit - Sandalias informales para mujer, Everso Armada , Sandalia KEEN Ana Cortez W para mujer Negro Negro , Bomba de vestido Nika BeautiFeel para mujer Floral negro Crocs Flip Crocband unisex Volt Green / White , Salvatore Ferragamo Womens Enfola Sella Alexandra Calf , Zapatos OLUKAI Pehuea - Mujer Bronce / Dk Java David Tate Wish Sandalia para mujer Armada Sandalia Campster de mujer Oboz Tomillo / Magenta , Sandalias de punta en punta con sandalias de mujer de YDN Sandalias de tacón bajo con cordones de zapatos de diapositivas rojo Sandalias con hebilla de botón Ridgeport para mujer Rockport Bronce KEEN Women s Uneek 8MM Sandal Negro Negro , Bomba Amie Naturalizer para mujeres Negro , L Artiste de Pump Step de Spring Step para mujer Turquesa Sandalia Debra de Josef Seibel 76444 Oceano , Imagine Sandalia de tacón IM-Dailey de Vince Camuto para mujer Oro suave Easy Street 30-8320 Sandalia airosa para mujer Estiramiento natural , Sandalia Jasmin de Dr. Scholl s Women Microfibra azul Chancleta Yoga Sling 2 Sanuk para mujer Cabaret Plano original de la mujer del Dr. Scholl s Collection Collection Kimberned Pointed Toe Vestido azul nubuck JSport by Jambu Women s Newbury-Water Ready Sandalia de pescador Blanco negro Earth Origins Selby Negro , Drew Womens Element Denim Blue Nubuck , Drew Womens Ariana Cuero de peltre Sandalias de cuero de Viv Viv de TOMS Cuero miel Sandalia de puño de 2 piezas Cobb Hill Women s Hollywood Caqui Vintage Sandalias EVA Gizeh de mujer Birkenstock Scuba 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 Zapatillas de running ASICS Gel-Nimbus 19 para mujer Negro / Onyx / Plata Skechers Women s Go MEB Razor 2 Blanco negro , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris Zapatillas de agua Merrell para mujer All Out Blaze Aero Sport Negro / Aventurina , Brooks Women s PureGrit 5 Peacoat / Passion Flower / Ceramic Zapatillas de running adidas Cloudfoam Ultimate para mujer Blanco / Blanco / Gris Tres , Zapato de running ASICS Gt-2000 5 para mujer Negro / Carbón / Flash Coral Zapatillas Erika Propet para mujer Hueso , Sandalias de diapositivas casuales Easy Street Womens Conca Fabric Open Toe Natural , PUMA Mujer Basket Corazón NS WN Sneaker Puma White-puma White Sandalia Dansko para mujer Minka Negro Full Grain , New Balance WW813 Hook and Loop Walking para mujer Blanco CLARKS Sandalia plana Brizo Sammie para mujer Microfibra perforada azul marino SAS Womens Sunny Bígaro , Zapatillas VI-S para fitness y yoga Vibram para mujer Gris Zapatillas Fm Cross-Trainer para mujer Tazon 6 WN de PUMA Puma Black-puma Team Gold , Zapato de tenis 7 de ASICS Gel-Resolution femenino Indigo Blue / Pink Glow / Paradise Green , Converse Chuck Taylor All Star Core Ox Salsa roja / blanca , Vans Auténtico Negro Negro Ballet de niña de flores Aerosoles femenino Negro exótico , nativo AP Mercury Liteknit Shell White / Shell White / Natural Rubber Clásicos de base Slip-on (Tm) para hombres de Vans Carbón Sandalia para mujer Ivanka Trump Earin Slide Natural , 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) 1659- Gold Stardust & Leather , Zapatilla Glamour de mujer verde Daniel Bone Kidskin Steve Madden Women s Gaze tobillo botín Patente negra , Trotters para mujer Dea Slingback Pump Desnudo Zapatilla Trista mujer BEARPAW Lino , Zapatillas de deporte lavables Walker Propet para mujer Sr Indigo Mocasines casuales Alpargata Deconstructed de TOMS para mujer Henna Red Washed Twill ,

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.

Estuardo Mira Cush Mule Mule de Stuart Stuart Weitzman Off White Cush Nappa a25a9df - 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 Estuardo Mira Cush Mule Mule de Stuart Stuart Weitzman Off White Cush Nappa a25a9df 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
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 970838
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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.