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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Tela / Hecho por el Hombre
Importado
Sintético único
Acuático entrenador cruzado con extraíble, perforado plantilla nitracel y puertos de drenaje en suela
Cloro material resistente
Adhesivo caucho para mejor tracción sobre mojado superficies
Correas la caja del mediopié proporciona medial y lateral apoyo; y malla superior en un agua híbrida entrenador
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,

Zapatos OLUKAI Pehuea - Mujer Pokeberry / Pokeberry , Mujer adidas Originals Ultraboost St Crayón Blanco / Negro / Aero Verde Zapatos OLUKAI Pehuea - Mujer Paprika / Paprika , Zapato de agua NRS Crush - para mujeres Gunmetal Calzado de agua Columbia Drainmaker II para mujer Cantera Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Lightgrey 1786 , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar 212gray Zapatillas de agua Merrell para mujer All Out Blaze Sieve Blue Crush Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Negro 1786 , Zapatos OLUKAI Pehuea - Mujer Negro Negro , Helly Hansen 11302 Bandera de sal para mujer F-1 Sneaker OFF BLANCO / SHELL PINK , Zapato de entrenamiento cruzado deportivo Hydro Sport para mujer RYKA Nube plateada / Niebla fría gris / Azul invierno / Rosa Speedo Women s Offshore Amphibious Pull-On Water Shoe Fucsia / Hawaiano Zapato de agua Tetrex Rapid Crest para mujer Merrell Rosa purpura , Zapatillas de deporte W Lillesand Fashion para mujer de Helly Hansen Azul marino / Cayena / Blanco apagado Zapatillas de deporte con diseño hidroeléctrico de mujer W Ahiga V3 de Helly Hansen Azul marino / Blanco apagado / Cayenne Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Lightgrey 1786 , Sandalias EVA Gizeh de mujer Birkenstock Rosa Eva Sandalias EVA Gizeh de mujer Birkenstock Khaki Eva , Zapatillas de deporte con diseño hidroeléctrico de mujer W Ahiga V3 de Helly Hansen PLUM / SHELL PINK , CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Faded Denim Zapato de agua Tetrex Rapid Crest para mujer Merrell Lima , NRS Desperado Wetshoe Azul negro Zapato de agua de la Isla Komomo para mujer de Tommy Bahama Black Print Zapato de agua Helly Hansen para mujer Aquapace 2 Uva magenta / roja Zapatos OLUKAI Pehuea - Mujer Pale Gray / Carboncillo Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Púrpura , Astral Mary Jay - Mujeres rojo , Sandalias EVA Gizeh de mujer Birkenstock Rosa Eva ,
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 Kinvara 7 Saucony para mujer Verde , Zapatillas de running Saucony PowerGrid para mujeres Gris / Coral , Zapatillas New Balance para mujer Fresh Foam Gobi Trail Running Blanco / oro rosa , Salomon Women s Speedcross 4 W Trail Runner Rojo tibetano / sangría / negro adidas Running Womens Supernova Sequence 9 Gris sólido / plata mate / blanco , Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Rosa táctil, negro, naranja fácil , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Gris oscuro Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Garza / Blanco / Luces púrpuras Calzado de senderismo Skechers Go s 15601 para mujer Blanco negro MOZO Zapatillas para mujer The Maven Sneaker Lunar Rock CLARKS Danelly Millie Sneaker para mujer Cuero blanco Sandalia de vestir de tela Leisa de nueve mujeres del oeste Gris , Birkenstock Unisex Arizona Shiny Snake Cream Sandalias Birko-flor - 8-8.5 B (M) US Mujeres - CLARKS Womens Un Reisel Myra Slide Sandal Cuero negro Apéndice Negro , BareTraps Women s Ann Marie Sandal Gris pardo Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer Púrpura Zapatillas Skechers Go Walk Lite-15360 para mujer Armada Zapatillas PUMA Tazon 6 Wide Wn para mujer Puma White-island Paradise-puma Silver Artemisa Coral Soft Buck Vans Herren Authentic Core Classic Sneakers Negro Zapatilla deportiva Kailee de mujer Ryka Negro / púrpura hielo / verano gris , Mujer NIKE SF AIR FORCE 1 MID Vast Grey / Vast Grey , Zapatillas de golf adidas Bounce para mujer, adidas, para mujer Gray Two Ftwr Blanco / Real Coral S , BareTraps Women s Cherilyn Slide Sandal Denim Blue Zapatillas de volante adidas Bounce 2 de Crazyflight para mujer Blanco / Plateado Metálico / Gris Dublin Ladies Foundation Zip Paddock marrón , Skechers You de Women s One Slip-on Shoe gris Trotters Liz Loafer para mujer Hueso

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 Zapato de/ entrenamiento cruzado deportivo mujer Hydro Sport Chrome para mujer RYKA Negro/ Berry/ Chrome Silver c3642ab 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 161256
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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.