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.

Zapatos Pointe rusos de media Pointe Rubin Pointe, caña media flexible Pointe, de U-Cut 5cc5b34




Ligero y tranquilo con un elegante plataforma sin puntera, cuero antideslizante único y excelente estabilidad, Rubin es el más adecuado a los menos llenos pie con cuadrado ligeramente cónico dedos de los pies.
Rubin presenta un ancho plataforma para los pies, baja corona y más lleno forma de talón para el pie con menos ahusamiento del metatarso desde el área hasta el talón.
Pre-arqueado construcción en todas partes el zapato imita la forma de la pie en pointe.
Próxima generación las espigas proporcionan flexibilidad y durabilidad.
U-Cut con cordón.
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


Zapatos Pointe rusos de media Pointe Rubin Pointe, caña media flexible Pointe, de U-Cut 5cc5b34 - movie-store.online

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

Zapatos Pointe rusos de media Pointe Rubin Pointe, caña media flexible Pointe, de U-Cut 5cc5b34

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.


Zapatos Pointe rusos de media Pointe Rubin Pointe, caña media flexible Pointe, de U-Cut 5cc5b34

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

50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2713- Ultra Silver Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7013- Satén gris plateado Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ Brownsatin 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) 1692- Cuero dorado claro / malla de carne , 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) 6006- Oro Pu y Sparkle , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ Adulto 2007 \ Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 6005- Tan satinado , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 7039 - Escala de plata 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) 1692- Cuero dorado y malla de carne , Así que Danca CH50 Character Shoe con 1.5 Heel Negro , Adulto 2007 \ , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1620- Cuero plateado , Rumpf Scooter 1575 Zapatilla de deporte Black Dance Split Sole Men Salsa Latina de Mujeres Jazz Hip Hop Fitness Zapato de ejercicio , Zapatillas Tap Tap Jazz Jazz de Bloch Dance para mujeres Negro Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7010-negro 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1620s- Cuero plateado 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) 7016- Oro 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) 6030 Stardust de oro y cuero , Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Satén marrón Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 6027- Black Pu y Black Mesh ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ S2806 Cobre y rojo y blanco Zapatillas Triple Cheer de adidas Performance para mujer Blanco / Gris granito / Gris claro Calzado Capezio Manhattan para personajes, Caramelo , Zapato de punto de equilibrio europeo Bloch para mujer Rosado , Así que Danca CH50 Character Shoe con 1.5 Heel Negro , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1620- Black Pu , Adulto 2007 \ Salón de baile para mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7005ESSEB Cómodo-Muy Bien [Paquete de 5] Serpiente Roja ,
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 de tirantes Sunstone para mujer Merrell Sodalita Zapatillas de running Adidas Performance Ultra s Boost M para hombre Núcleo Negro-blanco , Zapatillas para correr Nike Downshifter 6 para mujer Racer Blue / Black-Chalk Blue Sandalia NIKE Kawa Slide para mujer Negro / Vivid Pink Mujer adidas Originals Ultraboost St Indigo crudo / tinta noble / naranja de alta resolución Zapatillas NIKE Kids Revolution 3 (GS) para correr Negro / Negro / Blanco / Volt Zapatillas de running La Sportiva Helios 2.0 para mujer - SS18 Verde NIKE Men s Roshe Two Running Shoe Blanco / Blanco-Metálico Plata , Sandalia Tessa de mujer Miz Mooz Moho Supra Martine Run Skate Shoe para hombre Ladrillo rojo / blanco P W Minor Leisure Mujer Terapia Casual Extra Profundo Calzado Cuero Cordones Negro NIKE Women s Revolution 4 Running Shoe Blue Recall / Pure Platinum / Purple Slate , Sandalia Amanda de Miz Mooz para mujer Vaquero Zapatillas de skate auténticas Vans Kids Pewter / Black , Sandalias SAS para mujer, Nudu Slide Medianoche , Vans Auténtico Dark Slate , Sandalia Slip con nudo de mujer Soludos Flat Chambray Ariat Womens Scout Paddock Negro , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ , Bota de imitación para mujer SoftWalk Ante negro Minnetonka piel de venado mujer Gore Front mocasín Carmel Deerskin The North Face Womens Thermoball Microbaffle Bootie II Shiny Coffee Bean Brown / Dune Beige (Temporada pasada) , Cole Haan Women s 2.Zerogrand Laser Wing Oxford Ante azul marino TOMS Clásicos de mujer Lienzo negro clásico Aravon Women s Deliah - AR Flat Negro Bota western para mujer FRYE Sacha Moto Shootie Cognac-78001 Zapatillas de deporte Easy Spirit AP1 Sport Black Fab SoftWalk Womens Relax Black / Black Lycra / Microfiber , Zapatos antideslizantes clásicos de lona de mujer de TOMS rojo , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Raya roja tejida ,

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.

Zapatos Pointe rusos de media Pointe Rubin Pointe, caña media flexible Pointe, de U-Cut 5cc5b34 - 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 Zapatos Pointe rusos de media Pointe Rubin Pointe, caña media flexible Pointe, de U-Cut 5cc5b34 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.

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

Geometry Construction Reference

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and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.

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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.