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
sin costura superior proporciona soporte y estructura en un peso ligero paquete sin costuras Transfiera el espaciador seco malla mantiene los pies Fresco y cómodo
Biomecánicamente afinado e: motiontm entresuela características nuestro más acolchado y plataforma estable. Espuma de energía 1: 1 en cojines del antepié sobre el impacto y devuelve energía volver
energía 1: 1 espuma en cojines de talón en el terreno inicial impacto. publicar en el talón a través de medio pie para la pronación controlar
talón altura de la pila: 25 milímetro (incluye entresuela y suela). desplazamiento dinámico: 4 milímetros en contacto inicial a 7.5 milímetros en la mitad de la postura
Carbono caucho en la parte delantera del pie y la plataforma de choque ofrece amortiguación de larga duración y excelente abrasión resistencia
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


Calzado de frambuesa running Pearl iZUMi 14404 para B01HNFTDNO mujer Em Road H3 Gris/ rosa de frambuesa 266fa6b - 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

Calzado de frambuesa running Pearl iZUMi 14404 para B01HNFTDNO mujer Em Road H3 Gris/ rosa de frambuesa 266fa6b

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.


Calzado de frambuesa running Pearl iZUMi 14404 para B01HNFTDNO mujer Em Road H3 Gris/ rosa de frambuesa 266fa6b

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

Zapatillas de running NIKE para mujer Air Zoom Pegasus 34 (N) Mica Blue / White Racer Pink Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Blanco / Gris Oscuro , Zapatillas de running New Balance 790v6 para mujer Espacio exterior / guayaba Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo New Balance Vazee Running Shoe-Breathe Pack Zapatillas de deporte de moda para mujer Tóxico , Zapatillas de running Nike Air Zoom Elite 8 para mujer Hyper Turquoise / Black / Rio Teal / Volt , Zapato de running ASICS GEL-Zaraca 4 para mujer Aqua Mint / Silver / Onyx , Zapatillas de running New Balance 990v4 para mujer Dark Mulberry Zapatillas de running NIKE Air Max Thea para mujer Blue Spark / Summit White Amortiguación 635v2 New Balance para mujer Negro Zapatillas de running ASICS Gel-Kayano 24 NYC para mujer Veinte / Seis / Dos , Brooks Womens Adrenaline GTS 18 Negro Negro , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Azul gris / Coral Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado Zapatillas de running Under Armour Charged Bandit 3 para mujer Negro (001) / Glacier Grey Zapatillas para correr Zante V3 para mujer New Balance Thunder / Artic Fox Zapatillas de running adidas Alphabounce Beyond W, para mujer Gray Four / Carbon / Dark Solid Gray , Zapatillas para correr NIKE Womens Sock Dart Midnight Navy / Sail / Golden Beige Calzado de running NIKE Kids Roshe One (GS) Prisma rosa / blanco-seguridad naranja adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Verde Ceniza , Zapatillas de running adidas Ultraboost W para mujer Misterio Azul / Misterio Azul / Vapor Gris Zapato de running ASICS GT-1000 6 para mujer Insignia Azul / Plateado / Rojo Rouge , Zapatillas de running adidas Women s Edge Rc W Gris / Granito / Coral Fácil , NIKE Women s Tanjun Sport Fuchsia White Tart , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco / Negro-juego Royal Saucony Women s Guide 10 Zapatillas para correr Gray Teal , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Púrpura / rosa / verde azulado Zapatillas para correr Nike Flex Experience RN 5 Negro / Gris oscuro Heather
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 Skechers Performance On The Go 400 Tropical Flip Flop Negro / gris Sandalia Uneek O2-w de mujer KEEN Estate Blue / Harvest Gold Encaje de quimera de Columbia para mujer Ballena / Dark Mirage , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave , Salomon Sense Ride Running Shoe - Hombre Snorkel Azul / Añil Bunting / Sulphur Spring , Zapatilla de deporte Trail running wt690 para mujer de New Balance Teal / rosa Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro Zapatillas para caminar Skechers Performance para mujer Go Fit TR Negro / rosa Apis Mt. Negro , Tecnología Vionic Men s Orthaheel White / Navy Walker Gris Drew Shoe Women s Tulip Comfort Estiramiento negro / negro Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos Negro Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Tomate blanco , ara Sandalia mujer Tania Slide Rosegold Metallic , Sandalias de mujer Mia de Sandalias de Jerusalén Azul Very Fine Dance Shoes 2707 (Grado de competencia) 3 \ Oro , 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 6006- Tan Pu 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 7026- Red Velvet Trotters para mujer Dea Slingback Pump Black Micro / Satin Zapatos rusos de Pointe Rubin Pointe, caña media flexible de U-Cut , Puf lindo de Patricia Green para mujer Azul francés Botas de invierno Crocs para mujer Negro / Carbón , Wolverine Womens Original 1000 Mile Boot marrón Zapato de fútbol Maracana L W Diadora para mujer Amarillo negro Easy Street Lucky Oxford para mujeres Grey Suede Skechers Women s Reggae Fest Willows Flat Malla negra Splendid Midi Perf Vionic para mujer Slip-On Blanco , Easy Spirit Women s Traveltime Mule Patente negra / plateada Easy Spirit Women s Traveltime Mule Blanco negro TOMS Clásicos de mujer (Rojo) ,

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.

Calzado de frambuesa running Pearl iZUMi 14404 para B01HNFTDNO mujer Em Road H3 Gris/ rosa de frambuesa 266fa6b - 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 Calzado de frambuesa running Pearl iZUMi 14404 para B01HNFTDNO mujer Em Road H3 Gris/ rosa de frambuesa 266fa6b 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.

Carretera en funcionamiento
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 344462
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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.