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.

Salón de baile de mujeres boda Zapatos de Cuero baile 5] Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero plateado 265ed99




Cuero
gamuza único
Usted RECIBA el COMPLETO PAQUETE de CINCO (5) ARTÍCULOS de esto listado - BUNDLE de DANZA ZAPATOS, PROTECTORES DE TALÓN, Venta al por mayor exclusiva Regalos BOLSITA [cuidado con el imitador] para mantener la bolsita que refresca tu zapatos cuando no en uso, equipaje de mano ZAPATO Y DANZA ZAPATO SACHET que refresca tus zapatos Verifica todos al recibir su orden y notifíquenos si está incompleto.
QUÉ LOS PRO BAILARINES DESGASTE: Ligero par de zapatos de baile es una necesidad para profesionales bailarines. usando el mismo tipo de zapatos para tu competencia, salón de baile, salsa, latino, cha cha, rumba, costa este swing, bolero, mambo, foxtrot, Vals vienés, samba, paso doble, jive, quickstep, merengue, bachata, oscilación de la costa oeste, lindy hop, charlston, balboa, boogie-woogie, prisa, argentino tango, nghtclub 2 pasos, línea, fiesta, bodas, sociales bailar o bailar lecciones ahora puede.
HERMOSO Y ASEQUIBLE: esperará pagar más por esto asequible moderno, elegante, cómodo, femenino y con estilo zapatos de baile, pero debido a la fabricante ingenio, tú no es necesario Una increíble mejor regalo o presente para ti, tu amor uno o alguien especial. que los bailarines, solo como tú, usa ellos por hora y horas de baile.
CARACTERÍSTICAS: Pandeo de liberación rápida (hebilla y clip / gancho), mango mínimo para soporte y flexibilidad, plantilla acolchada para la absorción de choque y comodidad con buen agarre; solo para bailar
ORDEN calzado de calle (solo referencia; montaje no garantizado); uso de la tabla de tallas arriba no es aconsejable; MEDIANO ancho SÓLO, todos los demás lo harán no encaja bien
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


Salón de baile de mujeres boda Zapatos de Cuero baile 5] Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero plateado 265ed99 - 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

Salón de baile de mujeres boda Zapatos de Cuero baile 5] Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero plateado 265ed99

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.


Salón de baile de mujeres boda Zapatos de Cuero baile 5] Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero plateado 265ed99

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

Bota de mujer Pleaser Del1017rsf / c-s / sch Claro Plata / Plata Cromo , Zapatos de danza del salón de baile de las señoras muy finas EK9691 con 1.3 \ Satén marrón claro , 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) 2707- Satén blanco y adornos de plata # 119 , 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) 7012- Serpiente Blanca , LOVELY BEAUTY Lady s Ballroom fabricmps Zapatillas de baile Azul , 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 , Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 16612EB Cómodo-Muy Fino 2.5 \ de 5] Nubuck negro y malla negra , Very Fine Dance Shoes 2707 (Grado de competencia) 3 \ Plata blanca Así que Danca CH99 3 Heel con T-Strap Character Shoe Negro , Salón de baile para mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7005ESSEB Cómodo-Muy Bien [Paquete de 5] Cobra Zapatillas de salón de baile de mujer muy finas EKS2801 con tacón cubano de 1.2 \ Cuero 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 2719- Black Pu y Black Mesh , Capezio Dance Mujer Tiffany Rosa europeo Capezio Women s BR74 Practice 1 \ Negro , 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) 5009- Cuero dorado , 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) 1700- Trim oro / plata , 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 6030- Tan satinado 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) 2829- Blanco y plateado, tacón de aguja Calzado de ballet Freeform FF01 Femenino de Capezio Caramelo Zapato de bailarina Swing Tic-Tac-Toes para mujer Caqui Zapatillas de deporte unisex de bajo perfil Negro , Sandalia de plataforma Pleaser Flam809mg / b / m para mujeres Brillo negro / negro , Audición para adultos 2 \ Negro Zapatos rusos de Pointe Rubin Pointe, caña media flexible de U-Cut , 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 Broncearse 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 1720- 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 \ 2829- Blanco y 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) S92309 Trenza dorada y plateada ,
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 Orthofeet 872 Women s Comfort Diabetic Extra Depth Sandal Skechers Women s Go Ride 7 Running Shoe Carbón de leña / azul , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco / rojo / azul Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running New Balance para mujer Fresh Foam Zante V3 Breathe Pack Vivid Ozone Blue / White Converse Chuck Taylor All Star Seasonal Color Hola rojo , NIKE Mujeres Downshifter 7 Cool Grey / Lava Glow / Dark Grey / White , Zapatillas de trail running Ryka Women s Intent XT 2 SR Blanco / plata cromada Zapatillas de trail running Hiero V2 para mujer New Balance Espacio ultraterrestre / negro / amanecer blanqueado Zapatillas de running ST-2 para mujer Topo Athletic Azul blanco Naughty Monkey Mujer Hey Pony Sandal Blanco , Zapato deportivo ortopédico para mujeres Vionic Kona Negro / Fucsia L Artiste de Spring Step Women s Style pauletta Sandalia de cuero Gray Multi , Crocs Unisex Modi Sport Slide Negro / grafito Sandalia plana de mujer Dayna de Lauren Ralph Lauren Vainilla , Sandalia de gamuza Poppy de TOMS para mujer Black Suede / Chambray Sandalia CLARKS Wave Grip para mujer Negro sintético Zapatillas de deporte Skechers Sport Women s Empire Inside Look Blanco negro Calzado de mujer de fila Cress Walking Blanco / ciruela de azúcar Zapatos de entrenamiento Nike Free TR 6 para mujer Negro / Mtlc Red Bronze / Summit White , Zapato para caminar WW813 para mujer New Balance Blanco Sandalias de mujer Mia de Sandalias de Jerusalén Gris CLARKS Sandalia Leisa Lacole Slide para mujer Cuero azul marino , Zapatillas de deporte de la danza de la práctica de las mujeres para hombre Suela dividida VFSN016EB Cómodo -Muy bien {Paquete de 4} Púrpura Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Blanco negro , Botín de tobillo Lazor de Freebird para mujer rojo Sandalia de vestir Carrson de mujer Steve Madden Flor Multi Minnetonka, mujer, alpino, piel de oveja, moc Golden Tan Zapatos con cordones Wolky Comfort Gobly 40150 Gamuza Taupe Zapatillas Slip-On Roamer SAS para Mujer Lino

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.

Salón de baile de mujeres boda Zapatos de Cuero baile 5] Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero plateado 265ed99 - 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 Salón de baile de mujeres boda Zapatos de Cuero baile 5] Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero plateado 265ed99 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

Thirteen elementary straightedge 3157980
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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.