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 muy fino Salsa de tango latino Cepillo Zapatos de para de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo b3f38fc




Cuero, Tela
No Slip Suede Sole
Mínimo tablero de caña para soporte adicional y flexibilidad
Rápido pandeo de liberación (hebilla y clip / gancho)
Acolchado plantilla para shock absorción y comodidad
3 talón en la foto. Paquete de par de zapatos de baile y Danza Plegable Cepillo de zapatos
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 muy fino Salsa de tango latino Cepillo Zapatos de para de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo b3f38fc - 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 muy fino Salsa de tango latino Cepillo Zapatos de para de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo b3f38fc

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 muy fino Salsa de tango latino Cepillo Zapatos de para de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo b3f38fc

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

Bloch Dance Boost Mesh DRT Split Sole Dance Sneaker Rosado , 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 \ 2721- Silver Sparklenet & Silver Trim 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 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 Zapatillas de deporte Trinity Sneaker Dance de Bloch Negro 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) 5517- Satén blanco 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 Terciopelo negro 7001 , 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) 1606- Satén de marfil Very Fine Dance Shoes 2707 (Grado de competencia) 3 \ Negro Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 9624EB Cómodo-Muy Fino 2.5 \ de 5] Cuero negro , Zapato de baile femenino moderno de la gema del soleil AC de PUMA Puma Black / Puma Black 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) 5517- Satén blanco , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres S1002 2.5 Tacón de Inch + Plegable Paquete de cepillo Tan satinado , 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) 6027 cuero y malla de 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) S92309 Trenza dorada y plateada , Botas de mujer PLEASER BEAU-1018C / C / M - Sansha Hi-Step Dance Sneaker 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 \ 7039 - Escala de plata Zapato de bailarina Swing Tic-Tac-Toes para mujer Negro , Zapatillas Tap Tap Jazz Jazz de Bloch Dance para mujeres Negro Calzado de ballet Freeform FF01 Femenino de Capezio Caramelo , 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 1643- Black Weave Pu y Black Mesh 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 1670- Tan satinado Salón de baile de mujeres Zapatillas Dance Salsa Práctica latina Zapatos 1670cEB Cómodo-Muy Fino 1.5 \ de 5] Cuero 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) 2800- Gold Scale / Light Gold 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 \ 2707- Satén blanco y adornos de plata # 119 Jig Foo Latin Salsa Rumba Chacha Practice Ballroom Zapatos de baile para mujeres 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 6006- Tan Pu Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7017EB Cómodo-Muy Fino 3 \ de 5] Serpiente de oro ,
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 OLKAKAI Ohana - Hombre Cáscara / arcilla , Converse Chuck Taylor All Star Seasonal Color Hola Azul / blanco / rojo Zapatillas de running New Balance Fresh Foam Lazr V1 Blanco negro , Sandalias KEEN Clearwater CNX-W para mujer Dapple Grey / Dress Blue Chaco Women s Z2 Classic Athletic Sandal Brocade Pine Sandalia universal W Sanborn de Teva para mujer Nuez , Gel-Nimbus 19 de ASICS para mujer, azul Diva / Flash Coral / Aqua Splash Diva Blue / Flash Coral / Aqua Splash Zapatilla de deporte Trail running wt690 para mujer de New Balance Gris oscuro Sandalias EVA Gizeh de mujer Birkenstock Eva blanca NIKE Womens Lunar Force 1 Flyknit Workboot Trainer Zapatillas de moda Aceituna mediana / aceituna media , NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco , Zapatillas para correr Under Armour UA Speedform Fortis 2.1 para mujer Negro / blanco / azul veneciano , Zapatos de mujer Spira Classic Walker 2 con resortes Blanco Zapatillas para caminar Ryka Women s Dash 2 Azul marino / gris / morado Sandalias de cuero mujer Josef Seibel Tonga 25 Camello , Drew Shoe Women s Tulip Comfort Negro , Tierra Everwood Cuero de grano completo de almendra Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Negro Clásicos de base Slip-on (Tm) para hombres de Vans Baby Blue / Verdadero Tablero blanco Sandalia Maci de mujer Dansko Grano completo de frambuesa Sandalia Dolce Vita Poe Slide para mujer Cuero 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 \ 2701- Escala plateada de cuero y 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 8881- Black Nubuck & Black Trim Flip Flop FitFlop Carmel Toe para los pies de las mujeres Azul , Zapatilla Mesa de UGG para mujer Negro Glerups Modelo B Gris , Eddie Bauer Women s Laurel Chukka Negro , Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 27 TOMS - Slip-On de lienzo clásico para hombre Olivo militar , Zapato femenino Keli de Alegria Goth Cutie

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 muy fino Salsa de tango latino Cepillo Zapatos de para de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo b3f38fc - 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 muy fino Salsa de tango latino Cepillo Zapatos de para de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz rojo b3f38fc 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 1826920
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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.