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 muy finos con EK6829 de la señoras Marrón danza de salón de mujeres de las señoras con 2.5 \ Cuero Marrón Beige d063971




cuero y tela
gamuza único
Rápido pandeo de liberación (hebilla y clip / gancho)
Gamuza Suela
acolchada plantilla para shock absorción y comodidad
2.5 Talón
Talón Protectores o zapato Cepillo dependiendo en el tipo de talón
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 muy finos con EK6829 de la señoras Marrón danza de salón de mujeres de las señoras con 2.5 \ Cuero Marrón Beige d063971 - 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 muy finos con EK6829 de la señoras Marrón danza de salón de mujeres de las señoras con 2.5 \ Cuero Marrón Beige d063971

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 muy finos con EK6829 de la señoras Marrón danza de salón de mujeres de las señoras con 2.5 \ Cuero Marrón Beige d063971

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

Audición para adultos 2 \ Caramelo 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 , 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) 3030- Serpiente dorada , 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 , Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ marrón Bota para mujer Pleaser Bej3019rsf-7 / Bpu / Srs Silver Metallic Pu-silver, Silver Multi Rs Zapatillas de deporte unisex de bajo perfil Blanco Miller y Ben Tap Shoes; , Calzado Aria Pointe 121 para mujer Capezio Pétalo rosa 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 \ 1680- Cuero plateado Zapato de personaje de luz de posición de estudiante 650 de Capezio Negro , Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7017EB Cómodo-Muy Fino 3 \ de 5] Plata 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 \ 2719- Cuero plateado y malla de carne ~ 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 \ Cuero desnudo de cobre S9216 , 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 , Zapatillas para adultos con hebilla deslizante, T9200 Negro Zapato Tap-Flex para mujer, de Bloch Dance 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 8881- Brown Nubuck & Beige 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 \ 1683- Cuero plateado Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ Seradark Tan Satin , 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 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- Satén blanco brillo 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) 1662- Satén blanco Zapatillas de baile Zumba Fresh Street para mujer Azul 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) 6012- Cuero dorado , Zapatos muy finos Dance Sneakers Series SN016 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 8881- Silver Sparklenet (pedido personalizado, 4-6 semanas de tiempo de procesoAsí que Danca CH50 Character Shoe con 1.5 Heel 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 café 7001-oscuro
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 Lanzamiento de Brooks Womens 5 Negro / verde azulado / blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro Sandalia Terrell Ari Convert Sport de Merrell para mujer Paloma Zapatillas NIKE Women s Revolution 3 para correr Dark Purple Dust / White / Pink Blast , Sandalia deportiva Nolita para mujer Anne Klein Tela Coral Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Negro , Chaco Z / Canyon 2 Aguacate disperso Zapatillas de deporte trail running Hiero V2 para mujer New Balance Typhoon / Supercell / Alpha Pink Zapatillas de running La Sportiva Helios 2.0 para mujer - SS18 Verde , Zapatillas de running W1540V2 New Balance para mujer Blanco azul , Corredor de pista Arctic Talon 275 para mujeres Inov-8 Verde azulado / negro / rosa , Drew Shoe Women s Tulip Comfort Hueso Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Azul , Sandalia CLARKS Wave Grip para mujer Sage Nubuck , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Laguna Negra / Blanca-Volt-Blue , Sandalia de mujer Teva W Terra-Float Livia Morado oscuro , Zapato para caminar WW813 para mujer New Balance Blanco Vans Auténtico Cómics / Blk / Tru Skechers Womens Burst TR gris , Plataforma CLARKS Leisa Grace para mujeres Cuero rojo Zapato de voleibol de control de corte de gel femenino ASICS Uva / azul marino / blanco 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) S1001 Gold Scale & Dark Gold , Zapatilla Mesa de UGG para mujer Ópalo de fuego , Trotters para mujer Dea Slingback Pump Negro Negro Jeffrey Campbell Womens Lilith-2 Boot Nuevo Combo floral , Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra blanco rojo , Jacques Levine Pyramid Stud - Zapatilla de cuero con cuña de mujer Desnudo negro Mocasín de cuero con cordones para mujer Parodia de Caterpillar Negro Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 11 Blundstone EVA Rustic Black Boot ,

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 muy finos con EK6829 de la señoras Marrón danza de salón de mujeres de las señoras con 2.5 \ Cuero Marrón Beige d063971 - 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 muy finos con EK6829 de la señoras Marrón danza de salón de mujeres de las señoras con 2.5 \ Cuero Marrón Beige d063971 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.

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.