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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satén
Cuero único
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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


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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 de paloma Comfort de Art oro 50 tonos de de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1606- Black Pu 4cb2ce3

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.


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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 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) S9282 Trenza dorada y plateada , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres SERA7039 Tacón de 3 pulgadas + plegable Paquete de cepillo Escala de 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 \ 1661- Cuero plateado 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 7001-light Tan Velvet , 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) 6016- Cuero dorado y destellos , Calzado Capezio Manhattan para personajes, Negro Sansha Hi-Step Dance Sneaker Negro , ~ 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 \ S9206 Vegan Copper , 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- Flor blanca y azul Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Satén marrón claro y oro claro , 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 Dance Class Mujer CS401 Split Sole Jazz / Obstrucción Oxford Blanco Capezio Women s BR74 Practice 1 \ 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 \ 2702- Silver Stardust & Silver Leathe , 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) 2829- Oro Sparkle & Snake , 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 Zapato de bailarina Swing Tic-Tac-Toes para mujer Caqui Zapato de bailarina Swing Tic-Tac-Toes para mujer Estaño 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) 1625- Granito 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) 7013- Bronceado dorado , 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 \ 7017 - 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 \ 2829- Blanco y Plata Miller y Ben Tap Shoes; , Zapatos muy finos EKSA7001 de la danza del salón de baile de las señoras con el talón de 3 \ Terciopelo café oscuro , Calzado de ballet Freeform FF01 Femenino de Capezio Rosa claro , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ Salón de baile para mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7005ESSEB Cómodo-Muy Bien [Paquete de 5] Serpiente negra Mujeres muy finas Salsa Ballroom Tango Latin Dance Zapatos estilo 9691 Paquete con plástico Zapato de baile Protectores Cuero 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) 2707- Satén blanco y adornos de plata # 119
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 Sandalia Drew Angela para mujer Negro Sandalias KEEN Mujer Whisper-w Negro / Imán , Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Monumento , Chaco Mujer ZX3 Classic Sport Sandalia Melocotón de jardín , Sandalia de deporte Merrell, para mujer Terran Ari Backstrap Sodalita , Chaco Mujer Zcloud X Sandalia Deportiva Dahlia Eclipse , BONJOMARISA 2018 Verano Nuevo Elegante Sandalias de malla de la marca Decoración de metal Zapatillas Peep Toe Tacones bajos de mujer Tamaño grande 34-43 Calzado Oro Zapatillas de running New Balance Fresh Foam Lazr V1 Blanco negro Zapato de agua Tetrex Rapid Crest para mujer Merrell Negro Skechers Performance - Go Walk 3 Renew - Zapatillas de paseo sin cordones Rosado , Sandalia Barroll T Correa de Natural Soul para mujer Cuero blanco Skechers Sport Skech Air Run High Fashion Sneaker Negro / rosa caliente Alpargata de tela \ Negro Zapatillas de running New Balance Prism V2 para mujer Vintage Indigo / Alpha Pink , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gunsmoke / White-atmosphere Grey , Zapato de running ASICS para mujer Torrance Negro / Gris piedra Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata / Lima , Comfortiva Mujer, Faye Sandalia de tacón medio Whiskey Otago Nubuck , Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Lollipop / Peacoat / Bachelor Button / Blanco , Zapatillas de entrenamiento cruzado Realflex Train 4.0 de Reebok para mujer Acero / Seafoam Green / Ash Gray / White Supra Vaider LC Sneaker Gris / Serpiente / Blanco , Easy Spirit Deiny Women s Walking Natural Sandalias planas Alex de Steve Madden para mujer Leopardo Botas altas de lluvia de cocodrilo Crocs para mujer Café exprés Ellie Shoes Bota vaquera 418 vaquera para mujer Negro Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Denim azul claro , UGG Mujer Dakota Pom Pom mocasín castaña Total de sobre de movimiento total de Rockport para mujer Serpiente negra / múltiple Easy Spirit ON CUE Oxford para mujeres marrón , BareTraps Women s Bt Gahl tobillera Negro ,

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.

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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 de paloma Comfort de Art oro 50 tonos de de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1606- Black Pu 4cb2ce3 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
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