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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Importado
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

Sandalias Merrell de tirantes B071P5QLH7 Sunstone para mujer para Merrell Sodalita ffb858c

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,

Sandalias KEEN Mujer Ana Cortez Flip-W Concha de tortuga , Sandalia de susurro de mujer Keen Cascada / Stone Blue , ADRIENNE VITTADINI Calzado deportivo para mujer Celie Hook and Loop Cierre lateral Sandalia Oro rosa engastado metálico Sandalia KEEN Clearwater CNX para mujer Amarillo negro Drew Womens Cascade Nubuck negro , Skechers sandalia ancha de deporte 600-brilliancy para mujer Negro , Teva Terra-Float Active Lace - Mujeres Chispas de chocolate , Sandalia Uneek O2-w de mujer KEEN Opal azul / azul provincial Sandalia Terrell Post II Femenina de Merrell Aguanieve Sandalia Savannah de Ryka para mujer Gris blanco Skechers Performance On The Go 400 Vivacity Flip Flop Carbón de leña / rosa fuerte Drew Womens Cascade Nubuck marrón Chaco Zcloud 2 Sport Sandal para hombre Varsity Sun , Chancleta Eezay adidas para mujer Shock Pink / Collegiate Royal / Aero Blue , Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad , Sandalia original Teva Women s Original Deep Sea Coral , Sandalia Drew Angela para mujer Beige-multi Belldilar Cómodo y Walk Support Lady Sandal, Z-Strap Sport Slingback, Comodidad durante todo el día Sandalia al aire libre Para mujeres con un estilo Oro negro Sandalia Kea Vent mujer Columbia Marrón oscuro, Hawk Sandalia Terrell Post II Femenina de Merrell Dusty Olive , Skechers sandalia ancha de deporte 600-brilliancy para mujer Armada , Chaco Women s Z2 Classic Athletic Sandal Bombina Citrus Chaco Women s ZX2 Classic Athletic Sandal Credo Golden , Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Laguna Azul , Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Blanco FABULICIOUS COCKTAIL-501-5 Tobogán de plataforma con 5 \ Claro / Claro , Sandalia de tiras Enoki Twist de mujer Merrell Negro , Sandalia con flecos de cuero original Teva Women s W Olivo oscuro Sandalia Drew Angela para mujer Negro
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 Corkys Rock Mujer Slip On Azul Multi Zapatillas NIKE Women s Revolution 3 para correr Fresh Mint / White / Wolf Gray , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro , Zapatillas Merrell Women s Vapor Glove 2 Barefoot Trail Running Libertad , Zapatillas de trail running Inov-8 X-Talon 212 Negro / rojo / gris Sandalia deportiva Zvolv X de Chaco para mujer Pino Raglan , Zapatillas de running Vibram para mujer V Verde azulado / azul marino , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Porcelana Azul / Blanco / Asics Azul , Orthofeet más Cómodo diabético Tendón de Aquiles Dolor de pies extra Profundidad ancha 940 Sneakers de mujer Zapatos BLANCO MONTAÑA Zapatos Lorna Mujeres Sandalia Aceituna , Drew Womens Element Naranja quemada , Sandalia del vestido de Flaunt de las mujeres de LifeStride Oro suave , Entrenador cruzado Minimus 20v7 de New Balance para mujeres Fuschia Zapato de entrenamiento WX824 para mujer New Balance Flint Gray , Drew Shoe Women s Tulip Comfort Cobre Metálico , Converse Chuck Taylor All Star Core Ox Tejer rojo / negro / blanco , Zapatillas de deporte adidas by Stella McCartney Ultraboost Parley Sneakers de adidas by Stella McCartney Piedra / Núcleo Blanco / Azul Espejo nativo AP Mercury Liteknit Root Red / Shell White / Natural Rubber Onitsuka Tiger Asics Unisex Tiger Corsair Blanco verde , Sandalias de Jerusalén Sandalias de mujer Edna rojo 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- Tan satinado Zapatos muy finos EK1692 de la danza de salón de baile de las señoras de las señoras con 2.5 \ Cuero negro y malla negra , NIKE Women s Free 5.0 Tr Fit 4 Print Negro / Hyper Cobalt / Bright Mango / White , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Drizzle Grey Slub Chambray , Zapatillas Eloise de mujer Vionic Armada Sandalias de vestir Patti de Sam Edelman para mujer Candy Red Patent Easy Spirit ON CUE Oxford para mujeres Bronceado natural , Me Too Arena Sandalia para mujer New Luggage Goat Spore , Lan de Zerogrand de las mujeres de Cole Haan Ironstone Brush Off Cuero / Pavimento , CLARKS Jocolin Vista Boat Shoe para mujer Champagne Metálico Sintético

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 Sandalias Merrell de tirantes B071P5QLH7 Sunstone para mujer para Merrell Sodalita ffb858c 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2864384
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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.


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