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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Textil y sintético
Importado
Caucho suela
Plataforma mide aproximadamente 0.75 pulgadas
Seguro ajuste de captura de encaje sistema
Metatomical Plantilla de EVA
Non-marking suela de goma
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 Pastel KEEN Paloma Venice II H2-W/ para mujer Paloma/ Turquesa Pastel a955de4

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,

Sandalia Drew Element para mujer Denim Blue Nubuck Sandalia de mujer Bellini Nickel Negro , Sandalia Archie Fisherman del Dr. Scholl Nubuck negro , Adidas Women s Adilette CF + Logo W Sandalia deslizante Negro / Negro / Negro Chaco Mujer Zcloud Sport Sandal Agrupado azul , Sanda Kayenta para mujer Teva Negro , KEEN Hombres Newport H2 Sandal Azul marino / gris medio Sandalia de Hollyleaf de las mujeres de Merrell Rosa roja Sandalias KEEN Venice II-W para mujer Bleacher / Raven KEEN Hombres Newport H2 Sandal Cuervo / Aluminio Earth Origins Sophie Women s Sandal Sedona Brown Sandalias adidas de mujer Adilette Cf + Armad Athletic Slide Maroon / Ice Purple Colegiate Navy , Sandalias de encaje Ros Hommerson para mujer Oro , Sandalia original premier universal Teva W de mujer Aceituna , Sandalia deporte mujer Columbia Wave s Train Blanco negro , Sandalia Yucatan ECCO Mujer Shadow White Flip-flop Teva Women s Mush II Santori Tribal Neon Lime , Sandalia original Teva Women s Original Monterey Tan Sandalia de susurro de mujer Keen Poseidon / Blue Danube Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 melocotón Earth Origins Sophie Azul marroquí OluKai Ohana Piña / Negro Sanda Verra Teva para mujer Azul real , Chaco Mujer Zcloud X Athletic Sandalia Island Tango Sandalia de cuña Cape May de mujer Jambu Desnudo / negro , Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Azul Chaco Women s Z2 Classic Athletic Sandal Glaciar negro Teva Terra-Float Active Lace - Mujeres Chispas de chocolate
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 Zapatillas de running New Balance Zante v4 Fresh Foam para mujer Rosado , Zapatillas de running Kinvara 8 Saucony para mujer Gris / verde azulado / morado , Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco , Zapatillas de running adidas Ultraboost W para mujer Negro / Negro / Negro OluKai Ohana Platino / Negro , ALMA NATURAL Sandalia de pescador Beatrice para mujer Rosa Sandalia original Teva Women s Original Miramar Fade Coral Sand Multi NIKE Women s Tanjun Partícula rosa / blanco Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black Zapatillas anchas Skechers Go Walk Lite-15433 Wide Boat para mujer Armada , Bernie Mev Women s Mely Platform Bronce Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Blanqueado Lime Glo / Vintage Indigo , Zapatilla de deporte Rylee Fashion para mujer Taupe Puntikid Drew Shoe Women s Tulip Comfort Hueso Arche Womens Nitya Sand Nubuck , nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Caoba Metálica / Marrón Nocturno ALMA NATURAL Ballet femenino femenino Negro Propet Women s Fisherman Sandal ciruela Sandalias Skechers Reggae Slim-Vacay para mujer Negro Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón Apis Mt. Negro , Zapato deportivo Cross-Trainer 711V3 de New Balance para mujer Negro / Plata , Sandalia Chloe Slide de KENDALL + KYLIE Natural Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse Sandalias Emmerly Slide de Vince Camuto para mujer Metal Rose , 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 \ 7032- Silver Pu , Zapatillas de deporte de la danza de la práctica de las mujeres para hombre Suela dividida VFSN016EB Cómodo -Muy bien {Paquete de 4} Cuero negro , Bota de invierno UIG Mini Bailey Button Ii para mujer Negro , Zapatos muy finos EK1692 de la danza de salón de baile de las señoras de las señoras con 2.5 \ Cuero plateado y malla de carne TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Lienzo verde oliva ,

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 Pastel KEEN Paloma Venice II H2-W/ para mujer Paloma/ Turquesa Pastel a955de4 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

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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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