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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poliéster
Hecho en los Estados Unidos y Importado
Caucho único
Bungee sistema de captura de encaje
Características cuerda elástica que le da esto una sandalia segura ajuste
hidrofóbico malla de revestimiento. suela de goma. Compresión moldeada Entresuela de EVA
METATOMICAL DISEÑO DE FONDO. Este soporte interno el mecanismo es anatómicamente diseñado para proporcionar excelente arco soporte y cuna los contornos naturales del pie.
Todos KEEN sandalias de agua son lavables a máquina Usa una pequeña cantidad de detergente, lavado en ciclo suave y secar al aire.
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

Sandalia de susurro B01HFESN8E de mujer café Keen Licor de amarillo café/ amarillo b68fb66

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,

Chaco Women s Z2 Classic Athletic Sandal Edredón dorado , Drew Womens Cascade Sand Nubuck Naot Footwear Women s Brussels Hilos de plata de cuero , Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave , Flip-flop Teva Women s Mush II Companera Black Sandalia de deslizamiento ultra Comfort NIKE para mujer Blanco negro , Sandalia Savannah de Ryka para mujer Gris / Purpl , Wolky Comfort Jewel Caviar negro adidas Sandalia atlética Anyanda Flex Y W para mujer Medium Heather gris / Easy Coral / White , Chaco Mujer Zcloud X Sandalia Deportiva Echo Paloma , Chaco Mujer Zcloud 2 Athletic Sandal Berry Anemone Columbia Women s Barraca & shy; Sunlight Athletic Sandal Blanco negro Sandalia de Hollyleaf de las mujeres de Merrell Helecho Sandalia Uneek O2-w de mujer KEEN Blanco / oro de la cosecha Sandalias Fabulicious Women s Lip 101DM Slide Clear / Silver Multi Rhinestones Nike Benassi Duo Ultra Slide Blanco negro Sandalia Savannah de Ryka para mujer Gris / Purpl , Sandalias de cuña Hailey 33 de Josef Seibel para mujer Negro , Chaco Women s Zx2 Classic Athletic Sandal Apex Gray , Sandalia de mujer Bellini Nickel Broncearse , Sandalia Chaco Mujer Zvolv X2 W Lluvia Skechers Performance On The Go 400 Vivacity Flip Flop Azul marino / verde azulado Propet Women s TravelActiv Ss Sandal Púrpura / negro , Sandalia Terrell Post II Femenina de Merrell Blanco , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth OLUKAI Pua Sandal - Mujer Bubbly / Sahara MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Azul Skechers Performance On The Go 400 Vivacity Flip Flop Carbón de leña / rosa fuerte Sandalia KEEN Women s Uneek Slide-w Dapple Gray / Vapor ,
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 Zapato de running ASICS GT-1000 6 para mujer Gris / blanco / gris Zapatillas deportivas para hombre NIKE Lunar Apparent Negro / blanco-cool gris , OluKai Kulapa Kai Pale Jade Zapatillas NIKE Femenil Flex 2016 Rn Running Pink Blast / Blanco / Negro / Eléctrico Verde , NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Cool Grey / Black / Max Orange , Zapatillas de deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Coastal Blue 400 Chaco Women s Z2 Classic Athletic Sandal Con cuentas Wolky Comfort Jewel 12750 Cal y Nubuckleather Zapatilla de deporte Dansko para mujeres Gamuza marfil Sandalias ocasionales de mujer Tonga Josef Seibel Basalto Earth Origins Sizzle Azul marroquí Diapositivas PUMA FURNY x PUMA de piel sintética para mujer Cool Blue / Puma Silver SJP de Sarah Jessica Parker Mujer Meteor Sneaker Rose Gold / White Stripe , Zapatillas de viaje Proather para mujer II Gris pardo Supra Vaider LC Sneaker Negro / Rojo - Blanco , Zapatillas Skechers para mujer Go Walk Cool-15651 Sneaker Coral Zapatillas Hailee Ryka para mujer Azul marino , Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Carbón Vans Old Skool Unisex Adults Low-Top Trainers Mono Bumber / Red Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Asfalto , Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Gris Violeta / Blanco Zapatillas de golf con diseño deportivo Ignite Spikeless Sport para mujer de PUMA Negro / Plata / Sombra Oscura , Sandalia de mujer Lucienne Suede Slide de Nine West Dark Natural Suede Sandalia Dolce Vita Delana Slide para mujer Cuero rojo , Zapatillas de mujer australiana Overland Scarlett Australiano Merino Sheepskin Gold Sparkle Zapatos muy finos EK6829 de la danza de salón de mujeres de las señoras con 2.5 \ Satén marrón Zapatillas de ciclismo Smartodoors con suela de carbono o suela de tpu de nylon para carretera y MTB Blanco / Rojo B para carretera Zapatillas de mujer Haflinger AR Lizzy Slipper Gris Bomba de Angel II para mujeres estilo suave Elegancia de la Armada Bota Deia para mujer de TOMS Tinte de hilo natural

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 Sandalia de susurro B01HFESN8E de mujer café Keen Licor de amarillo café/ amarillo b68fb66 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 1615410
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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.