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.

Zapatillas ONEMIX para hombre, C-blanco rojo para al mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco/ rojo 35ea076




El gran agujero de malla vampiro transpirable el diseño está de acuerdo a la aerodinámica eso te hace sentir como caminar en la brisa.
El interno está hecho de chinlon imitado + hilo elástico, agradable para la piel, suave y elástico.
Alto Sandwich estándar malla suave y piel amigable, suave y moda.
Usando fabricación de clase de lujo tecnología, y usando el avanzado Cuatro agujas seis Tecnología de líneas, que revela calidad en detalles.
Elástico 3D que hace punto interno se puede frotar cuando lavado y secado rápido, que te da un vigorizante experiencia.
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


Zapatillas ONEMIX para hombre, C-blanco rojo para al mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco/ rojo 35ea076 - 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

Zapatillas ONEMIX para hombre, C-blanco rojo para al mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco/ rojo 35ea076

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.


Zapatillas ONEMIX para hombre, C-blanco rojo para al mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco/ rojo 35ea076

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

Sperry Womens Fathom Blanquecino Zapatillas para caminar Ww840v2 New Balance para mujer Gris Easy Spirit - Zapatillas Faisal2 para mujer Rosado , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Cordón negro / blanco , Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Cuero negro Zapatillas Slip-on Keds Womens X Design Love Fest Triple Decker Natural , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Vino rojo Zapatillas para caminar al aire libre Go Skechers Performance para mujer Chocolate Zapatillas para caminar adidas outdoor Terrex Voyager CW CP W para mujer Misterio Ruby / Dark Burgundy / Black , New Balance WW411v2 Hook and Loop Walking para mujer Blanco , Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On Cuero negro , Skechers Women s Go Step Lite-Indulge Loafer Flat Carbón , Earth Origins Tova Negro Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 Blanco / Juego Real Prop & eacute; t Zapatillas para caminar TravelFit para mujer Naranja / Rosa Skechers Women s Go Walk Revolution Ultra Sneaker Negro / Multi Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Negro / Blanco / negro Zapatos ocasionales diabéticos Maggy X Black de Dr. Comfort para mujer Negro Zapatillas para caminar Nell Ryka para mujer Gris / amarillo , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata , Zapatillas para caminar Propell Mujer Travellite MJ rojo , Orthofeet 851 Comfort Diabetic Therapeutic Profth Shoes para mujer Zapatillas para caminar ProFit TravelFit Slide para mujer Azul negro Skechers Women s Go Step Lite-Indulge Loafer Flat Gris pardo , Zapatos de tierra Masio Negro Zapato de caminar sin tocar Skechers Performance para mujer Go Step Carbón Artemisa Dólar amarillo suave , Earth Orígenes Sizzle para mujeres marrón Easy Spirit Twist2 Sneaker para mujer Mezclilla ,
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 Saucony Women s Triumph ISO 4 Niebla / gris Zapato de running ASICS para mujer Torrance Carbon / Pink Peacock / Black Zapato de senderismo KEEN Mujer Arroyo II Chocolate Chip / Sap Green , Zapatillas de running Under Armour Speedform Intake 2 para mujer Merlot / Tropic Pink / Merlot , Zapatillas de running Reebok para mujer Collegiate Royal / White , Zapatillas de running adidas Ultraboost X Parley para mujer Carbono, Carbono S, Azul Espíritu S , Zapatillas náuticas para mujer Saucony Originals Menta / azul , Zapatillas de running NIKE Unisex Flyknit Racer Volt / Black-secoya , Zapatillas de running Flex Ride 630v5 para mujer New Balance Gris Zapatillas de running NIKE Lunartempo 2 para mujer Racer azul / negro / blanco Kenneth Cole REACTION Sandalia de cuña con correa baja en el tobillo de mujer Maravilla , Sanda Verra Teva para mujer Negro , Zapatillas de viaje Proather para mujer II Negro , Vionic Women s Chill Larrun Loafer Serpiente negra , Zapatillas de running Pure Cloudfoam de adidas para mujer Negro / Negro / Blanco Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco Drew Shoe Women s Tulip Comfort Croc negro Sandalia de cuña Logan Walking Cradles para mujer Negro , SAS Mujer, Pampa Sandals Negro Zapatillas de tenis adidas Asmc Barricade 2017 para mujer Blanco / Universo / Electricidad , Diadora B. Original Sneaker para hombre Blanco / Blanco / Blanco Converse Chuck Taylor All Star Core Hola Negro / Garza / Delfín Birkenstock Arizona Sandalia de ante con suela suave Sfb Pink Nubuck , Apretón de jóvenes y adultos Zapato de grifo blanco y negro para espectadores , Noble Outfitters Womens Muds Stay Cool 6 Deep Turquoise , Ariat Womens Volant S Tall Zip Tall Riding Negro Bota Ecuestre Xandywc LifeStride para mujer Gris oscuro Wc Pantuflas Mahabis Classic para hombre gris Gris oscuro / Rojo de Rjukan , Trotters para mujer Dea Slingback Pump Patente perlada multi acolchada / negra Zapatillas de deporte Arizona Trotters para mujer 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.

Zapatillas ONEMIX para hombre, C-blanco rojo para al mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco/ rojo 35ea076 - 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 Zapatillas ONEMIX para hombre, C-blanco rojo para al mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco/ rojo 35ea076 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.

Para caminar
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.