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.

Bota de moda de lona negra Winsted Dr. Dr. moda Martens 19846 Negro 3d1fe65




Tela o Textil
Importado
Sintético único
Eje mide aproximadamente tobillo-alto de arco
Vulcanizado construcción
Goma lengua con zorro, incluyendo una crujiente raya amarilla alrededor la entresuela que se hace eco del Dr. Martens amarillo costura
Hecho con un peso pesado Lienzo de 14 oz
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


Bota de moda de lona negra Winsted Dr. Dr. moda Martens 19846 Negro 3d1fe65 - 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

Bota de moda de lona negra Winsted Dr. Dr. moda Martens 19846 Negro 3d1fe65

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.


Bota de moda de lona negra Winsted Dr. Dr. moda Martens 19846 Negro 3d1fe65

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

Corral Boots Womens A3461 Negro / Hueso Earth Orígenes Raven Womens Bootie Negro Arcopedico 1361 Shocks 5 Botas para mujer Noche estrellada negra , Blundstone Hombres 1464 Nubuck Boots Gris Nubuck Bota Darcy para mujer Easy Street Granito bruñido , 055398210 Timberland PRO - Botas de trabajo TiTAN para mujer - Café marrón SOREL Womens Farah Tall Kettle / Jet FRYE Bota de Combate con Cremallera Frontal Julie para Mujer Cepillo completo de grano Cognac , Botas Harley-Davidson Womens Adrian, de cuero, negras, de corte medio Negro , Bota con forro de piel de oveja de Duckfeet Arhus Desnudo , CORRAL Botín perforado tobillo de mujer - G1379 rojo Zapatos posteriores de Dr. Scholl s Women para mujeres Estuco de microfibra Botas de cuero Rieker para mujer Negro , Mocasines con suela de goma con cordones de cuero Twisted X para mujer - Marrón / Turquesa Marrón / Turquesa , Naot Footwear Groovy para mujer Piel de olivo aceituna / Cuero humo vintage , Dr. Comfort Vigor Mujer diabético terapéutico Extra profundidad senderismo botas de cuero de encaje Negro Botín de la linterna de las mujeres de Seychelles Nubuck rosa claro , Guante de mujer EMU Apollo Bay Índigo , TOMS Womens Desert Wedge Bootie Gris pardo Glerups Unisex G-01 - Botas de fieltro Púrpura Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 17 , Botas Carmina Hardy para mujer Mezcal negro Cole Haan Women s 2.Zerogrand Laser Wing Oxford Marca Brown , Spring Step - Chino para mujer Púrpura , Bota Chelsea Madden M-Graye para hombre Ante negro , CLARKS Cheyn Work Tobillera para mujer Cuero negro Glerups Unisex G-01 - Botas de fieltro rojo Sandalias con cordones Joanie de SOREL para mujer VERDE OLIVA Zapatillas Mujer Teva W DE LA Viña Dos 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 Calzado de running New Balance para mujer 530v2 de New Balance Ártico Fox / amanecer blanqueado , Zapatillas de running NIKE Unisex Flyknit Racer Blanco / White-sail-pure Platinum Encaje de quimera de Columbia para mujer Zinsson , KEEN Hombres Newport H2 Sandal Tinta / óxido de la India Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Europa Purple , Drew Womens Cascade Sand Nubuck Zapatillas para correr Under Armour UA Flow Criss Cross para mujer Acero , Sandalia deportiva Teva W Hurricane 4 para mujer Hazel Black Zapato de running ASICS Gel-Quantum 360 cm para mujer Punto medio gris / glaciar / coral Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Coñac , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón Sandalia atlética de la sandalia del deporte de Lila 6 de las mujeres de MBT Gris / verde azulado / melocotón / blanco , adidas Athletics 24/7 Tr M Cross Trainer Noble Rojo / Noble Rojo / Cristal Blanco , Skechers Women s Microburst Showdown Walking Sneaker Negro , Calzado SCARPA Kailash Plus GTX para mujer Marron oscuro , JBU by Jambu Women s Gwen Rain Shoe Cazador Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Gris Stuart Weitzman Landslid femenino Jeans ante Sandalia Dolce Vita Delana Slide para mujer Cuero natural , Sandalia Finn Comfort Sansibar para mujer Nappa negro Zapatilla Ascot de hombre UGG Té de China , Sandalia para colgar edredones de mujer Anne Klein Tela negra / blanca Reebok Zapato de baile Pilox 1.0 para mujer Negro / Vitamina C / Gris Ceniza / Blanco , Bomba Lloret para mujer J.Renee Beige , Bota para mujer Apolo Earthies Terciopelo burdeos , Sandalia Birkenstock de cuero Florida para mujer Negro Botas tobilleras Adaline Bluff de Merrell para mujer Negro , Propet Walkers Evolution Oxford lavable de las mujeres Gris / rosa TOMS Clásicos de mujer (Lienzo natural) Sam Edelman Loraine Loaine para mujer Arena

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.

Bota de moda de lona negra Winsted Dr. Dr. moda Martens 19846 Negro 3d1fe65 - 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 Bota de moda de lona negra Winsted Dr. Dr. moda Martens 19846 Negro 3d1fe65 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.

Botas
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1280779
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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.