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.

ASICS para Plata mujer Gel-Fit Negro Yui 2 SE Negro/ ASICS Limelight/ Plata 9a457d4




Malla ^ Sintético ^ Textil
Importado
Traer algo de calor extra al estudio con el ASICS Gel-Fit Yui 2 SE.
Diseñado para el estudio atleta.
Ligero, malla transpirable, textil y delgado parte superior sintética materiales.
Cordones cierre.
Totalmente integrado ajustar las características del sistema una envoltura ajustada contador de talón para una sensación personalizada. Lazo de tracción frontal. Ligeramente lengua acolchada y collar. forro de tela ofrece un gran zapato sensación. Ortholite y reg; ofrece todo el día comodidad que mecha lejos la humedad y te mantiene seco. GEL & reg; sistemas atenuados choque durante el impacto y fases de despegue y despegue.Durante AHAR y reg; Alto Abreo Caucho) suela.Importado. Mediciones: Peso: 7 oz Producto mea
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


ASICS para Plata mujer Gel-Fit Negro Yui 2 SE Negro/ ASICS Limelight/ Plata 9a457d4 - 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

ASICS para Plata mujer Gel-Fit Negro Yui 2 SE Negro/ ASICS Limelight/ Plata 9a457d4

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.


ASICS para Plata mujer Gel-Fit Negro Yui 2 SE Negro/ ASICS Limelight/ Plata 9a457d4

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

Zapatos de entrenamiento Nike Free TR 6 para mujer Negro / Blanco-Hiper violeta Entrenador cruzado de malla 77V2 para mujer New Balance Negro / Plata , Zapatillas de entrenamiento cruzado Froma FM de PUMA Tazon 6 para hombre Puma Black Zapatilla de deporte Fierce Bright Cross-Trainer para mujer de PUMA Pink Glo / Pink Glo , Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Naval Blanca Zapatillas de cross-trainer Fierce Bright Mesh para mujer de PUMA Aruba azul , Under Armour Threadborne Push para mujer Negro (003) / Marea Tropical Entrenador campeón Gusto Cross Trainer Negro Negro , Zapato Cross-Trainer X Conviction Women de ASICS Aluminio / Diva Rosa / Gris Glaciar New Balance 711v3 Heather Cross Trainer para mujeres Acero / Negro , Zapatillas Cross-Trainer CrossFit Speed ​​Tr para mujer Reebok Negro / gris ceniza / blanco clásico / goma de goma / peltre Zapatillas de entrenamiento cruzado Froma FM de PUMA Tazon 6 para hombre Puma Black-puma Black ASICS Womens Weldon X Negro / Carbono / Blanco , Zapatillas Adidas de mujer, atletismo y rebote Cross-Trainer Gris claro Heather / Vapor Gris / Blanco Calzado de entrenamiento casual Comfort WX409V3 para mujer New Balance Blanco azul , New Balance 711v3 Heather Cross Trainer para mujeres Blanco negro K-Swiss Tubos de mujer Infinity CMF Cross Trainer Titanio / Oro Artesanal , Zapato de entrenamiento Cross-Trainer para entrenamiento Newonic Flexonic WX99V2 para mujer Negro / Champagne Metálico Zapatillas de entrenamiento cruzado Froma FM de PUMA Tazon 6 para hombre Periscope-puma Silver , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Lobo gris / gris oscuro / blanco / azul glaciar , Zapatillas de entrenamiento Fierce Eng Mesh Cross-Trainer para mujer de PUMA Royal Blue / Puma White PUMA Mujer Ignite Flash Evoknit WN Sneaker Island Paradise-whisper White-puma White , Zapato deportivo Cross-Trainer 711V3 de New Balance para mujer Gráfico de velocidad de pigmentos / rayas , Entrenador cruzado NIKE Flex 8 femenino Negro / Blanco-antracita Skechers Sport Skech Air 2.0 City Love Fashion Sneaker Carbón de leña / menta Adidas Adipower Weightlift Cross Trainer Trace Cargo / Trace Cargo Zapatillas Superga 2750 Cotu para mujer Lt amarillo , Zapatillas de entrenamiento cruzado FM Tazon 6 WN de PUMA para mujer Puma Blanco / Fucsia Púrpura / Puma Plata , Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Ciruela / Plateado / Rojo Rouge
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 Lanzamiento de Brooks Womens 5 Negro / Púrpura / Imprimir Zapato deportivo ASICS GT-2000 3 Lite-Show para mujer Tomate Cherry / Lite / Amarillo de Seguridad , Chaco Mujer ZX3 Classic Sport Sandalia Chuckwalla Aqua , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Negro Sandalia Yucatan ECCO Mujer Abedul , Wolky Women s Rio Sandalia Black Crash Suede , Sandalia deportiva Chaco Zvolv X2 para mujer Pino Diamante , Salomon Women s Speedcross 4 W Trail Runner Rojo tibetano / sangría / negro , Zapatillas NIKE Mujer Air Max Sasha Black Negro Blanco Negro Blanco Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Gris crudo, Gris crudo, Gris ceniza , Prop & eacute; t Olivia Oxford para mujeres Hueso , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata , Sam Edelman Sandalias de vestir Yardley para mujer Masilla de gamuza , EN Zapatilla Cloudster para mujer Magenta / Berry Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea , Toalla de vestir Dulce para mujer Touch Ups Champagne Shimmer Apéndice Negro Zapatillas para caminar Go Step Lite-Petal 14723 de Skechers Performance para mujer gris Sandalia plana Nanci de Naturalizer para mujer Mezclilla , Zapatos atléticos para diabéticos Meghan Turquoise de Dr. Comfort para mujer Azul , NIKE Blazer Mujer Low SE Premium BareTraps Sandalia para mujer Flossey Slide Rosa roja , Zapato de voleibol Gel-Tactic 2 para mujer de ASICS Rendimiento Negro / Pixel Rosa 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2829- Oro Sparkle & Snake , Zapatillas Zeppa en suela francesa para mujer marrón Sandalia Birkenstock de cuero Florida para mujer Blanco FRYE Melissa Pull on Fashion Boot para mujer Fumar Tory Burch Coco Espadrille rojo Zapatillas de deporte Trolide Derby Trainer Fashion Rockport para mujer Gris oscuro Martillo Mujeres Vixen Wide bowling Shoes- Blanco / Negro / Naranja Blanco / negro / naranja ,

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.

ASICS para Plata mujer Gel-Fit Negro Yui 2 SE Negro/ ASICS Limelight/ Plata 9a457d4 - 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 ASICS para Plata mujer Gel-Fit Negro Yui 2 SE Negro/ ASICS Limelight/ Plata 9a457d4 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.

Fitness y entrenamiento cruzado
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1208586
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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.