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.

Desafío B0751PL94Q femenino de Desafío Brooks cerámica 9 Antracita/ cerámica/ flor de la pasión 3711f63




Malla ^ Sintético
Hecho en EE. UU.
Usted te rebela! y salir a la calle en el jadeo digno amortiguación y la comodidad de la Brooks & reg; ¡9 zapatillas para correr!
Creado para el neutral corredor a leve sobrepronador mirando para una amortiguación de felpa con un toque de estabilidad.
Ligero, transpirable, control de la humedad Malla de elementos y parte superior sintética materiales.
Ajustable sillín de medio pie para mayor soporte.
Acolchado lengua y cuello Cierre con cordones. Transpirable, que absorbe la humedad Revestimientos de elementos mantener los pies más frescos y secador.Removable plantilla de espuma moldeada. Brooks de larga duración ADN entrega un paseo personalizado a los corredores de todos diferentes tamaños, formas de pie y velocidades.DRB y reg; Acel como la caña unidad promueve torsional rigidez en el medio pie, mientras permitiendo el talón y la parte delantera del pie para actuar de forma independiente. Segmentado Crash Pad asegura transiciones más suaves. Construcción estable de Pod - Entresuela y suela
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


Desafío B0751PL94Q femenino de Desafío Brooks cerámica 9 Antracita/ cerámica/ flor de la pasión 3711f63 - 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

Desafío B0751PL94Q femenino de Desafío Brooks cerámica 9 Antracita/ cerámica/ flor de la pasión 3711f63

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.


Desafío B0751PL94Q femenino de Desafío Brooks cerámica 9 Antracita/ cerámica/ flor de la pasión 3711f63

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

Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Blanco / Negro-volt-cloro Azul , Saucony Women s Ride 10 zapatillas para correr Blanco azul , Zapatillas de running NIKE Lunarglide 9 para mujer Negro / blanco / gris oscuro / gris lobo NIKE Women s Wmns Air Max 270, Ocean Bliss / Blanco-Negro-Hot Punch Ocean Bliss / White-black-hot Punch Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Blanco , ASICS Kanmei Shoe Casual para mujer Negro / Negro / Blanco Zapatillas de running ISO para mujeres de la guía Saucony Pizarra / Melocotón Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , Zapatillas para correr Mizuno Women s Wave Creation 15 Rosa / negro / rosa Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Zapatillas de running Saucony para mujer Omni 16 Gris azul Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro , Zapatillas de running Saucony Women s Freedom ISO Berry | , Zapatillas de running Mizuno Wave Viper para mujer Negro-fucsia Pruple-plata HOKA ONE ONE Clifton 3 para mujer Zapato de running Citrus / Blue Jewel Size 11 M US - Zapato de running ASICS Gel-Quantum 360 cm para mujer India Ink / Flash Coral / Mid Grey NIKE Women s Roshe Two Print Binario Azul / Estampado Floral , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco negro Nike Hombres Air Zoom Pegasus 33 Juego Royal White Black 402 , Zapatillas de running Swift para mujer adidas Originals Hielo rosa / blanco / negro Zapatillas de running NIKE Lunarglide 9 para mujer Cool Grey / Black-pure Platinum , Zapatillas para correr New Balance Rushv3 para mujer Trueno Negro Zapatillas de running New Balance 990v4 para mujer Plata menta / plata menta EN la zapatilla de deporte Running Cloud para mujer Blue Haze , Under Armour Women s Thrill 3 Grosella negra / blanco / rosa penta , Zapatillas de running adidas Ultraboost W para mujer Still Breeze / Still Breeze / Black , NIKE W Air Huarache City Low Women Ah6804-006 Arena del desierto / Obsidiana oscuro-blanca Zapatillas de deporte Reebok Print Run Dist para hombre Tiza Rosa / Porcelana , Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco 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 Zapatillas de running Questr Cc W de mujer adidas Aero Pink S, Aero Pink S, Carbono S , Zapatillas Drew Shoe para mujer Paige Ternero de hueso Sandalias KEEN Venice II H2-W para mujer Skydiver / Blue Opal , Sandalia original Teva Women s Original Campo Negro / Blanco , Under Armour Speedform Velociti para mujeres Rhino Gray (076) / Negro Sandalia de mujer KEEN Rose Boysenberry / Red Violet Sandalia de deslizamiento ultra Comfort NIKE para mujer Obsidiana / azul de cloro New Balance Women s w990v4 Running Shoe Gris / rosa Zapatillas de trail running Inov-8 Roclite 290 (W) para mujer Azul morado adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Verde Ceniza Sandalia de mujer de nivel medio Aerosoles Tela negra Zapatillas de running Flash para mujer New Balance Blanco / Alfa Rosa / Negro Zapatillas Nike Free TR Fit 3 para mujer 555158-402 Dark Armory Blue / Black / Gym Red PUMA BTS x Collaboration Court Star (366202) , Bomba Jude para mujer Adrianna Papell Satén negro Nike Tech Xtreme Cadet Glove Negro / Blanco - Bluecap - Blue Glow Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Negro Zapato de tenis 7 de ASICS Gel-Resolution femenino Aqua Splash / Blanco / Diva Blue Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Black / Pro Purple Ridgerock , adidas Men s Barricade 2017 Minimalism Tenis Zapatos sin teñir- (CG3091-H17) Blanco , Zapatilla de deporte de gamuza bala para mujer Gola Azul marino / grisáceo Converse Chuck Taylor All Star Core Ox Blanco negro , Zapatillas de golf Skechers Go Golf Elite Ace Jacquard para mujer Azul marino / rosa Sandalia Davin Slide de Lucky Brand para mujer Canyon Rose , Sandalia de mujer Lucienne Suede Slide de Nine West Suede verde claro Incrustaciones y cristales de 14 pulgadas en color blanco Off Glitter de Corral, para mujeres Hueso , 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 \ 5008mirage- Party Sparkle & Ultra Silver , 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) 6030 Stardust de oro y cuero , Bota de combate de lentejuelas Gia Mia Dancewear para mujer Negro , Botas de águila pescadora Earth para mujer Cuero cepillado marrón oscuro

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.

Desafío B0751PL94Q femenino de Desafío Brooks cerámica 9 Antracita/ cerámica/ flor de la pasión 3711f63 - 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 Desafío B0751PL94Q femenino de Desafío Brooks cerámica 9 Antracita/ cerámica/ flor de la pasión 3711f63 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 1037555
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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.