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.

Calzado de W running Amphib XA mujer Amphib W Trail para mujer Salomon Potente púrpura 4070ccf




Textil y sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
construcción única de Kurim pone un suave, malla de skinguard contra tu pie, mientras drenan bien y secado más rápido, entonces puede entrar y salir del agua sin sacrificando la comodidad o rendimiento.
El sistema Quicklace hace que sea fácil ajustar el ajuste sobre la marcha, y sostiene el pie cómodamente en colocar, mojado o seco.
Luz y fácilmente compresible parte superior hace esto zapato genial para cosas en tu fin de semana bolsa para un tropical vacaciones.
Salomon 2 años limitados garantía
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


Calzado de W running Amphib XA mujer Amphib W Trail para mujer Salomon Potente púrpura 4070ccf - 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

Calzado de W running Amphib XA mujer Amphib W Trail para mujer Salomon Potente púrpura 4070ccf

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.


Calzado de W running Amphib XA mujer Amphib W Trail para mujer Salomon Potente púrpura 4070ccf

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 adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Rosa táctil, negro, naranja fácil , Zapatillas Altra AFW1755F Mujeres Lone Peak 3.5 Púrpura / Naranja Zapatilla de deporte de velocidad vertical para mujer Vasque Gárgola / verde , Zapatillas de running New Balance Gobi v2 Fresh Foam Trail para mujer Rosa / negro Zapatillas de running MT-onemix Unisex Modern City Lightweight Running Jogging Gris oscuro Zapatillas de trail running Saucony para mujer Grid Excursion Tr10 Gris / Azul / Coral , Columbia Montrail Women s Fluidflex X.S.R. Phantom Purple, Nappa Green , Zapatillas Adidas Terrex Trailmaker GTX - Mujer Super Blush / Negro / Blanco , Zapatillas de running Inov-8 2017 Roclite 325 GTX Trail para mujer - Negro / Gris - 000561-BKGY-M-01 Negro Zapatillas de trail running TIMP IQ de Altra para mujer Carbón de leña / azul Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Vestido Blues / Neutral Gray , Saucony Women s Xodus Iso Trail Runner Púrpura / rosa / algodón , Salomon Women s X-Mission 3 W-w Aruba Azul / Azul marino Blazer / Verde Flash Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Negro / Blanco / Stealth / Cool Gray Zapatillas de trail running WT910V3 para mujer New Balance Azul negro , Zapato de trail WT1210 para mujer New Balance Gris / rosa , Zapatillas de trail running Salomon para mujer Techamphibian 3 W Vintage Kaki Saucony Women s Xodus Iso Trail Runner Denim / negro / rosa Zapatillas de trail running WT910V2 New Balance para mujer Azul púrpura , Salomon Sense Ride Running Shoe - Mujer Blue Bird / Deep Lagoon / Blazer azul marino Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Trueno , Zapatillas de agua Merrell para mujer All Out Blaze Aero Sport Muy uva , Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker W para mujer Aero Blue / Trace Purple / Semi Frozen Yellow , Zapatillas de trail running Saucony para mujer Grid Excursion Tr10 Gris / Azul / Coral , Columbia Montrail Women s Fluidflex X.S.R. Phantom Purple, Nappa Green Zapatillas de trail running X Scream W para mujer Salomon Boss Blue / Score Blue / White Zapatillas de trail running NIKE Dual Fusion 2 para mujer Negro / rosa atómico / gris fresco , Zapatillas de trail running Salomon para mujer Techamphibian 3 W 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 Chaco Women s Z1 Classic Athletic Sandal Trellis Teal Zapatillas de running ASICS Gel-Kahana 8 para mujer Negro / Verde hielo / Naranja caliente , Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre , Sandalia para mujer Terrell Ari Wrap de Merrell Muy uva , Salomon Women s Speedcross 4 W Trail Runner Poseidón adidas Sport Performance Terrex para mujeres Sendero CMTK Gore-TEX Zapatillas de deporte, gris, Malla, Textil, Caucho, 8 M - Zapatos lindos del rastro de las calcomanías del Fox de la historieta para las zapatillas de gimnasia de Nmd de las mujeres , Sandalia Savannah de Ryka para mujer Azul marino / verde Calzado de trail running In-X-Talon 212 (S) para hombres Inov-8 Negro / Neón Amarillo / Gris Sandalias Joanie II de mujer SOREL Harina de avena Sandalias de cuero de Viv Viv de TOMS Chambray negro texturizado , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Estaño , Zapatillas de running NMD_Racer Primeknit de mujer adidas Gris ceniza / azul tinte , Reggae de Skechers Slim-Forget Web anudada Gore Open Toe Slingback Sandal Taupe / Coral Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Walking Cradles Womens Henna Azul marino Nubuck Estampado Diadora N9000 Men Toe Snthetic Blue Sneakers Castillo de roca , Diadora Unisex EVO Aeon Negro Vans Auténtico Blue Depths / Very Berry Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / gris , Sandalia Dolce Vita con diadema para mujer Tejido negro Sam Edelman Women s Slide Sandalia Pewter / Putty Metallic Boa / Kid Suede Leather , Zapatillas de voleibol Mizuno Wave Wave Tornado X para mujer Negro / púrpura , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres SERA1683 Tacón de 3 pulgadas + plegable Paquete de cepillo Negro Ferrini Ladies Maverick Square Toe Boots Broncearse J. Rene233; Plácido azul , Zapatillas de ciclismo Pearl XZUMi W X-ALP Drift IV para mujer Rojo Rouge , Bota de lluvia Pamp Sport Cuff Wpn para hombre de Palladium Chocolate / hierro forjado TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Toffee recubierto de lino / raya , Zapatillas de bolos SST 8 LE de Dexter para mujer Blanco / rojo / 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.

Calzado de W running Amphib XA mujer Amphib W Trail para mujer Salomon Potente púrpura 4070ccf - 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 Calzado de W running Amphib XA mujer Amphib W Trail para mujer Salomon Potente púrpura 4070ccf 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 1824523
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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.


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