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.

Zapato deportivo ASICS Gel-Cumulus 18 B0163GIM0Y Acuario para mujer Acuario/ 18 Flash Coral/ Blue Jewel bbd0f4e




Sintético
Importado
Caucho único
Impacto Sistema de orientación (I.G.S) (Corriendo) : Diseño de ASICS filosofía que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.
FluidRide: FluidRide proporciona la combinación definitiva de rebote y amortiguación propiedades con peso reducido y excepcional durabilidad.
Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.
Orientación Trusstic: Esto Sistema Trusstic: integra orientación Construcción de línea para una marcha mejorada eficiencia mientras proporcionar el mediopié integridad estructural.
sin costura Construcción
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


Zapato deportivo ASICS Gel-Cumulus 18 B0163GIM0Y Acuario para mujer Acuario/ 18 Flash Coral/ Blue Jewel bbd0f4e - 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

Zapato deportivo ASICS Gel-Cumulus 18 B0163GIM0Y Acuario para mujer Acuario/ 18 Flash Coral/ Blue Jewel bbd0f4e

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.


Zapato deportivo ASICS Gel-Cumulus 18 B0163GIM0Y Acuario para mujer Acuario/ 18 Flash Coral/ Blue Jewel bbd0f4e

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 de running adidas Ultraboost W para mujer Pizarra negra / negra / oscura Zapato de running ASICS GT-2000 4 para mujer Carbón de leña / Flash Coral / Flash Yellow , Zapato de running Saucony Zealot ISO 2 para mujer Azul marino / azul / sil , Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado , Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Deep Lake / Lime Punch , Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul , Zapatillas de running Mizuno Wave Rider 20 para mujer Negro / Blanco / Rosa Glo Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Wolf Gray / Metallic Rose Gold , Zapatillas de running NIKE Free Run para mujer - SU16-10 - Blanco Mango / Sunset Glow Zapatillas de running Reebok Women s Run Supreme 2.0 Mystic Maroon / Poison Pink / White Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral , NIKE Men s Flex 2017 RN Negro / Hematita Metálica / Antracita / Gris Oscuro , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Negro-corredor Rosa-blanco , Calzado deportivo New Balance para mujer Vazee Rush V3 Blanco / Vintage Indigo , Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima , NIKE Mujeres Air Max Thea Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout , Under Armour Threadborne Fortis para mujer Negro (001) / Stealth Gray , Brooks Womens Hyperion Peacoat / azul marino / cacatúa Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral , 361 Zapatillas de running para mujer 361-Enjector Negro / Blanco / 0900 , Zapatillas de running adidas Ultra Boost para mujer Azul , Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapatillas de entrenamiento Nike Womens Metcon 3 Cool Grey / Solar Red-black-pure Platinum , Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-blanco-negro , New Balance Women s w940v3 Running Shoe Plata , Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Azul
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 Sandalia Savannah de Ryka para mujer Gris blanco Converse Chuck Taylor All Star Seasonal Color Hola Pistola Metal / Blanco , Sandalia Drew Angela para mujer Beige Multi Python Print Leather , Zapatillas de running adidas Cloudfoam Xpression W para mujer Claro Onix / Mate Plata / Blanco , Salomon Women s Speedcross 4 W Trail Runner Bird Off Paradise Sandalia de cuña Jubilee Propet para mujer Denim Multi , Zapatillas para correr Nike Downshifter 6 para mujer Deep Royal Blue / Blue Glow Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Negro / antracita / blanco , Zapato de agua Tetrex Rapid Crest para mujer Merrell Rosa purpura , Calzado de caminar para la salud WW840 de New Balance para mujeres Negro Zapatillas de running ASICS Gel-Nimbus 20 para mujer Azul / blanco , Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Púrpura / plateado , Sandalias SAS para mujer, Nudu Slide Armada , Sandalia de mujer UGG Holly Azul Multi , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Pewter Suede , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Equipaje Tan , Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Gris pardo , Telacos NARUTO Anime Hyuga Hinata Zapatos de cosplay Botas por encargo Modo de chakra de nueve colas Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Gris , Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Oro / Metálico Sandalias de mujer Mia de Sandalias de Jerusalén Miel SJP de Sarah Jessica Parker Women s Charlie Slide Sandal Ebano Satinado Sam Edelman Women s Bay Slide Sandal Peltre Metálico Cuero , Lauren by Ralph Lauren Sandalia mujer Davan Slide Cuero Súper Suave Negro , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7010-melocotón , ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ Cuero desnudo de cobre S9216 , 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 \ 7013- Satén gris plateado , CLARKS Women s CloudSteppers Sillian Jetay Flat Microfibra perfilada roja

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.

Zapato deportivo ASICS Gel-Cumulus 18 B0163GIM0Y Acuario para mujer Acuario/ 18 Flash Coral/ Blue Jewel bbd0f4e - 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 Zapato deportivo ASICS Gel-Cumulus 18 B0163GIM0Y Acuario para mujer Acuario/ 18 Flash Coral/ Blue Jewel bbd0f4e 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 3722157
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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.