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 deportivo ASICS Gel-Excite 4 para mujer Gel-Excite Azul índigo 13405/ azul/ orquídea 72ab572




Sintético
Hecho en EE. UU. o importados
Caucho único
Eje mide aproximadamente Parte baja del arco
Parte trasera del pie GEL acolchado Sistema: Atenúa choque durante el impacto fase y permite para una transición sin problemas a medio cuerpo
Desmontable Plantilla: una plantilla que puede ser eliminado para acomodar una ortesis médica
AHAR Suela: acrónimo para ASICS High Abrasion Rubber. Colocado en crítico áreas de la suela por excepcional durabilidad
Para kilometraje moderado corredores, el GEL-Excite 4 zapatos ofrecen comodidad y durabilidad. Nuestra firma Rearfoot y gel para el pie tecnología de amortiguación suaviza el impacto enérgico, mientras nuestro ahar objetivos de suela zonas de alto impacto para una mayor resistencia úsese y tírese. En 1949, el Sr. Kihachiro Onitsuka comenzó su calzado deportivo compañía (Onitsuka) Co., Ltd.) por fabricación zapatos de baloncesto fuera de su vida habitación en Kobe, Japón. Él eligió el nombre ASICS para su compañía en 1977, basado en un famoso latín frase
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 deportivo ASICS Gel-Excite 4 para mujer Gel-Excite Azul índigo 13405/ azul/ orquídea 72ab572 - 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 deportivo ASICS Gel-Excite 4 para mujer Gel-Excite Azul índigo 13405/ azul/ orquídea 72ab572

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 deportivo ASICS Gel-Excite 4 para mujer Gel-Excite Azul índigo 13405/ azul/ orquídea 72ab572

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 ISO para mujeres de la guía Saucony Gris / menta Nike Men s Air Presto Essential Rojo universitario, negro-negro Skechers Performance Women s Go Run 400 Obstruct Frambuesa Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Negro-corredor Rosa-blanco , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Púrpura / rosa / verde azulado Brooks Womens Hyperion Negro / Diva Pink / Diamond Yarn Zapatillas de running NIKE Air Max 90 para mujer Vela / radiante esmeralda negro Under Armour Women s Dash 2 Steel / Icelandic Rose / Msv , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Wolf Gray / Volt-black Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul Zapatillas de running ASICS Gel-Nimbus 20 para mujer Rosa negra / helada , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Negro / Cool Gray / Volt / White Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-Gris Oscuro Zapatillas de running TOPO FLI-Lyte 2 para mujer Amarillo negro , Zapatillas de running adidas Women s Edge Rc W Gris Dos / Plata Metálica / Energy Aqua , Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-negro Zapatillas de running ASICS Gel-Contend 4 para mujer Podar / Plata / Negro , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Multi Color 999 , Zapatillas de entrenamiento Nike Free Tr 8 Amp para mujer Gris oscuro / Blanco-negro , Zapatillas de running Mizuno Women s Wave Catalyst 2 Turquesa / Coral Fiery Zapatillas de running Cavu para mujer HOKA ONE ONE Negro / Bluebird , Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado , PUMA Men s Roma Basic Sneaker Negro / Blanco / Plata , HOKA ONE ONE Zapatillas para mujer Arahi Sunny Lime / Blue Topaz , Zapatillas de running adidas Adistar Boost ESM para mujer Gris claro / verde lima PUMA Women s Meteor WN s Running Sneaker Asphalt / Puma Silver Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino blanco / negro-sigiloso puro ,
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 Mizuno Wave para mujer Inspire 13 Blarney / Eléctrico Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello Brooks Womens PureFlow 7 Gris / negro / rosa , Zapatillas de running NIKE Air Max Flair para hombre Negro / antracita-negro , Chaco Z / Canyon 2 Cottage Poppy , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Porcelana Azul / Blanco / Asics Azul , Zapatillas de trail running para correr en tierra suave Primus Lightweight de Vivobarefoot para mujer Negro , Zapatillas de running La Sportiva para mujer Trailido Bushido Gris / menta MT-onemix Women Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Beige / Blanco I , Zapatillas adidas Ultraboost X Shoe para mujer , Zapatillas para caminar Skechers Performance Go 4 Inspire para mujer Gris pardo , jiandick Rhinestone de las mujeres del tobillo Bootie Prom Sandalias de tacón de noche Vestido de noche Stiletto High Heel Negro-plata , MOZO Zapatillas para mujer The Maven Sneaker Lunar Rock , Melissa Women s Puzzle Peep Toe Flats Blanco Adrianna Papell Mujeres Natalia Ballet Flat Malla de colmena de gunmetal , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Gris blanco Zapato de entrenamiento Ryka Women s Dream Negro / Gris hierro / blanco , Sandalias planas de mujer de Easy Street Cocodrilo blanco , Sam Edelman Women s Slide Sandalia Yute / Caramelo Dorado Birkenstock Arizona Sandalia de ante con suela suave Negro Zapatillas de deporte unisex de bajo perfil Rosado , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Black Metallic Woven Botas Ellie Shoes para mujer 421-Groove Engineer Blanco , Trotters Lucy Ballet para mujer Cerezo negro , JENN ARDOR Stiletto zapatos de tacón alto para mujer: puntiagudo, cerrado dedo clásico Slip On Dress Bombas Desnudo Coach Womens Jessie cerrado tobillo Botines de moda Lt Ft Gray Botas Pearlie de Cole Haan para mujer Cuero negro , Ariat Womens Volant S Tall Zip Tall Riding Negro , TOMS - Slip-On de lienzo clásico para hombre Lienzo multi negro impreso TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Ash Classic

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 deportivo ASICS Gel-Excite 4 para mujer Gel-Excite Azul índigo 13405/ azul/ orquídea 72ab572 - 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 deportivo ASICS Gel-Excite 4 para mujer Gel-Excite Azul índigo 13405/ azul/ orquídea 72ab572 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 4959537
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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.