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.

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Cuero
Importado
Click aquí para explorar DREW estilos que son adecuados para su dolencias.
El Cascada deportiva proporciona la comodidad y se necesita estabilidad para alcanzar nuevas alturas!
Premium parte superior de cuero.
Tres puntos ajustabilidad con Velcro & reg; sujetadores de gancho y lazo en el dedo del pie, empeine, y talón.
Suave y transpirable revestimientos de tela.Generosamente amortiguado extraíble plantillas de plantilla soporte metatarsal y está abrochado con un gancho y bucles para acomodar ortesis prescritas. Diseñado con el doble profundidad agregada de un zapato tradicional. Ideal para extremo condiciones del pie. Peso ligero Entresuela con cuña de EVA.Características un acero templado caña para agregar estabilidad.Ligero y caucho duradero la suela es antideslizante.Importado.Medidas: Peso: 9 oz Mediciones del producto fueron tomadas usando
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


Drew Womens Cascade Drew Nubuck negro c9df75a - 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

Drew Womens Cascade Drew Nubuck negro c9df75a

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.


Drew Womens Cascade Drew Nubuck negro c9df75a

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

Sandalia Teva W Terra-Float Nova de Teva para mujer Tacion Gray / Multi Sandalia Deportiva Jordan Spires de Via Spiga Albaricoque gamuza Sandalia deportiva Tirra para mujer Teva Teal / Dark Shadow Earth Origins Sophie Women s Sandal Frost Grey Skechers Performance On The Go 400 Vivacity Flip Flop Gris pardo , Wolky Comfort Jewel Caviar negro Sandalias de encaje Ros Hommerson para mujer Oro Orthofeet 872 Women s Comfort Diabetic Extra Depth Sandal , Chaco Mujer ZX1 Classic Sport Sandalia Dart Peach Sandalia KEEN Clearwater CNX para mujer Celestial / Vapor , Earth Origins Sophie Negro , Teva Terra Sandal para hombre Rocio Olive Sandalia Drew Angela para mujer Dusty Multi Metallic Leather Romika Women s Hollywood 02 Sandalia Deportiva Negro , Sandalia de cuña Jubilee Propet para mujer Corteza múltiple , Clorts Women s Whisper Sandal Outdoor Seaside Senderismo Zapatillas de agua SD207 Rosa Sandalias para mujer Teva Women s W Hurricane XLT2 Kerne Blue / Multi Chaco Mega Z / Cloud Pyramid Eclipse Wolky Comfort Jewel 20620 Cuero Rose Viejo , Sandalia Yucatan ECCO Mujer Blanco / estampado de flores , Sandalia deporte mujer Columbia Wave s Train Cloudburst, Blanco , Chaco Women s ZX2 Classic Athletic Sandal Varsity Pine Sanda Verra Teva para mujer Deep Teal , Sandalias Bali KEEN para mujer Ágata gris / aceituna oscura , Sandalias Swiftwater Mesh de Crocs para mujer Negro , Sandalias Newport H2 de KEEN para mujer Poseidón / Capri , Sandalia Drew Element para mujer Nubuck azul marino , Skechers Performance On The Go 400 Tropical Flip Flop Negro / gris , Sandalia atlética de Rka Women s Glance Azul / verde azulado
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 para correr Nike Women s Free Rn Distance 2 Hyper Violet / White Dark Iris , Chaco Women s Z2 Classic Athletic Sandal Edredón dorado , Under Armour Women s Speedform Slingride 1.1 Gris azul Amortiguación 635v2 New Balance para mujer Negro Zapatillas de running ASICS Gel-Nimbus 19 para mujer Diva Blue / Flash Coral / Aqua Splash Zapatillas de running Saucony Women s Triumph ISO 4 Vizi Rojo / Blanco , Zapatillas ONEMIX para correr con aire para mujer, diseño único de la suela del zapato Negro Encaje de quimera de Columbia para mujer Ballena / Dark Mirage , CLARKS Womens Wave Grip Sabio Salomon Women s Sonic Pro naranja Zapatillas de deporte DMX Lite para mujer Reebok Negro / Gris Estaño / Icono Rosa / Blanco , NIKE Men s Benassi Just Do It Athletic Sandal Sepia Stone / Sepia Stone / Summit White FitFlop Womens Serene Python Print Slip-On Negro , Sandalia Dominica de Ryka Coral / gris Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Naturalizer Womens Canary Cuero blanco , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede , Sandalia Marevick para mujer OTBT Nueva Taupe Inov-8 Mujer F-Lite 235 V2 (W) Cross Trainer Coral rojo , Converse Chuck Taylor All Star Core Ox Salsa roja / blanca Sandalia Rose Slide de mujer Yosi Samra Piel sintética gris / fucsia , Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Azul marino / naranja Zapatillas UGG Womens Florencia Nocturn Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Black Nubuck & Black Trim , Zapatos OLUKAI Pehuea - Mujer Teal / Teal , Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Cáscara Zapato deportivo Fiat Fusion para mujer Brezo , Zapato de baile Militaire de Bloch Dance Negro , Zapatillas Slip-On Roamer SAS para Mujer Negro CLARKS Women s CloudSteppers Sillian Jetay Flat Gris púrpura ,

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.

Drew Womens Cascade Drew Nubuck negro c9df75a - 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 Drew Womens Cascade Drew Nubuck negro c9df75a 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2038621
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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.