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.

Bota vaquera con punta en punta mujer de cocodrilo de de en mujer Ferrini Negro 4191eb6




100% Cuero
Importado
Cuero suela
Cuero único
Cuero vacuno superior
Square dedo del pie
12 pulgadas altura
Bloque tacón
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


Bota vaquera con punta en punta mujer de cocodrilo de de en mujer Ferrini Negro 4191eb6 - 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

Bota vaquera con punta en punta mujer de cocodrilo de de en mujer Ferrini Negro 4191eb6

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.


Bota vaquera con punta en punta mujer de cocodrilo de de en mujer Ferrini Negro 4191eb6

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 caminar Vivobarefoot Women s Tracker FG L Negro Botín de tobillo Ramona para mujer de Bella Vita Camel Burnish , Bota de bota estilo LYLIANA FRYE para mujer Dusty Rose Soft Oiled Suede , Botín de cuero mujer Gravati Brown 2 , Bota Saela de mujer UGG Driftwood Glerups Unisex G-01 - Botas de fieltro Gasolina Palladium Pallabrouse Baggy Mujeres Sahara / Ecru , Bota de flecos clásica Minnetonka para mujer Softsole Brown Cole Haan Women s 2.Zerogrand Laser Wing Oxford Serpiente negra / blanca , Ovation Celebrity Slim Secret Asiento completo Monaco Blue Botines cortos de señorita Macie Women s Brook Brook Short Booties - U2010-02 marrón , Aquatalia Womens Fabienne Castaño de gamuza UGG Womens Fluff Momma Mongolia Bota Blanco , Bota Chelsea Duckfeet Roskilde Cuero negro Coach Womens Jessie cerrado tobillo Botines de moda Nappa sedosa negra , Zapato de uniforme de cuero Bates para mujer Negro Botines cortos de señorita Macie Women s Brook Brook Short Booties - U2010-02 marrón , Old Gringo Womens Vittoria rojo , Nine West Sweeorn Suede para mujer Tela negra / negra , UGG Womens Gracen Whipstitch Negro Stuart Weitzman Women s Vigor Patente negra FRYE Sacha Moto Shootie para mujeres Tan Soft Oiled Suede , CLARKS Mujer Caddell Hop Boot Negro Plataforma FLY London Yoxi755fly para mujeres Antracita / Negro , GOGO, Botas altas elásticas de 3 \ Rosado Bota de traje seco Scubapro - Negro , Bota de senderismo impermeable X Alp MTN GTX para hombre Salomon Negro / Asfalto / Pulga Bota de trabajo Cactus 6 \ Marrón claro Bota de moda negra Dr. Martens Rakim Black Ajax
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 Under Armour Women s Drift 2 Sigilo gris (008) / rosa islandesa , Arche Drick Noir Flat Noir Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas Cardo rojo / luz solar , Earth Origins Sophie Sedona Brown , Sandalia deportiva Nolita para mujer Anne Klein Tela Coral , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Gris / Gris / Gris oscuro Heather , Zapatillas de running Salomon Speedcross Vario para mujer Deep Dahlia / Black / Cosmic Purple , Scott Supertrac RC femenino negro / amarillo 6.5 M EE. UU. - , Zapatillas de trail running Pearl NZUMi W 2012 Trail N 1 para mujer Scuba Blue / Shadow Gray , Zapatillas de running NIKE Air Max Thea para mujer Blanco / Blanco-blanco Zapatillas de running NIKE Womens Huarache Run Ultra PRM para mujer Blanco azul Zapatillas para correr Saucony Koa Tr para mujer Vio / Aqua SAS Mujer, Pampa Sandals Fantasía Sandalia Pavi de estilo suave para mujer Mid Brown Vitello Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Black Suede / White Mesh Sandalia Drew Cascade para mujer Nubuck marrón , Zapatos de entrenamiento Nike Free TR 6 para mujer Purple Shade blanqueado Lila 502 , Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Carbón , Zapato de skate casual Heathrow DC para hombre Negro / Turquesa / Blanco Sandalia Bobbi Slide de Dolce Vita para mujer Ante amarillo Plataforma CLARKS Leisa Grace para mujeres Cuero de arena , Kenneth Cole New York Sandalia para mujer Janie Slide Multi , Vince Camuto Sandalias Ellanna Slide para mujer Moody Blues Sandalia de diametro de moda Manaan de 3 bandas para mujer de KAANAS Gris pardo , Zapatos muy finos EKSA7001 de la danza del salón de baile de las señoras con el talón de 3 \ Terciopelo azul Propet Cush N Foot Slipper & Oxy Cleaner Bundle Negro , Mocasín Sabrina Moc II Lamo para mujer Franja beige Trotters Lucy Ballet para mujer Desnudo , Bomba de vestido LifeStride Sevyn para mujer Luxe Navy 4 , Zapatillas SAS Women s Tour con cordones Zapato de confort activo Plata ,

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.

Bota vaquera con punta en punta mujer de cocodrilo de de en mujer Ferrini Negro 4191eb6 - 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 Bota vaquera con punta en punta mujer de cocodrilo de de en mujer Ferrini Negro 4191eb6 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2732036
4405451
5227084
5988294
6219362
6841518
9744033
11983476
14436975
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.


Warning: file_get_contents(https://www.google.com/webmasters/sitemaps/ping?sitemap=http://) [function.file-get-contents]: failed to open stream: HTTP request failed! HTTP/1.0 400 Bad Request in /home/ftp/m/mg186/wwwroot/index.php on line 1735