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.

Sandalia Fara Sandalia mujer Slide de Donald J Pliner para mujer Pliner Capri fdd4b63




Cuero
Importado
Talón mide aproximadamente 2 pulgadas
Slide en fácil y de moda estilo con estos elegantes zapatos Fara.
Superior hecho de brillante cuero con decorativo detalles del hardware.
Fácil diseño sin cordones.
Forro hecho de cuero.
Ligeramente amortiguado estacionado plantilla. Plataforma style.Synthetic único. Mediciones: Altura del tacón: 2 en peso: Plataforma de 11 oz Altura: 1 1 y frasl; 4 en mediciones de productos fueron tomadas usando tamaño 9.5, ancho M. Tenga en cuenta que las medidas puede variar según el tamaño. Peso de calzado es basado en un solo artículo, no un par
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


Sandalia Fara Sandalia mujer Slide de Donald J Pliner para mujer Pliner Capri fdd4b63 - 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

Sandalia Fara Sandalia mujer Slide de Donald J Pliner para mujer Pliner Capri fdd4b63

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.


Sandalia Fara Sandalia mujer Slide de Donald J Pliner para mujer Pliner Capri fdd4b63

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

Birkenstock Mujer Mayari Birko-Flor Sandal Plata Sandalia Plataforma Coletero Zerogrand de Cole Haan para mujer Roccia Snake Print , Sandalia Fara Slide de Donald J Pliner para mujer Plata Sandalia Harlotte Woven Slide de Via Spiga para mujer Cuero multicolor Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Gris oscuro , Birkenstock Arizona Sandalia de ante con suela suave Cuero Habana Aceitado , Sandalia Ryana Marilyn Slide para mujer Blanco negro , Plataforma CLARKS Women s Delana Liri Cuero blanco Chanclas Novy Slide FitFlop para mujer Estaño , Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro , Sandalia Bobbi Slide de Dolce Vita para mujer Ante azul marino , Circus de Sam Edelman Sandalia de mujer Raquel Slide Patente negra , Sandalia Carlos s Sanmie Slide de Carlos Santana para mujer Negro / Multi , Stuart Weitzman Sandalia Rosita para mujer Dusty Pink Tpu Sam Edelman Women s Bay Slide Sandal Cuero metálico pulido de yute , Sandalias de mujer Amalia de Donald J Pliner Negro Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Plata / Metálico , Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno azul , Koolaburra de UGG Women s W raychel Slide Sandal Ánfora , Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse , Sandalias de mujer Mia de Sandalias de Jerusalén Gris , Sandalia de diagonal cósmica de Melissa para mujer Marrón oscuro , Sandalia Trotters Ellie Wedge Slide para mujer Gris , Sandalias de mujer Mia de Sandalias de Jerusalén marrón , Sandalias de plataforma deslizante con plataforma Forella de Gentle Souls para mujer Marrón / Multi Sandalias veganas Arizona Arizona Birkenstock Vegan Stone Pull Up Birko-flor , Sandalia Helina Slide de mujer kensie Floral negro Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Frost Grey Mesh , Romika Fidschi 22 Sandalia de vestir para mujer 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 Zapatillas de running NIKE para hombre con versatilidad de tren gratis Gimnasio Rojo / Blanco / Negro , Zapatillas para correr Zante V3 para mujer New Balance Thunder / Artic Fox NIKE Mujeres Air Zoom Pegasus 33 Platino / Negro / Gris / Rosa explosión adidas Womens Pureboost 2.0 Clima Ash Pearl , Sandalia Savannah de Ryka para mujer Gris oscuro Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Blanco / Antracita / Lobo Gris / Sigilo Zapatillas para correr Saucony Women s Cohesion 10 Gris / negro / azul , Zapatillas de deporte trail running Hiero V2 para mujer New Balance Typhoon / Supercell / Alpha Pink Sandalia plana Halle de Walking Cradles para mujer Serpiente negra / plateada Zapatillas de running ISO para mujeres de la guía Saucony Gris / menta Baabuk Wool Sneaker - Mujer Gris claro / naranja Zapatilla deportiva Kailee de mujer Ryka Frost Grey / Cotton Candy / Summer Grey Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Blanco negro , Bomba de cuña Lansing para mujer Trotters Lagarto negro Diadora Unisex Intrepid NYL Blue Estate , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Negro Sandalias con tachuelas Gale para mujer Donald J Pliner Armada Sandalia KAANAS Sausalito Bow Espadrille Slide para mujer Gris , Vince Camuto Sandalia para mujer Carran Slide Picante , Sandalias planas de mujer de Easy Street Coñac Sam Edelman Sandalia Bernice Slide para mujer Candy Red Zapatillas de verano mahabis Agen Aqua / Borgen Blue 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 16612- Black Pu y Black Mesh Tangas Birkenstock Moca , Bomba de vestido de pasión Easy Street para mujer Patente rojo brillante , Bota de mujer Kamik Rogue 9 Marron oscuro , Hush Puppies Womens Chowchow WT Oxford Gamuza añil vintage , Nine West Leather Lilianne Leather para mujer Vino Zapatillas de deporte de mujer Alegria Essence Siempre tuyo , Easy Spirit Women s Traveltime Mule Suede claro natural / blanco ,

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.

Sandalia Fara Sandalia mujer Slide de Donald J Pliner para mujer Pliner Capri fdd4b63 - 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 Sandalia Fara Sandalia mujer Slide de Donald J Pliner para mujer Pliner Capri fdd4b63 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 965590
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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.