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 para mujer Terrell Wrap Ari Baja Wrap de 12530 Merrell Baja Azul 6f7e1be




Cuero y Textil
Importado
Sintético suela
Plataforma mide aproximadamente 0.5 pulgadas pulgadas pulgadas
Cuero Y parte superior tejida
EVA Plantilla para cojín
Merrell Air Cushion en el talón absorbe choque y agrega estabilidad
Neopreno para comodidad acolchada
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 para mujer Terrell Wrap Ari Baja Wrap de 12530 Merrell Baja Azul 6f7e1be - 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 para mujer Terrell Wrap Ari Baja Wrap de 12530 Merrell Baja Azul 6f7e1be

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 para mujer Terrell Wrap Ari Baja Wrap de 12530 Merrell Baja Azul 6f7e1be

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

KEEN Hombres Newport H2 Sandal Midnight Navy / Feather Grey Chaco Mujer Zcloud X Sandalia Deportiva Blazer Green , Wolky Comfort Jewel Brandy Smooth Leather , Sandalias Newport H2 de KEEN para mujer Cáñamo / azul provincial , Sandalia original Teva Women s Original Sun / Moon Insignia Blue , BONJOMARISA 2018 Verano Nuevo Elegante Sandalias de malla de la marca Decoración de metal Zapatillas Peep Toe Tacones bajos de mujer Tamaño grande 34-43 Calzado Plata Sandalia KEEN Women s Uneek Slide-w Dapple Gray / Vapor , Skechers Performance Women s The Go 400 Radiance Flip Flop Carbón Merrell Women s Around Town Print Sandalia atlética Fiery Red Flip-flop Teva Women s Mush II Tiki azul marino / frambuesa , Sandalia deportiva Nolita para mujer Anne Klein Tela Coral Teva Terra-Float Active Lace - Mujeres Azul Sandalia de cuero Velope de nueve mujeres del oeste Negro , CLARKS Sandalia plana Brizo Ravena para mujer Armada Earth Origins Sophie Women s Sandal Azul marroquí , Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino , Skechers sandalia ancha de deporte 600-brilliancy para mujer Natural Flip-flop Teva Women s Mush II Tiki azul marino / frambuesa Sandalia Helen de Propet para mujer Negro Sandalia Merrell para hombre All Out Blaze Aero Sport Vapor Sandalia Terrell Post II Femenina de Merrell Pizarra , Skechers Women s-Go-600-15315 sandalia ancha de deporte Gris pardo , Chaco Women s ZX2 Classic Athletic Sandal Camper Purple , Sandalia Drew Angela para mujer Negro , Sandalias mujer Teva Womens W Original Universal Premier-Leather Sport Medianoche negro Sandalia deportiva Tirra para mujer Teva Gris oscuro Sandalias KEEN Venice II H2-W para mujer Negro / gris acero , Sandalia KEEN Women s Uneek Slide-w Negro Negro , Rieker Womens Velcro Sandal Marrón / Azul / Flores Marrón / azul / flores ,
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 Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino Under Armour Speedform Velociti para mujeres Rhino Gray (076) / Glacier Grey Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente Sandalia de mujer KEEN Rose Imán / Gárgola , Zapatillas de running NIKE Air Huarache para hombre Gris oscuro / antracita , Zapato atlético para mujeres con neuropatía de Ped-Lite - Adele Blanco , Drew Shoe Athena para mujeres Blanco Linea Paolo Felicia | Ante negro Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Sandalia CLARKS Wave Grip para mujer Gris Claro Nubuck Nike Hombres Air Zoom Pegasus 33 Racer Blue / Navy / Blue Glow UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Rojo blanco Zapatillas de entrenamiento Fierce Eng Mesh Cross-Trainer para mujer de PUMA Red Blast / Puma White , Adidas Sport Performance Mujeres Terrex CC Voyager Sleek Sneakers Aero Blue / Chalk Blue / Aero Pink , Zapatilla Reebok para mujer ROS Workout TR 2-0 Cross-Trainer Niebla / Blanco / Meteorito / Puro , Zapatilla Reebok para mujer ROS Workout TR 2-0 Cross-Trainer Niebla / Blanco / Meteorito / Puro , Zapatillas de deporte Adidas Adipure Sport para mujer Blanco / Aero Blue / Chalk Coral Sandalia del vestido de la inversión de LifeStride de las mujeres Tierno Taupe , Sam Edelman Sandalia Glenn para mujer Cuero nuevo y caído negro , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Oro negro , Vans Unisex Old Skool Zapatillas clásicas de skate C & l Chambray Blue , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5004- Gold Sparkle 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 1612- Black Pu Bota de campo Ovation Flex Sport para mujer Negro , Sandalias de vestir Patti de Sam Edelman para mujer Grey Kid Suede Leather , Zapatilla de mujer TOMS Tela escocesa roja / negra , Pantuflas Mahabis Classic para hombre gris Gris Claro / Rojo Rjukan Ovation Celebrity Slim Secret Knee Patch Carbón , Vionic Womens Syra Armada , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Costura de la mantita a rayas Chambray

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 para mujer Terrell Wrap Ari Baja Wrap de 12530 Merrell Baja Azul 6f7e1be - 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 para mujer Terrell Wrap Ari Baja Wrap de 12530 Merrell Baja Azul 6f7e1be 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 4787917
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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.