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.

MICHAEL Gunmetal Michael MICHAEL Kors Womens Keiko Slide Gunmetal Slide/ Silver eb4362f




Sintético
Importado
Talón mide aproximadamente 0.75 pulgadas
Su el estilo es como la magia en el aire!
Hecho por el hombre superior.
Slip-on construcción.
Marca logotipo en el paso.
Open-toe.Synthetic forro. Ligeramente plantilla acolchada. Hecho por el hombre suela.Importado.Medidas: Altura del tacón: 3 y frasl; 4 en Peso: 8 oz Altura de la plataforma: 1 & frasl; 2 in Mediciones del producto fueron tomadas usando tamaño 8, ancho M. Tenga en cuenta que mediciones pueden variar por tamaño
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


MICHAEL Gunmetal Michael MICHAEL Kors Womens Keiko Slide Gunmetal Slide/ Silver eb4362f - 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

MICHAEL Gunmetal Michael MICHAEL Kors Womens Keiko Slide Gunmetal Slide/ Silver eb4362f

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.


MICHAEL Gunmetal Michael MICHAEL Kors Womens Keiko Slide Gunmetal Slide/ Silver eb4362f

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 plana de mujer Dayna de Lauren Ralph Lauren Broncearse , Earth Origins Selby Azul marroquí , Sandalia de mujer Teva W Terra-Float Livia Negro Negro , Havaianas Logo delgado femenino Metallic Flip Flop Aceituna verde Sandalia de cuña Bolivia SoftWalk para mujer Negro , Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Ante rojo Bota de tobillo Yama para mujer de FLY London Ante rojo , Birkenstock Betula con licencia de la mujer Mia Birko-Flor & iquest; Blanco básico , Vionic Women s Chill Larrun Loafer Serpiente de marfil Propet Women s Fisherman Sandal marrón , Chaco Mujer Aubrey Sandal Aleación , Sandalia Drew Cascade para mujer Sand Nubuck , Bota de tobillo Yama para mujer de FLY London Lodo Aceitado , Chanclas de mujer Havaianas Animal Arena gris / rosa , Sandalia Sharley Slide Vince Camuto para mujer Red Hot Rio , Sandalias de vestir Korthina de mujer de Vince Camuto luz de la luna , KEEN Kids Seacamp Ii CNX Water Shoe Blue Depths / Gargoyle , Mocasín de mujer RIALTO Vette Marrón / E-print , La sandalia para mujer FLEXX Shoreline Black Saratoga CLARKS Sandalia de cuña Arla Jacory para mujer Elástico Fucsia Profundo Kate Spade New York Women s Ivonna Sandal Lienzo de rayas azul marino / crema Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja Sandalia con tachuelas Tali Mini Bow de Cole Haan para mujer Piedra pómez Sandalia deportiva Marc Jacobs Women s Somewhere Rojo / Multi , CLARKS Sandalia Arla Kaydin para mujer Tela elástica negra NIKE Men s Benassi Just Do It Athletic Sandal Blanco / Variedad Royal White Plano original de la mujer del Dr. Scholl s Collection Collection Kimberned Pointed Toe Henna Multi Impreso Terciopelo Sandalia con correa en T para mujer ECCO Paloma Salvaje / Paloma Salvaje , Sandalia Piper Jambu Mujeres Azul marino / mezclilla ,
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 retro Gel-Lyte III para mujer ASICS Azul clásico / azul clásico , Saucony Women s Ride 10 Running Shoe Gunmetal / rosa , BZees Sandalia con cuña de beso para mujer Blue Multi Print Sandalia de mujer Drew Alana Blanco negro , Under Armour Women s Charged Rebel Negro (001) / Rhino Gray , Calzado deportivo New Balance para mujer Vazee Rush V3 Vivid Coral / Fiji Zapatillas de trail running Xa Pro 3D W para mujer Salomon Azul Medieval Sandalia plana de mujer Dayna de Lauren Ralph Lauren Vainilla Zapatillas para correr Nike Zoom All Out Flyknit para mujer Bright Crimson / White-tm Rd-volt Zapato de running COMUTORA para mujer ASICS Mar tempestuoso / Mar tempestuoso / Blanco , Zapatilla deportiva Kailee de mujer Ryka Insignia Blue / Yucca Mint / Summer Grey Zapatilla de running 33-FA para mujer de ASICS Vapor / Diva Pink / Melon , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Negro , Sandalias EVA Gizeh de mujer Birkenstock Blanco Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Champagne / Metálico , Calzado de senderismo Go Joy 15603 de Skechers para mujer Armada , Zapatillas para caminar Wren Propet para mujer Negro , New Balance 711v3 Heather Cross Trainer para mujeres Acero / Negro , Sandalia Juniper Mujer Chaco Nutria , Bomba Amie Naturalizer para mujeres Negro , Zapatillas para caminar Reebok Women s Work N Cushion 3.0 Negro , 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) 1606- Pearl Gold , Ariat Womens Volant S Tall Zip Tall Riding Negro Zapatilla deslizante para mujer UGG Negro / Natural Bota Willa Tweed para mujer de Caterpillar Azúcar morena para mujer , J. Rene233; Lemmon Botas Karel para mujer de UGG castaña , Botas altas Erykah Geox para mujer Negro ED Ellen DeGeneres Womens Casie Fósil / Rosado Champán Gamuza / Tejer , Mephisto Sabatina Tuxedo Oxford para mujer Patente negra

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.

MICHAEL Gunmetal Michael MICHAEL Kors Womens Keiko Slide Gunmetal Slide/ Silver eb4362f - 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 MICHAEL Gunmetal Michael MICHAEL Kors Womens Keiko Slide Gunmetal Slide/ Silver eb4362f 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
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1695720
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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.