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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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


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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 Merlot Montana B01N3P0J9V Earth para Earth mujer Merlot 61388f0

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.


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

Bota de cuña de ante Leigh para mujer Crocs Negro Zapato deportivo Fiat Fusion para mujer Azulejo , Bota alta cosida a la rodilla de mujer FRYE Charcoal-72068 Kodiak Women s Claire impermeables con cordones botines , Bota MuckBoots Rover II (Niño pequeño / Niño pequeño / Niño grande) Camo , Jeffrey Campbell Botines de tacón de bloque de sirena de mujer Neopreno negro Caminante de Flyroam de las mujeres de Timberland Jet Black Dt Suede , Wedge del desierto de las mujeres de TOMS Cuero crudo gamuza / kiltie Blondo Womens Vivi Waterproof Seta de gamuza , Mocasín con flecos Minnetonka para mujer Negro BareTraps Women s Bt Rylen Tobillo Botín Marron oscuro , Bomba de cuña Cleopatra Bernie Mev para mujer Negro / Metálico , Botas de moda veganas Castel Indigo para mujer de Dr. Martens Índigo BURGAN 871 TAC Force Bota de cremallera táctica de 8 pulgadas Negro Prop & eacute; t Blizzard de mujer Zip de invierno con cremallera Tejido negro / azteca , Botas cortas de cuero Mordenmiss para mujer Zapatos nuevos Estilo 1 paño grueso y suave negro , Sandalia de vestir OTBT Free Spirit para mujer Nuez dura , Kaitlyn Pan Block Heel Microsuede Botas sobre la rodilla ajustadas Caramelo , Wedge del desierto de las mujeres de TOMS Suede marrón chocolate con piel de oveja , Ovation Celebrity Slim Secret Knee Patch Gris 055398210 Timberland PRO - Botas de trabajo TiTAN para mujer - Café marrón Bota de moto Fulton para mujer Roxy Negro , Bota de rastro Eddie Bauer para mujer Esprsso (marrón) Bota de combate Myra Lug para mujer FRYE Forest Soft Oiled Suede , Botas de mujer Paul Green Jada Negro Bota de Imogene para mujer Cole Haan II Harvest Brown Walking Cradles Ante Ante de las mujeres Cuero negro Jeffrey Campbell Womens Lilith-2 Boot Nuevo Combo floral SOREL Bota de cuero baja grande para mujer Mayor / Fósil
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 Flyknit 2017 Dark Purple Dust / Black , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta Zapatillas de running NIKE Air Max Thea para mujer Blanco negro , Calzado de running con punta de material compuesto, resistente al deslizamiento, Meiera 2, de Fila, para mujeres Monumento / Castlerock / Safety Yellow Merrell Women s Bare Access Flex Trail Runner Hielo Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul , Zapatillas de running Mizuno Women s Wave Hitogami 3 Cítricos / Pensamiento , New Balance Vazee Summit V2 Trail Running Shoe para mujer Negro / verde militar Zapatillas de mujer Salomon X Ultra 3 GTX para mujer Imán , Zapatos OLUKAI Pehuea - Mujer Granadina / Granadina Zapatos de lona OLUKAI Waialua - Mujer Dusty Olive / Dusty Olive Zapatos para caminar de cuero Stretch Walker de Pedors Womens Negro Zapatillas clásicas Vionic para mujer Walker Blanco / rosa Zapatillas de running NIKE Lunartempo 2 para mujer Racer azul / negro / blanco Sandalias de vestir de piña de mujer de Jack Rogers Brillante rosa / dorado , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro , Sandalia atlética de la sandalia del deporte de Lila 6 de las mujeres de MBT Gris / verde azulado / melocotón / blanco Sandalia LifeStride para mujer Eden Negro , Bota de mujer Ciara de Musse & Cloud Negro , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Paillette Scintillate Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre (Van Doren) Sklsnwfk / Rcngr , Sandalia Cato Dolce Vita para mujer Orchid Suede Sandalia de cuña Balen de Charles by Charles David para mujer Azul , Zapatillas de beisbol PowerAlley 4 W TPU de adidas Performance para mujer Real-blanco , Cobb Hill Rockport - Aubrey Dress Pump para mujer Armada Zapatillas Spin Biking SPD de montaña / montaña Tommaso Terra 100 para mujer Negro / verde azulado Cloud Aline Fantasía negra , Bota Darcy para mujer Easy Street Granito bruñido Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento rojo T.U.K. borgoña ,

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.

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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 Merlot Montana B01N3P0J9V Earth para Earth mujer Merlot 61388f0 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

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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.