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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Cuero
Importado
Goma suela
Medidas de talón aproximadamente 2
Medidas de la plataforma aproximadamente 0.75
Tobogán de cuña sandalia con cuero de tiras superior con floral apliques y cuentas clúster
Flexible, suela duradera
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

CLARKS Lexi Myrtle CLARKS Sandal Sandal para mujer B00LWABROU marrón 5ee3329

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,

Sandalia de cuero Merrell Dwayka Seaway Post para mujer Negro Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero negro , Naughty Monkey Mujer Hey Pony Sandal Blanco Sanda Verra Teva para mujer Rayas modernas negro / blanco , Sandalias de gamuza con tacón de aguja Lulu Aztek de FitFlop para mujer Caramelo , Zapatos OLUKAI Pehuea - Mujer Arcilla / arcilla Imagine Sandalia de tacón IM-Dailey de Vince Camuto para mujer Oro suave JBU by Jambu Women s Wildflower Sandal Carbón , Sandalia Terra FI Lite de Teva para mujer City Lights Black / Multi Earth Origins Hadley Hibiscus Red , NAOT Kayla Elegant Sandalias de mujer Cuero Buffalo Nueve West Women s Lynneah Leather Mule Cuero natural oscuro , Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero Taupe Drew Womens Element Naranja quemada Sandalia plana de mujer Dayna de Lauren Ralph Lauren Vainilla , XYD Prom Dancing Shoes Elegant Open Toe Strappy Sandalias de tacón de tobillo Wrap vestido de bombas para las mujeres Plata NIKE Men s Benassi Just Do It Athletic Sandal Binario azul / menta fresca Skechers Mujer Paso a Paso-Markers-Stretch Vamp Sling Back Sandal Negro , CLARKS Sandalia Saylie Moon para mujer Peltre Metálico Cuero , Sandalia con cuña Verona de mujer Alegria Yehaw Black J.Renee Womens Soncino Cuero metalizado Taupe Sandalia de mujer David Tate Violet Negro Sandalia Misty Fisherman de Ryka para mujer Negro / Yucca Mint Sandalia Dionay Calvin Klein para mujer Rojo carmesí Sandalia ajustada FitFlop para mujer Mezclilla Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro , Marc Fisher LTD Womens Alida Espadrille Wedge Ante rojo Bomba Jude para mujer Adrianna Papell Marfil , Bomba de plataforma Kiara para mujer Badgley Mischka Blanco ,
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 Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout , Calzado de running ft. Perpetua ft de memoria Fila Women s Castlerock / Sugarplum / White , Zapato NIKE Roshe One femenino Negro / antracita / negro , Zapatillas de running New Balance para mujer 590v3 Reflection / Vintage Indio OLUKAI Honoka a Sandals - Mujer Sahara / Sahara Slingback de Seleste de las mujeres naturales del alma Cuero negro , Sandalias de tacón de soñador Easy Street para mujer Pewter Metallic CLARKS Sillian Stork Fisherman Sandal para mujer Negro Easy Street 30-8320 Sandalia airosa para mujer Estiramiento rojo , LifeStride Women s Camille Sandal Rosa caliente Easy Spirit Women s Traveltime Mule Patente negra / plateada , CLARKS Zapatillas CloudSteppers para mujer Sillian Tino Zapatillas con cordones Tela gris aterciopelada , Bernie Mev Women s Gummies Gem Flat Paja , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede Vans Auténtico (Tarde en la noche) Hot Pink / Cupcakes , Sandalia para mujer Ivanka Trump Earin Slide Ante amarillo Diapositivas de Sam Edelman Gertrude para mujeres entrecruzadas Negro , Bota de montar Xandy para mujer LifeStride Bronceado oscuro Dakota femenina de UGG Café exprés Zapatilla clásica Dualyz Unisex Kush con suela extraíble Gris oscuro / gris claro , ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ 5008m Rainbow Sparkle & Snake Copper Bota de tobillo de lona Martins Winsted Mid Olive de Dr. Martens Aceituna Sandalia Birkenstock de cuero Florida para mujer Negro , FRYE Bota corta Renee Seam para mujer Slate-72066 Regarde Le Ciel Giovanna-Camel Multi Laced Oxford - Skechers Women s Bikers Winner Sneaker Negro , Zapatillas de deporte lavables Walker Propet para mujer Sr Black , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Frost Grey / True White TOMS - Slip-On de lienzo clásico para hombre Azul eléctrico Dexter Bowling Deanna derecha zapatos de bolos Gris 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.

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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 CLARKS Lexi Myrtle CLARKS Sandal Sandal para mujer B00LWABROU marrón 5ee3329 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 643257
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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.