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 único
Eje mide aproximadamente parte baja del arco
Cuero superior
Corte bajo diseño
troquelado Entresuela de EVA
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

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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,

Drew Shoe La alegría de las mujeres Combo marrón Easy Spirit Mujer Ezcool Mule Gris / Gris Multi Gamuza , Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón Prop & eacute; t Zapatillas para caminar TravelFit para mujer naranja , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Equipaje Tan Ballet de niña de flores Aerosoles femenino Serpiente gris , Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 Blanco / Puro Platino / Plata Metálica , Zapatillas para caminar Propell Mujer Travellite MJ Plata Zapatillas de viaje Proather para mujer II Armada , Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer Teal Zapato para caminar WW411v2 New Balance para mujer Blanco Trotters Mujer Simmy Mary Jane Flat Azul oscuro Zapatillas para caminar Go Step Lite-Petal 14723 de Skechers Performance para mujer Armada , Skechers Women s Walk Walk Lite-Daisy Loafer Armada Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro Zapatillas de deporte Dr. Comfort Sandy para mujer Gris Zapatos de mujer Spira Classic Walker 2 con resortes Negro , Zapatillas para caminar Ryka Women s Devo Plus 2 Azul rosado , Zapatillas para caminar adidas outdoor Terrex Voyager CW CP W para mujer Gray Four / Black / Chalk White , Drew Shoe Women s Tulip Comfort Black Print , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco Easy Spirit Women s Motion Lace up Oxford Blanco Trotters Mujer Simmy Mary Jane Flat Sabio Sperry Womens Fathom Blanquecino , Hush Puppies Men s Gus Oxford Marron oscuro , Easy Spirit - Zapatillas Faisal2 para mujer Rosado Zapatillas para caminar Brooks Women s Addiction Walker Blanco , Calzado de espuma fresca New Balance para mujer 1165v1 Púrpura , Zapatillas para caminar Skechers Go 4 Merge para mujer Multi ,
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 Wolky Women s Rio Sandalia Azul , Zapato NIKE Roshe One femenino Negro / antracita / negro , Juego de mujer Diadora Low Mirror Sneaker Ice Brook NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Negro / Gris Oscuro / Verde Palm Zapatillas de running Swift para mujer adidas Originals Base Verde / Blanco / Negro , Sandalia deportiva Merrell Adhera Three para mujeres Armada Zapatillas de deporte casual Gazelle de adidas para hombre Collegiate Royal / Running White / Cream White , Zapatillas de running adidas Aerobounce W para mujer Indigo crudo / plata metálica / naranja de alta resolución Inov-8 X-Talon 225 Trail Runner Rojo / negro / gris , Zapato de agua Tetrex Rapid Crest para mujer Merrell Lima , Skechers Performance Go Walk 4 Luxury para mujeres Blanco , Sandalias planas Stevie para mujer Dansko Cuero blanco , Zapatos de tierra Camino Cuero negro premium , Under Armour Women s Street Precision Low Lux Bayou Blue (953) / Marfil Nike Tech Xtreme Cadet Glove Pure Platinum / Blanco / Volt / Black Under Armour Threadborne Push para mujer Gris Nublado (102) / Rosa Tropic Zapatillas Superga 2750 Cotu para mujer Total de Marina Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Blanco negro , Zapatillas de golf NIKE Explorer 2 de mujer AA1846-101 Blanco / Blanco / Gris Lobo / Plata Metálica , Zapatillas de golf Adidas, sin puntas, adidas Women s W Ftwr Blanco / Plateado Metálico / Baja Azul-tmag , BareTraps Womens Jimina Open Toe sandalias de diapositivas casuales Negro Birkenstock Arizona Sandalia de ante con suela suave Cuero de cobre Dr. Martens Womens Eastern Art Pascal Boots Grano de paja revés oriental TOMS Clásicos de mujer Gris claro Cuello de piel con cuentas y conejo de piel de conejo Laurentian, pantuflas de ante negro FRYE Womens Billy Short Marrón oscuro lavado antiguo tire hacia arriba Baabuk Wool Sneaker - Mujer Azul marino / blanco , Zapatillas de fútbol MW-Tech RB R TF para césped Diadora para hombre Negro / amarillo fluorescente Socofy Block Heel Botines, para mujeres Color bohemio Patrón de empalme Cremallera lateral tobillo Botas de cuero Azul Easy Spirit Women s Traveltime Mule Cuero negro / negro ,

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 Zapato de caminata caminata Reebok para Dust/ mujer Cl Lthr Desert Dust/ White 44862ba 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 126332
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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.


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