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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Tela
Importada
Ir para un relajado y aspecto deportivo en el Easy Spirit & reg; Deiny.
Tejido parte superior con un punto diseño y una ronda dedo del pie.
Ligeramente lengua acolchada y collar.
Cordones cierre para un seguro y ajustable ajuste.
tela suave revestimientos. Acolchado plantilla para agregar soporte.Ligero suela hecha por el hombre.Importado.Medidas: Peso: 6 oz Mediciones del producto fueron tomadas usando tamaño 7.5, ancho M. Tenga en cuenta que las medidas puede variar según el tamaño. Peso de calzado es basado en un solo artículo, no un par
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,

Skechers Go Walk Lite-15350 Wide Sneaker para mujer Negro Dr. Comfort Annie-X Mujer diabética terapéutica Doble Profundidad Zapato Lycra Velcro Negro , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Plata , Skechers Synergy Positive Outcome Mujer Slip on Walking Sneakers Negro Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer gris , Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 Blanco / medianoche azul marino , Trotters Mujer Simmy Mary Jane Flat Ante azul marino Calzado de mujer de fila Cress Walking Fila Navy, Monumen, Blanco Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Pacífico / Plata , Prop & eacute; t Propet Zapato para caminar Eden para mujer Blanco / azul polvo , Skechers Women-14960 Sneaker Amarillo Reebok Women s Work N Cojín caminando zapato Negro Prop & eacute; t Zapatillas para caminar TravelFit para mujer Púrpura , Zapatillas para caminar Go Flex 2-14992 de Skechers Performance para mujer Azul marino / gris Calzado de senderismo Go Joy 15603 de Skechers para mujer Negro / gris , Zapatillas de deporte Romy de Easy Spirit para mujer Vapor-gris , Sandalias de mujer Ros Hommerson Mellow Armada , Zapato deportivo artesanal de mujer superfeet Addy Yuca Apis Mt. Negro , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Negro , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Naranja / Rosa , Easy Spirit - Zapatillas Faisal2 para mujer Azul oscuro , Zapatillas de deporte Arizona Trotters para mujer Negro Trotters Mujer Simmy Mary Jane Flat Gris claro , Sesto Meucci Womens Elki Sand Roxy Skechers Women s Go Walk Joy-15615 Sneaker Azul marino / Multi Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata , Walking Cradles Womens Henna Nubuck estampado rojo
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 OluKai Ohana Tropic Blue / Dark Java , Zapatillas de running adidas Alphabounce Rc W para mujer Indigo crudo / Gris Uno / Noble Indigo , Zapatillas NIKE Air Huarache Run SE (GS) Girls Fashion 904538 Amarillo / Blanco / Platino Zapatillas NIKE WMNS Air Max Thea 599409-701 para mujer Oro , Under Armour Women s Drift 2 Gris Nublado (941) / Blanco Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , Zapatillas de trail running Ryka Women s Intent XT 2 SR Negro / Plata cromo / Carbón Calzado deportivo New Balance para mujer Vazee Rush v2 Gris oscuro , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Negro , Zapatillas de running adidas Women s Edge Rc W Gris / Granito / Coral Fácil , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Lobo gris / negro-antracita Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Negro / Heritage Canvas / Diamond Tribal , Zapato de entrenamiento Cross-Trainer para entrenamiento Newonic Flexonic WX99V2 para mujer Negro / Champagne Metálico Clásicos de base Slip-on (Tm) para hombres de Vans (Checkerbord) Estate Blue / True White Sandalias planas para piscina Walking Cradles para mujer Envoltura Metálica Multi / Cork , Sandalia de cuero Lucili de Nine West para mujer Negro / Cuero Multi , Dublin Ladies Foundation Zip Paddock Negro Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Blanco negro , New Balance Sm4040n1 para mujeres Azul marino / blanco Zapatillas de ciclismo Smartodoors con suela de carbono o suela de tpu de nylon para carretera y MTB Blanco / Rojo B para carretera , Botín Colibri Nf63 para mujer El Naturalista Rioja / penacho , Bota Superflex para mujer FitFlop Negro Sandalias de la diapositiva del dedo del pie abierto de la moda de las mujeres de XYD Deslizamiento de cuero en las zapatillas de la zapatilla del vestido de la puerta Plata , Zapatillas terapéuticas Cuddle de Dr. Comfort Women Camello UGG Womens Riley Adular , Entrenador Lena Talones para mujer Niebla , Sandalias Jerusalem Sandal Sandalias Elan de goma para mujer Negro / Tostado , Coach Womens - Botas de punta de tobillo a puntera de Montana Mink Soft Vegan Leather , MICHAEL Michael Kors Mujeres Caroline Sling Toffee Patent , Bota de mujer Toms Alpa Cuero 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 Easy Spirit 19921 Deiny Women s Easy Walking Spirit Vapor 4ee366c 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 406383
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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.