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
Goma suela
Plus Fitting Sistema - Dos extraíbles Camas para agregar y doble profundidad. Plantilla de poliuretano con Drilex Topcover.
Flexible, Ligero, poliuretano Suela exterior
Cuero Revestimiento. Tobillera.
Templado Vástago de acero.
Capacidad Ortótica prescrita.
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

Drew Smiles Slip Microdot para Smiles mujer Negro para/ Microdot a342723

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.


Drew Smiles Slip Microdot para Smiles mujer Negro para/ Microdot a342723

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 Vionic Shelley Slide para mujer Negro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul Zapatillas deportivas para caminar Spira CloudWalker para mujer con resortes Blanco negro , Drew Shoe Athena para mujeres Blanco azul , Zapatillas de neopreno para caminar Pedors Womens Classic MAX Negro , Calzado para caminar CUSH + WA365v1 New Balance para mujer Blanco negro Rieker Womens 41346 Doris 46 Blanco apagado / Metálico / Hielo , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Acuarela / Iridiscente Zapatillas para caminar Cinch de mujeres Devotion Plus de Ryka Azul negro , NIKE Men s rsquo; s Zapatos casuales Darwin Zapatillas deportivas ligeras Confort Running Negro / blanco eléctrico verde Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Acuarela floral Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata blanca , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada , Zapato de caminar WW769V1 para mujer New Balance marrón , Zapato para caminar Skechers Go Joy para mujer Azul marino / blanco , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Pewter Suede , Trotters Mujer Simmy Mary Jane Flat Sage Veg Tumbled Leather , Sandalias SAS para mujer, Nudu Slide Oro Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Piedra , Easy Spirit Solana Oxford para mujeres Gris oscuro , Hush Puppies Mujeres Power Walker Ii Loafer Cuero blanco PUMA Mujer Basket Corazón NS WN Sneaker Puma Black-puma Black , Zapatillas para caminar Go Walk 4 Achiever de Skechers Performance para mujer Negro , La alegría de la mujer de la tierra Jengibre , Saucony Mujer Grid Omni Walker-W Plata blanca Zapato para caminar Metrolyte para mujer ASICS Negro / Regatta Blue / Regatta Blue , Zapatillas para caminar Brooks Women s Addiction Walker Blanco , Zapatillas para caminar Wren Propet para mujer Negro , Hush Puppies Mujeres Power Walker Ii Loafer Cuero de abedul ,
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 Converse Chuck Taylor All Star Seasonal Color Hola Pulir Sandalia deportiva W Sirra de mujer Teva para mujer Azulejo azul Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco Blanco , Converse Chuck Taylor All Star Seasonal Color Hola Beach Glass Zapato de running ASICS GEL-Venture 5 para mujer Frost Grey / Grey / Silver / Magenta NIKE Mujeres Air Zoom Pegasus 33 Rosa fuego / blanco / brillante Correpasillos Merrell para mujer 3 Trail Runner Punch ácido Sbicca Women s Salem Sandal Estaño Pantalón largo clásico NIKE Golf para mujer Tech Essentials Pale Gray / Sail-light Bone Propet Walker de la estabilidad de las mujeres Gris , Zapato de caminar sin tocar Skechers Performance para mujer Go Step Gamuza de carbón , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Naranja / Rosa Sandalias de mujer Lola Sabbia Lily rojo , Finn Comfort Women s Ikebukuro Oxford Cigar Luxory Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Blanco azul Entrenador cruzado 611v1 de New Balance para mujeres Silver Mink / Seafoam Zapatilla deportiva perforada con láser Mephisto para mujer Hielo blanco liso / plateado , Zapatillas adidas Kids Cloudfoam Race K Sneaker Azul / Solar Azul / Blanco Zapatillas adidas Barricade Classic Bounce Tennis, de adidas, para mujer Blanco / Plata Metálico / Gris Sólido Ligero Heather Vans Unisex Old Skool Zapatillas clásicas de skate (Classic Tumble) Negro / True White Zapatillas de golf Skechers Women s Go Golf Birdie Blanco negro , Sandalia HEATHER2 Slide de Via Spiga para mujer Pelo de ladrillo pony , Trotters Lucy Ballet para mujer Blanquecino Zapatillas de deporte unisex de bajo perfil Negro Vionic con Orthaheel Indulge Gracie Zapatillas para mujer Azul claro Zapatilla de cuña de cuero para mujer Jacques Levine 1221 Blanco Zapatillas de tenis Grandpro Tennis de Cole Haan para mujer Vapor Gris Nubuck / Roccia Snake , Bota de mujer Toms Alpa Ante negro FLY London Yoss de mujeres Luxor negro Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Plata

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 Drew Smiles Slip Microdot para Smiles mujer Negro para/ Microdot a342723 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

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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.