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.

Zapatillas Praga mujer de deporte de cuero de Praga Drew de Shoe para mujer Estaño c30020d




Cuero
sintético único
suave, piel genuina superior
Suave forro de tela y cierre de cordones
Plus Sistema de ajuste para mayor profundidad y doble profundidad
Extraíble, Dri-Lex forrado plantilla con AEGIS Microbe Escudo
Templado mango de acero y flexible, ligero, suela de poliuretano
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


Zapatillas Praga mujer de deporte de cuero de Praga Drew de Shoe para mujer Estaño c30020d - movie-store.online

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

Zapatillas Praga mujer de deporte de cuero de Praga Drew de Shoe para mujer Estaño c30020d

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.


Zapatillas Praga mujer de deporte de cuero de Praga Drew de Shoe para mujer Estaño c30020d

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

Zapato para caminar WW411v2 New Balance para mujer Gris , Easy Spirit Solana Oxford para mujeres Gris oscuro Nubuck múltiple Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Plano extra ancho Feet Breeze Womens Zapatos para caminar , Zapato para caminar Skechers Go Joy para mujer Carbón Bernie Mev Women s Gummies Gem Flat Reflectante negro Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Heather negro / gris SAS Womens Sunny Mora Zapatillas para caminar Ryka Women s Devo Plus 2 Azul rosado Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Blanco , Sandalias de mujer Ros Hommerson Mellow Arena Zapatillas para caminar Wallet Lenex X821 de Aetrex para mujer Blanco Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Gris / Franela Zapatillas para caminar Propell Mujer Travellite MJ rojo Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Carbón , Zapatillas Skechers para mujeres Go-Sandy Boating Azul (Nvw) Easy Spirit - Zapatillas Faisal2 para mujer Roca gris , Skechers Women s Walk Walk Lite-Daisy Loafer Gris pardo , Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer gris , Skechers Women s Walk Walk Lite-Daisy Loafer Armada , Zapatillas de viaje Proather para mujer II Blanco Apex Women s Ambulator Biomechanical Negro , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro , Easy Spirit Mujer Ezcool Mule Gris / Gris Multi Gamuza , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Negro Zapatillas Slip-on Keds Womens X Design Love Fest Triple Decker Natural Toni Pons Womens Vic Black Sedavi Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Gris / Franela Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Pewter Suede ,
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 NIKE Mens N98 Tribute Track Jacket Armería azul marino / blanco , Zapatillas NIKE Air Huarache Run SE (GS) Girls Fashion 904538 Plata Drew Womens Ariana Dusty Gold Leather Calzado de running NIKE Kids Flex 2017 RN (GS) Pure Platinum / Tough Red Merrell Siren Edge Hiker para mujer Aluminio , Salomon Unisex S-Lab Wings 8 SG Mesh, Textile Running Sneakers Negro , FitFlop Womens F-Sporty II Python Print Negro , Sandalia Natalia Ryka para mujer Blanco negro Zapatillas de running Brooks Women s Glycerin 16 Road Púrpura / rosa / gris , Zapatillas de viaje Proather para mujer II Blanco , Sandalia de vestir Bayleen II de mujer de Cole Haan Néctar / naranja picante Zapatillas de running Under Armour Micro G Pursuit D para mujer Antracita (100) / Negro , Zapatillas de running W630v5 para mujer New Balance Negro / Alpha Pink / Plata Metálica Sandalia Harman Bandolino para mujer Tormenta Alpargata Chambray Clara denim de mujer TOMS Drizzle Grey Hemp / Multi Tribal Zapatilla Ascot de hombre UGG Ohia Red / Dark Java , Entrenador cruzado NIKE Flex 8 femenino Negro / Blanco-antracita , Zapatos atléticos diabéticos Katy Turquoise de Dr. Comfort para mujer Turquesa Diadora Unisex Intrepid NYL Negro Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Blanco FootJoy Colección Casual para Mujer Zapatos de Lino de Golf Medianoche / tela de mezclilla , Zapatillas para caminar Reebok Ultra V DMX Max para mujer Negro / gris plano Sam Edelman Women s Bay Slide Sandal Cuero negro Sandalia Fabia Slide de Spring Step para mujer Blanco / Multi Sandalia de cuero Merrell Dwaykair Seaway para mujer Helecho Sandalia Jula de Donald J Pliner para mujer Negro , Sam Edelman Sandalia Glenn para mujer Marfil moderno , Zapatillas de deporte de mujer Easy Spirit de Easy Spirit Gris oscuro Nubuck múltiple Propet Women s Alice Oxford Negro Mocasines casuales Alpargata Deconstructed de TOMS para mujer Ante azul marino

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.

Zapatillas Praga mujer de deporte de cuero de Praga Drew de Shoe para mujer Estaño c30020d - movie-store.online


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 Zapatillas Praga mujer de deporte de cuero de Praga Drew de Shoe para mujer Estaño c30020d 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 1366269
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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.