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 de running/ W adidas Alphabounce Em W para mujer Em Easy Coral/ Black/ Easy Orange 9fcaed2




Textil / Sintético
Importado
Sintético único
Runner tipo: neutral
BOUNCE proporciona energizado comodidad para todos deportes todo el día
sin costura FORGEDMESH superior diseñado con áreas de apoyo y estirar para ayudar asegurar una costumbre ajuste que se adapta para cada movimiento
Textil forro para mayor comodidad
ADIWEAR ofertas de suela lo último en durabilidad de alto desgaste
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 de running/ W adidas Alphabounce Em W para mujer Em Easy Coral/ Black/ Easy Orange 9fcaed2 - 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 de running/ W adidas Alphabounce Em W para mujer Em Easy Coral/ Black/ Easy Orange 9fcaed2

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 de running/ W adidas Alphabounce Em W para mujer Em Easy Coral/ Black/ Easy Orange 9fcaed2

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

Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas de running Reebok Print 2.0 para mujer Nube gris / azul cielo / negro / azul gable / blanco / plata Zapatillas de running W1540V2 New Balance para mujer Blanco azul , Zapatillas de running New Balance 990v4 para mujer Dark Mulberry Skechers Performance Women s Go Run 400 Obstruct Rosa caliente Zapatillas para correr ASICS Gel-Venture 6 para mujer Negro / Island Blue / Pink , Zapatillas para correr Altra Women s Torin 3.0 Azul / Coral , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco-negro-corredor Azul Zapatillas de running Saucony para mujer Omni 16 Teal / Citron , Zapatillas para correr Nike Women s Free Rn Distance 2 Hyper Violet / White Dark Iris Zapatillas de running NIKE Zooms Winflo 4 para mujer Antracita / gris oscuro / negro , Saucony Women s Guide 10 Zapatillas para correr Azul claro / azul , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Phantom / Black Zapatillas de running Mizuno Wave Rider 20 para mujer Gunmetal / Fuchsia Purple Zapatillas de running Wstro para mujer New Balance Alpha Pink / Black / Lime Glow Zapatillas de running adidas Performance Powerblaze W para mujer Gris / Metálico / Plata / Menta viva , Zapatillas de running NIKE para mujer Epic React Flyknit Negro / gris oscuro Brooks Womens Neuro 2 Negro / Púrpura Cactus Flor / Blanco adidas outdoor para mujer Terrex Swift R2 GTX Grey Three / Gray Two / Chalk Coral Zapatillas para correr NIKE Womens Sock Dart Midnight Navy / Negro-blanco Zapatillas de running Reebok para mujer All Terrain Super 2.0 Azul Neón / Azul Práctico / Azul Instinto / Amarillo Solar Zapatillas de running ASICS Gel-Nimbus 19 para mujer Azul Púrpura / Violeta / Airy Blue Zapatillas para correr New Balance Rushv3 para mujer Acero / Negro Zapatillas de running adidas Aerobounce Pr W para mujer Core Black, Silver Met, Ftwr White ASICS Kanmei Shoe Casual para mujer Negro / Negro / Blanco , Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Onyx / Carbono Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Espuma de mar / cielo cubierto Under Armour Speedform Velociti para mujeres Rhino Gray / Glacier Grey / Black Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Estratos
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 Calzado de running Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle , Zapatillas de running A7000WX12 para mujer Apex gris Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro CLARKS Lexi Myrtle Sandal para mujer Cuero rojo , BCBGeneration Mujer Addie Sandal Sonrojo , Earth Womens Allegro Ballet Flat Oro rosa , Sandalia Sharley Slide Vince Camuto para mujer Floral Prop & eacute; t Zapatillas para caminar TravelFit para mujer Lluvia Turquesa Zapatillas deportivas sintéticas Drew Shoe para mujer Estampado floral fucsia Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Blanco azul , Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 Voltaje Verde / Tiza Azul-blanco , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro Supra Vaider LC Sneaker Carbón blanco Calzado deportivo DAWGS para mujer Golf Spirit Blanco / Naranja Sandalia de girasol Aerosoles para mujer Oro , Kaepa Youth Stellarlyte Cheer Shoe (par) Blanco Sandalia de plataforma de playa Coast de Seychelles Rosa Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7017EB Cómodo-Muy Fino 3 \ de 5] Cobre Bomba Sadie Wedge 40MM de Cole Haan para mujer Roccia Snake , Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Serpiente Glerups Modelo A Gris Zapatilla Oh Ewe II de mujer Acorn Carbón Zapatos de mujer VANELi Allure Pumps Nappa negro / elástico negro , Botín de tobillo Hadley de Bella Vita para mujer Ante negro , Zapatilla Kendyl de mujer UGG Negro Calvin Klein Women s Dayo Oxford metálico plano Gris , Corso Como Women s Galaxy Oxford Denim Stripe Laser Leather , CLARKS Zapatillas Tri Etch para mujer Cuero blanco Mocasín de gamuza Kilty de Minnetonka para mujer Gris pardo ,

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 de running/ W adidas Alphabounce Em W para mujer Em Easy Coral/ Black/ Easy Orange 9fcaed2 - 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 de running/ W adidas Alphabounce Em W para mujer Em Easy Coral/ Black/ Easy Orange 9fcaed2 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.

Carretera en funcionamiento
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 845462
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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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