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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Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Todo nuevo Mizuno Smooth Paseo: nuevo máximo tecnología de ondas de nube junto con un de cuerpo entero, articulado La entresuela U4icX ofrece un peluche y flexible paseo
nuevo más grueso y anatómico superior calcetín para mejor ajuste y comodidad
U4icX Forro de Strobel para mayor comodidad
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 para correr Mizuno Women de s Wave/ Sky Azulejo Women azul/ plata/ amarillo de seguridad bd45a90 - 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 para correr Mizuno Women de s Wave/ Sky Azulejo Women azul/ plata/ amarillo de seguridad bd45a90

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 para correr Mizuno Women de s Wave/ Sky Azulejo Women azul/ plata/ amarillo de seguridad bd45a90

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

Zapatillas de running NIKE Zooms Winflo 4 para mujer Ruido Aqua / Blue Force / Glacier , Zapatillas de running adidas Women s Edge Lux W Black / Vapor Gray Metallic / Orchid Tint Zapatillas de running Kinvara 5 Saucony para mujer Menta / cereza Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello Zapatillas de running Saucony Women s Freedom ISO rojo , Calzado de running NIKE RN 2017 2017 para mujer Wolf Grey / Sunset Glow / Cool Grey , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco Zapatillas de running NIKE para mujer Lunarepic Flyknit Shield Verde Brillo / Mtlc Rojo Bronce-gris oscuro , Nike Hombres Air Zoom Pegasus 33 Loyal Blue / White-black , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Begonia Pink / Begonia Pink / Baton Rouge , NIKE Mujeres Downshifter 7 Blanco negro Zapatillas de running NIKE Lunarglide 9 para mujer Pure Platinum / Chrome , Zapatillas de running Kinvara 8 Saucony para mujer Berry White , Zapatillas de running Adidas Women s Energy Cloud WTC W Fácil menta / blanco / negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Blue Fox / Cerulean-college Navy Zapatillas para correr New Balance W840V3 para mujer Trueno Zapato deportivo para acolchado 635v2 New Balance para mujer Gris , NIKE Women s Tanjun Coral Stardust / Sail Brooks Womens Adrenaline GTS 18 Azul / Menta / Rosa , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Mica Blue / White / Racer Pink / Sport Fuchsia Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco azul Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono Nike Air Zoom Mariah Fk Racer para mujer Carbono ligero Zapatillas de running ASICS Gel-Cumulus 19 para mujer Gris / plata / lima , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul marino / gris , Short NIKE Tempo para mujer Volt / Black / White / Matte Silver , Calzado de running NIKE RN 2017 2017 para mujer Carbono ligero / Carbono ligero , Zapatillas de running adidas Women s Edge Lux W Ash Pearl / Chalk White / Chalk Coral , Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / Hola Lite ,
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 Zapatillas de running NIKE Womens Huarache Run Ultra PRM para mujer Blanco negro Sandalia Terrell Convertible II Merrell para mujer Tierra oscura , BZees por Naturalizer Dream Negro , NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Pure Platinum / Cool Grey-noise Aqua Mujer adidas Originals Ultraboost St Indigo crudo / tinta noble / naranja de alta resolución NIKE Men s Lunar Fingertrap Cross Trainer Negro / Gimnasio Azul-blanco , Zapatillas de running Mizuno Wave Rider 20 para mujer Blue Atoll / Silver Zapatillas de running Cavu para mujer HOKA ONE ONE Negro / Bluebird Sandalias Swiftwater Mesh de Crocs para mujer Fumar , Calzado deportivo Inov-8 Roclite 315 para mujer Verde azulado / negro , Zapatillas para caminar Skechers Go Step Lite-14750 para mujer Azul marino , Zapato para caminar WW411v2 New Balance para mujer Blanco Sandalias con plataforma SAS de cuero con punta abierta para mujer Negro , Trotters Mujer Simmy Mary Jane Flat Cuero caído Veg negro , Zapatillas de deporte cruzadas WX822V3 de New Balance para mujer Negro / Moire Graphic Bota de senderismo Gore-Tex de media caña alta para mujer Merrell Moab Beluga Trotters Mujer Simmy Mary Jane Flat Azul oscuro , Zapato de malla Nohea de OLUKAI - Hombre Negro Zapatillas de deporte Nike Womens Metcon 2 AMP Running 843972 Sneakers Shoes Azul resplandor / Negro Volt negro , Calvin Klein Women s Rivita Sandal Platino blanco Vans Unisex Old Skool Zapatillas clásicas de skate Boom Boom Black / True White Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Peltre negro , Sandalia tobillo de piscina con correa individual Iona de Gentle Souls Ternero de pelo negro , 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 7012- Serpiente Blanca , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 7004- Plata TOMS Clásicos de mujer Lienzo rojo , Botines Chelsea de Intendencia 2976 de Dr. Martens para hombre Negro , Zapatilla de deporte Bromly Oxford Fashion Aravon para mujer Black Multi Sanuk Women s Pair O Dice Flat Aceituna Sanuk Women s Donna Paige Flat Natural ,

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 para correr Mizuno Women de s Wave/ Sky Azulejo Women azul/ plata/ amarillo de seguridad bd45a90 - 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 para correr Mizuno Women de s Wave/ Sky Azulejo Women azul/ plata/ amarillo de seguridad bd45a90 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

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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.