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 para correr Mizuno Women s Wave para Rider 21 Gray 19086 High Rise/ Gray Stone 8840c9d




Textil y sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Nube onda ofrece un suave, animado y liviano sentirse
Diseñado Malla superior para un calcetín como todavía ajuste transpirable
u4iCx la amortiguación mejora sensación de nube
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 s Wave para Rider 21 Gray 19086 High Rise/ Gray Stone 8840c9d - 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 s Wave para Rider 21 Gray 19086 High Rise/ Gray Stone 8840c9d

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 s Wave para Rider 21 Gray 19086 High Rise/ Gray Stone 8840c9d

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

Camiseta adidas Swift Run W para mujer Trace rosa / gris tres / cristal blanco Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Negro En la ejecución de calzado de mujer en la nube Denim / Blanco Calzado de running NIKE RN 2017 2017 para mujer Negro / blanco / antracita / gris lobo Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / gris antracita-cool Zapatillas de running NIKE Lunar Skyelux para mujer Celadon-azul Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono , Zapatillas para correr Mizuno Women s Wave Creation 15 Azul Zapatillas de running New Balance 990v4 para mujer Plata menta / plata menta , Zapatillas de running adidas Alphabounce Beyond W, para mujer Núcleo negro, núcleo negro, tela gris cinco Desafío femenino de Brooks 9 Antracita / cerámica / flor de la pasión Zapatillas de running Altra Women s Intuition 4 Púrpura / negro Zapatillas de running Hurricane ISO 2 Refl de Saucony para mujer Gris blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Blanco / Negro-volt-cloro Azul Saucony Women s Ride 10 zapatillas para correr Blanco azul Zapatillas de running NIKE Lunarglide 9 para mujer Negro / blanco / gris oscuro / gris lobo , NIKE Women s Wmns Air Max 270, Ocean Bliss / Blanco-Negro-Hot Punch Ocean Bliss / White-black-hot Punch Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Blanco ASICS Kanmei Shoe Casual para mujer Negro / Negro / Blanco Zapatillas de running ISO para mujeres de la guía Saucony Pizarra / Melocotón Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , Zapatillas para correr Mizuno Women s Wave Creation 15 Rosa / negro / rosa Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace Zapatillas de running Saucony para mujer Omni 16 Gris azul , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro , Zapatillas de running Saucony Women s Freedom ISO Berry | Zapatillas de running Mizuno Wave Viper para mujer Negro-fucsia Pruple-plata HOKA ONE ONE Clifton 3 para mujer Zapato de running Citrus / Blue Jewel Size 11 M US - ,
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 ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Black White Hot Punch 006 Zapatillas de running para correr adidas outdoor Kanadia 8 TR para mujer Lino / Col. Rookite 325 Trail Runner de mujeres Inov-8 Gris oscuro , Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos castaña Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black Zapatillas de viaje Propet para mujer Denim Taupe DAWGS Womens Arch Support Z Sandals Bienvenido a Las Vegas Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Blanco , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Blanco NIKE Men s Roshe Two Running Shoe Wolf Gray / White / Volt Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black , Zapatillas de deporte Romy de Easy Spirit para mujer Blanco azul claro Calzado de entrenamiento casual Comfort WX409V3 para mujer New Balance Negro Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Pure Platinum / Night Purple / Wolf Gray , Zapatillas Cross-Trainer de entrenamiento Newonic Flexonic WX66V2 para mujer Gunmetal / Alpha I-RUNNER Eliza Women s Therapeutic Athletic Extra Profundidad Zapatilla de cuero / de malla con cordones Gris azul , Diadora N9000 Men Toe Snthetic Blue Sneakers rojo Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Blanco invierno Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Malt Taupe Action Leather Sandalia Dian J Pliner Mujer Ginie Slide Oro , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Gold Sparklenet (Orden personalizado, 4-6 semanas de tiempo de proceso) 50 tonos de oro 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) 7013- Bronceado dorado , Steve Madden Mujeres Hugh Flat Terciopelo azul Skechers Bikers para mujeres - Roamer Ante negro relajada , Naot Women s Kumara Flat Onxy cuero / hilos de plata de cuero Cole Haan Women s 2.Zerogrand Laser Wing Oxford Nectar / Vapor Grey , Original S.W.A.T. Negro TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Drizzle Grey Systhetic Suede ZXD Men Chic Low Heel Slip On Mocasines Planos de punta redonda Casual Zapatos de barco Estampado de leopardo ,

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 s Wave para Rider 21 Gray 19086 High Rise/ Gray Stone 8840c9d - 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 s Wave para Rider 21 Gray 19086 High Rise/ Gray Stone 8840c9d 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 2365029
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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.