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 ciclismo Bicycle ciclismo para mujer B07CN49MJS Sidi Alba Road Bicycle Negro 2db9064




Material superior: Cierre de Politex
: dial, gancho y lazo
Suela: Millenium 4 carbon Compuesto
Grapa Compatibilidad: Carretera de 3 hoyos
Recomendado Uso: ciclismo de ruta
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 ciclismo Bicycle ciclismo para mujer B07CN49MJS Sidi Alba Road Bicycle Negro 2db9064 - 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 ciclismo Bicycle ciclismo para mujer B07CN49MJS Sidi Alba Road Bicycle Negro 2db9064

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 ciclismo Bicycle ciclismo para mujer B07CN49MJS Sidi Alba Road Bicycle Negro 2db9064

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 ciclismo para mujer Sidi Alba Road Bicycle Negro Zapatillas de ciclismo de carretera Shimano para mujer RP2 Azul negro Zapatillas de ciclismo para mujer Sidi Alba Road Bicycle Fucsia / Blanco Zapatos de ciclismo Vittoria Ikon Blanco Zapatillas Shimano SH-AM7 MTB Gris azul , Fizik R5B Donna BOA Zapato con Fuschia Trim Antracita , Zapatos de ciclismo Vittoria Ikon Negro Zapatillas de ciclismo Pearl iZUMi Tri Fly Pro v3 Lime Punch / Black Smartodoors MTB / Road Cycling Shoes Racing Ciclo calzados para hombres Mujeres SD01-MTB-Blanco / rojo , Zapatillas de ciclismo SPI X-Alp Launch SPD Pearl iZUMi para mujer Perla ahumada / Highrise , Zapatos MTB Giro Riela R para mujer Dark Shadow / Bright Pink , Zapatos urbanos de mujer Louis Garneau para mujer Negro / rosa Zapatillas de ciclismo Smartodoors con suela de carbono o suela de tpu de nylon para carretera y MTB Blanco / Rojo B para carretera , Zapatillas Spin Biking SPD de montaña / montaña Tommaso Terra 100 para mujer Negro / verde azulado Zapatillas de ciclismo Shimano SH-RP9 - Hombre Armada , Zapatillas Shimano SHRP9 Flagship Road Performance Shoe para hombre Blanco Zapatillas de ciclismo SPI X-Alp Launch SPD Pearl iZUMi para mujer Perla negra / ahumada , Zapatillas de ciclismo Pearl iZUMi Tri Fly Pro v3 Lime Punch / Black , Zapatillas de ciclismo de carretera Shimano para mujer RP2 Azul negro , Zapatillas de carretera Shimano SH-RT5W para mujer marrón , W X-ALP ENDURO IV Blanco / Morado Haze Zapatillas de ciclismo para mujer Sidi Alba Road Bicycle Blanco Zapatillas de ciclismo X-Road Fuel IV Pearl iZUMi para mujer Negro / Rosa Glo Zapatillas de ciclismo de carbono Tri Fly V para mujer Pearl iZUMi Blanco negro Zapatillas de ciclismo Pearl XZUMi W X-Alp Canyon para mujer Ahumado Perla / Niebla Verde Zapatos de ciclismo para mujer Giro 2017 Petra VR Dirt Titanio / joya azul Zapatos de bicicleta de MTB Maltese Falcon Clipless para mujer Five Ten Granito , Zapatillas de ciclismo Shimano SH-RP9 - Hombre Negro , Zapatos de ciclismo para mujer Giro 2017 Petra VR Dirt Blanco
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 Fresh Life Arishi V1 para mujer New Balance Blanco / Piscis , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Rosado , Saucony Women s Guide 8 Running Shoe Blanco / Crepúsculo / Rosa Calzado casual tejido NIKE NIKE para hombre Bronce Negro Cumbre Blanca 700 Short NIKE Tempo para mujer juego real / vívido rosa / plata mate Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Lobo gris / blanco brillante mango Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Acero Zapatillas de running NIKE para mujer Air Presto Gris , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Púrpura , Zapatillas para correr NIKE Womens Air Presto negras / blancas / negras 878068-001 talla 6 - Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco , Skechers Women-14960 Sneaker Negro , Sandalia Drew Cascade para mujer Nubuck azul marino Bomba de vestido Nika BeautiFeel para mujer Negro Negro , Caminante atlético ancho de las mujeres de Aetrex Gris blanco Vionic Women s Flex Sierra con cordones Amarillo , Zapatillas Superga 2750 Cotu para mujer Burdeos , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada Clásicos de base Slip-on (Tm) para hombres de Vans (Verificador primario) Verdadero azul / blanco Deflector de gravedad Alivio del dolor comprobado Desafío G de las mujeres Mighty Walk - Mejor Zapatos para el talón Dolor, dolor en el pie y fascitis plantar Gris, rosa , Supra Method Skate Shoe Negro Negro BareTraps Women s Beverly Slide Sandal Mezclilla Sam Edelman Women s Slide Sandalia Denim Blue Kid Suede Leather Pantuflas para mujer Cincasa Menorca Galdana Slipper / Womens Armada , Acorn Women s Cozy Ewe Negro , TOMS Clásicos de mujer Gris claro Trotters para mujer Dea Slingback Pump Gris claro CLARKS Cheney Madi Slip-on Loafer para mujer Cuero 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 de ciclismo Bicycle ciclismo para mujer B07CN49MJS Sidi Alba Road Bicycle Negro 2db9064 - 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 ciclismo Bicycle ciclismo para mujer B07CN49MJS Sidi Alba Road Bicycle Negro 2db9064 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 119025
1297524
2945230
5007230
9267058
9820745
11194609
11674650
18540092
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.