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
Rearfoot Tecnología GEL Sistema de amortiguación atenúa el choque durante la fase de impacto y permite una transición sin problemas a medio cuerpo
Sockliner que puede ser eliminado para acomodar una ortesis médica
AHAR Suela: acrónimo para ASICS High Caucho abrasivo que se coloca en áreas críticas de la suela por excepcional durabilidad
Invertido las orejetas proporcionan cuesta arriba y tracción cuesta abajo en todos los tipos de terreno
Promedio peso por par: 1 lb. 3 oz
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


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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/ ASICS Gel-Venture 1216 6 para mujer Gel-Venture Lima negra/ carbón/ neón e482eee

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.


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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 ASICS Gel-Kayano 23 para mujer Poseidón / Plata / Cacatúa , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow Zapatillas de running Mizuno Wave para mujer Wave Horizon Azulejo azul-amarillo New Balance Women s w990v4 Running Shoe Gray / Castlerock Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul NIKE Women s Tanjun Blanco negro , Brooks Women s PureFlow 6 Heather / Marañón anacardo / cocodrilo , Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Armería Marina / Mint Foam / Glacier Grey / Black Zapatillas de running Sublite Authentic 2.0 MT para mujer Reebok Negro / blanco / plata mate / gris medio Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Negro , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Brt Mng / Wlf Gray-pch Crm-plst P , Zapatillas de running NIKE Mens Air Zoom Structure 19 Wolf Gray / Wolf Gray-dark Grey , Skechers GOrun 5 Zapatos de mujer Morado Púrpura Zapatillas de running NIKE Lunar Skyelux para mujer Dark Purple Dust / White / Loyal Blue Zapatillas para correr Under Armour UA Speedform Fortis 2.1 para mujer Negro / Gris Glaciar / Gris Glaciar Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blue Vapor / Camp Smoke Zapatillas para correr Nike Women s Free Rn Flyknit 2017 TRABAJA AZUL / OSCURO OSCURO-CLORO AZUL Skechers Women s Go MEB Razor 2 Rosado , Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa Zapatillas de running Saucony Women s Triumph ISO 2 Azul / azul claro , Saucony Triumph ISO 3 Running Sneaker para mujer Púrpura / azul / Citron Zapatillas para correr NIKE Air Huarache Run Ultra SE para mujer LT OREWOOD BRN / LT OREWOOD BRN , Zapatillas de running W775V2 para mujer New Balance Gris / morado Calzado casual NIKE Air Max 90 Ultra 2.0 Flyknit para mujer Negro / blanco-negro-antracita , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo Zapato deportivo ASICS Gel-Excite 4 para mujer Gris medio / blanco / verde Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Blanco negro , Brooks Womens Levitate Negro / Ébano / Plata ,
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 Sandalias de encaje Ros Hommerson para mujer Oro Zapatillas de entrenamiento de vuelo Jordan Nike para hombre Cool Gray / Cool Gray / Wolf Gray Zapatillas NIKE Women s Revolution 3 para correr Negro / Fresh Mint / White Converse Chuck Taylor All Star Seasonal Color Hola Roadtrip Blue / White / Black Zapato de running NIKE Kids Tanjun Print (GS) Antracita / Mtlc Platino Sandalia de susurro de mujer Keen Imán / Sangría Chaco Women s Zx3 Classic Athletic Sandal Lima gris , Zapatillas de trail running Inov-8 Trailtalon 235 (W) para mujer Teal Sandalia NIKE Benassi JDI para mujer Blanco negro Zapatillas de trail running Xa Pro 3D W para mujer Salomon Azul Medieval Zapato de agua Easy Spirit para mujer First Walker Armada Vivien Flat de Bella Vita para mujeres champán Bomba de plataforma Kiara para mujer Badgley Mischka Blue Radiance Zapato de malla Nohea de OLUKAI - Hombre Dark Shadow / Dark Shadow Crocs Flip Crocband unisex Naranja (naranja / blanco) , Clásicos de base Slip-on (Tm) para hombres de Vans Llama pared negro negro Vans Auténtico (Índigo Tropical) Azul / Blanco Verdadero , Zapatillas de golf adidas W Climacool II para mujer Clear Aqua / Running blanco / plata mate Sam Edelman Sandalia Bernice Slide para mujer Candy Red Zapatilla de deporte Blair Fashion de Naturalizer para mujer Dover Taupe Leather Rumpf Scooter 1575 Zapatilla de deporte Black Dance Split Sole Men Salsa Latina de Mujeres Jazz Hip Hop Fitness Zapato de ejercicio , Birkenstock Mules para mujeres Marron oscuro TOMS Kids 10009918 Alpargata-K Paletas blancas , Botas de águila pescadora Earth para mujer Cuero cepillado marrón oscuro , Dolce mujeres Riley Oxford Negro , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Nubuck / cuerda negro TOMS Clásicos de mujer Pink M Lbty Amelie P Trotters Jake para mujer plana Tejido elástico azul claro TOMS - Slip-On de lienzo clásico para hombre Azul eléctrico Zapatos de bowling Elite Right Platinum White - para mujer ,

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.

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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/ ASICS Gel-Venture 1216 6 para mujer Gel-Venture Lima negra/ carbón/ neón e482eee 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.