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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Cuero
Hecho en los EE. UU.
Caucho suela
ABZORB talón
Caminar sendero de huelga
Reflectivo detalle
Montado en los EE. UU.
Nuevo Balance llama a zapato 'ensamblado en los Estados Unidos cuando el contenido nacional es menos del 70%
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

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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,

Calzado SCARPA para mujer Hydrogen GTX Gris medio / Agua estancada Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro rojo , Sandalias Harper Huarache de SoftWalk para mujer Rojo / Tostado , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro Drew Shoe Women s Paradise II Velco Slip-on Blanco Trotters Mujer Simmy Mary Jane Flat Grafito Zapato deportivo terapéutico Kellie para mujer de Dr. Scholl, ancho ancho - Prop & eacute; t Propet - Zapato para caminar con correa Eden para mujer Blanco / azul polvo Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Mora , Drew Shoe Women s Tulip Comfort Ternero blanco , CLARKS Womens Trigenic Flex Ante negro , Orthofeet 877 Women s Comfort Diabetic Therapeutic Extra Profundidad Calzado Lycra Velcro P W Minor Leisure Mujer Terapia Casual Extra Profundo Calzado Cuero Cordones Negro Calzado de espuma fresca New Balance para mujer 1165v1 Gris , Atletismo femenino de Apex Blanco / Vinca Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Blanco , Zapatillas de paseo Skechers Go Walk Lite-15430 para mujer Natural , Zapatos de mujer RYKA Sky Walking Negro / rosa , Zapatillas Slip-on de mujer Skechers You Inspire Negro Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Negro , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Moca Drew Shoe La alegría de las mujeres Hueso , New Balance WW813 Hook and Loop Walking para mujer Blanco Finn Comfort Women s Ikebukuro Oxford Espresso Metal Leather / Jasmin Okapi Leather Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco , Zapatillas clásicas Vionic para mujer Walker Blanco / rosa / gris , Zapatillas Erika Propet para mujer Chili Red Zapatillas para caminar Ilex Easy Spirit para mujer Grey Multi Suede
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 NIKE Women s Revolution 3 para correr Gris oscuro / blanco / negro Propet Women s TravelActiv Ss Sandal Negro plata Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Azul Cloro / Antracita / Gris Fresco , Sandalia Yucatan ECCO Mujer Shadow White Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris blanco Zapato para caminar Eden Propet para mujer Blanco Tierra Daylily Nubuck rojo Zapatillas para correr Nike Downshifter 6 para mujer Gris oscuro / Hyper naranja / blanco / Copa La sandalia de gladiador FLEXX Women s Band Together Cherry Nubuck Zapatillas clásicas Vionic para mujer Walker Blanco azul Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Pewter Suede Under Armour Charged All Around Speedknit True Ink (918) / Blanco Zapatillas Ryka Dynamic 2.5 Cross-Trainer para mujer Blanco negro Zapato para caminar WW813 para mujer New Balance Blanco Zapatillas de deporte CrazyMove Pro de adidas by Stella McCartney Core Blanco / Hi-res Blue S18 / Semi Solar Red , PUMA Mujer Ignite Flash Evoknit WN Sneaker Puma White Vans Auténtico (Digi Hula) Blanco clásico / Blanco verdadero , Diadora Unisex Intrepid NYL Limoges / Dahlia White Under Armour Ua ClutchFit Drive Ii para hombre Púrpura / blanco Sandalia plana Kip de Trotters para mujer Blanquecino , Zapatillas de voleibol WH-NY para mujer Wave Bolt 4 de Mizuno para mujer Naval Blanca , Ariat Womens Volant S Tall Zip Tall Riding Negro , Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Bronceado oscuro Foamtreads Barry Masculina Negro Zapatillas de deporte unisex de bajo perfil Blanco Zapato de personaje de luz de posición de estudiante 650 de Capezio Caramelo , Burton Limelight Snowboard Boots para mujer - , Asolo Women & fnof; _Ts Falcon GV Hiking Boots Gris / piedra con calcetines SOREL Sandalia casual con cordones Ella para mujer Negro Twisted X Ladies Turquoise Lace Driving Mocs Bombardeo ,

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 New Balance WW577 Hook Walking and WW577 Loop Hook Walking para mujer Negro 902091c 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 5680493
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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.