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
Suela de goma
Eje mide aproximadamente 2.5 desde el arco
cierre automático
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,

Zapatillas de running ASICS Gel-Nimbus 19 para mujer Indigo Blue / Paradise Green / Pink Glow , Zapatillas de running NIKE Air Huarache para hombre Mango brillante / blanco Zapatillas de running ASICS Gel-Kayano 23 para mujer Blanco / Plata / Acuario , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Ember Glow Team Red Black 802 Zapatillas de running Wstro para mujer New Balance Alpha Pink / Black / Lime Glow , Zapato de correr ideal para rejilla de mujer Saucony Negro / Gris / Pr Zapatillas para correr Saucony Women s Cohesion 10 Azul marino , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro NIKE Men s Benassi Just Do It Athletic Sandal Blanco / puro platino / negro / blanco Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco / Bahía / Plata , Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Negro / Gris Sólido Zapatillas de running New Balance Boracay V3 para mujer Supercell / Ozone Blue / Bleached Sunrise , Brooks Womens Neuro 2 Noche Azul / China Azul / Coral Fiery , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Acero / Trueno , Zapatillas de running Saucony Women s Freedom ISO Azul Zapatillas de running Gel-Moya para mujer de ASICS Azul blanco , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris negro NIKE Men s Roshe Two Running Shoe Negro , Calzado de running NIKE Kids Roshe One (GS) Blanco / Blanco / Negro , Nike Hombres Air Zoom Pegasus 33 Midnight Navy / Blanco-negro-puro Platino Zapatillas de running Flash para mujer New Balance Blanco / Alfa Rosa / Negro Zapatillas de running ASICS Gel-Nimbus 19 para mujer Indigo Blue / Paradise Green / Pink Glow , Zapatillas de running adidas Springblade Razor para mujer Blanco Zapatillas de running adidas Duramo 3 para mujer Correr Blanco / Visión Azul / Plata Metálica , NIKE Women s Tanjun Sport Fuchsia White Tart Zapatillas de running New Balance 990v4 para mujer Dark Mulberry Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Azul marino / rosa / verde azulado Zapatillas de running adidas Women s Edge Lux w Misterio Ruby / Bahia Magenta / Calzado Blanco , Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello ,
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 negro , Saucony Triumph ISO 3 Running Sneaker para mujer Gris morado , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Lino Verde 6367 , Zapatillas de running Saucony para mujer Peregrine 8 Negro / Menta Sandalia de cuña Rheta Sam Edelman para mujer Ante negro , Zapatillas de running Kineta Relay para mujer Saucony Gris / Heather / Coral , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Armada , Zapatos de lona OLUKAI Waialua - Mujer Negro Negro Zapatillas para caminar Skechers Performance para mujer Go Step Lite Ultrasock gris , Sandalia Terra FI Lite de Teva para mujer Cremallera doble negra Zapatillas para caminar WW669V1 New Balance para mujer Gris , Sandalia Cyprine Slide de mujer Naturalizer Negro Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Negro , J.Renee Womens Soncino Rhinest plateado , Sandalia Terra FI Lite de Teva para mujer Agua , SAS Womens Sunny Mora , Zapato de lona KEEN Sienna MJ para mujer Corona azul Zapatillas de viaje Proather para mujer II Coral Mesh Dr. Comfort Annie-X Mujer diabética terapéutica Doble Profundidad Zapato Lycra Velcro Beige , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Crema / Combo , ara Sandalia mujer Tania Slide Rosegold Metallic Stuart Weitzman Sandalia Dohickey Slide para mujer Mojave Suede , Zapatillas de voleibol adidas Crazyflight X 2 de mujer Blanco / Negro / Poder Rojo , Zapatillas de ciclismo X-Road Fuel IV Pearl iZUMi para mujer Aqua Mint Zapatillas de running Nike Free Tr2 para hombre Seta / Seta Negra Zapatilla Samantha de Kate Spade New York para mujer Gris claro Easy Spirit ON CUE Oxford para mujeres marrón , Botas para mujer Demonia SWING-221 Blk Canvas-vegan cuero CAMSSOO Mujeres Moda Peep Toe Lentejuelas brillantes Muslo alto sobre Rodilla Pupms Heel Fiesta de Navidad Botas de baile Oro , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Russet Orange Heritage Canvas

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 azul 19182 Balance W880gb6 Balance para mujeres Gris azul 145a65b 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 3034146
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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.