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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Ligero lavado Parte superior de lona.
Importado
Caucho único
interno Plantilla con ventilación para la comodidad y transpirabilidad.
Sólido caucho que no deja marcas suela.
Ligero lavado lona superior. Forro de lona.
Abierto malla en el talón para una mejor transpirabilidad.
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

Zapato de vestir 11969 uniforme al aire Cornstalk libre Vulc N Columbia Vent Slip para mujer de Columbia Acero, Cornstalk ee9cb40

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,

Chaco Mujer Diana Sport Sandalia Biselado negro KEEN Hombres Newport H2 Sandal Imán / Capuchina , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth , Wolky Comfort Jewel Estampado de flores negro Teva Terra Fi 4 - Mujeres La Manta Black Sandalia NIKE Kawa Slide para mujer Hyper Punch / Negro / Blanco , Sandalia KEEN Women s Venice H2 Midnight Navy / Coral caliente , Chaco Women s Zx2 Classic Athletic Sandal Parcheado de ámbar , OluKai Kulapa Kai Negro Negro , Skechers Performance On The Go 400 Tropical Flip Flop Marina tropical Zapatilla de deporte Sina 11 de Josef Seibel para mujer Negro Chaco Mujer Zcloud X2 Sandalia Deportiva Popline Boa Sandalia de tiras Enoki Twist de mujer Merrell Azul claro , Encaje de quimera de Columbia para mujer Zuraqua , Chaco Mujer Zcloud X2 Sandalia Deportiva Plata metálica Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Baja Azul , adidas Performance Mujer Adilette CF Ultra Stripes C W Sandalia deportiva Negro / Blanco / Negro , Sandalia deportiva Tirra para mujer Teva Trufa de ciruela Sandalia Drew Element para mujer Nubuck marrón CLARKS Sandalia plana Brizo Ravena para mujer Armada Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Negro / Negro / Blanco Tiza , ALDO Mujer Zarella. Negro Varios , Sandalia Terrell II de Merrell para mujer Dusty Olive Sandalias Newport H2 de KEEN para mujer Cáñamo / azul provincial Sandalias KEEN Mujer Whisper-w Vapor / acero gris Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7668 negro azul , Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Sandalia KEEN Women s Uneek Slide-w Aqua Sea / Turquesa Pastel
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 Wolky Comfort Jewel Choque de Multi Picasso , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Plata negra / metálica Sandalia deportiva Zvolv X de Chaco para mujer Arco caliente Chaco Mujer Zvolv Athletic Sandal Pane Angora , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Dark Grey Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris claro / blanco Converse Chuck Taylor All Star Seasonal Color Hola Blanco negro , Merrell Women s Mqm Flex Monumento , Salomon Xa Pro 3D GTX W Trail Runner Remolacha roja / sangría / negro , Zapatillas Skechers Go s Lite Lite-Mystic para caminar Negro Zapato de caminar sin tocar Skechers Performance para mujer Go Step Carbón , Bota de chukka Bootie Bootie de cuero de mujer Comfort de Cara Comfort para mujer Negro Zapatillas de running W775V2 para mujer New Balance Gris / morado Sandalia Janny de BareTraps para mujer Rosa roja , Zapatillas para correr Nike Flex Experience RN 5 Wolf Gray / Voltage Green / White / Pink Sandalias mujer La Plume Jen Black Croco , Sandalias con hebilla de botón Ridgeport para mujer Rockport Plata , Sandalias SAS para mujer, Nudu Slide Oro New Balance 711v3 Heather Cross Trainer para mujeres Gris / Vapor de agua New Balance WX20v6 Cross Trainer para mujeres Gunmetal / Ozone / Thunder , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Blanco negro , HEAD Zapatillas para mujer Sprint Pro Court Blanco / gris / plata Clásicos de base Slip-on (Tm) para hombres de Vans True White (Lienzo) , Zapatillas de golf PUMA Golf PG Blaze Disc, para mujer Puma White-bright Rose ALMA NATURAL Sandalia Kiki Slide para mujer Color de malva , Sandalia tobogán moonstruck de mujer muy volátil Oro , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ , Blundstone Super 550 Series Boot Nuez , Botas de parche de textura Claremont de Ann Creek para mujer Blanco negro , TOMS - Slip-On de lienzo clásico para hombre Canal Blue 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 Zapato de vestir 11969 uniforme al aire Cornstalk libre Vulc N Columbia Vent Slip para mujer de Columbia Acero, Cornstalk ee9cb40 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 202637
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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.