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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Textil y sintético
Importado
Caucho único
Seguro ajuste de captura de encaje sistema
Metatomical Plantilla de EVA
Non-marking suela de goma
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

Sandalia Uneek O2-w de mujer KEEN KEEN Aura/ Astral B00CWB0OVK/ Libertad 086d7a7

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.


Sandalia Uneek O2-w de mujer KEEN KEEN Aura/ Astral B00CWB0OVK/ Libertad 086d7a7

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

Sandalia para mujer Teva Terra-Float Travel Knit Paloma Salvaje , Sandalia Savannah de Ryka para mujer Moonrock / Snowline Ecru Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Zorro ártico , Xero Shoes Z-Trail Sandalia ligera - Descalzo-Inspirado Ruta de senderismo, Correr sandalias deportivas - De las mujeres Carbón Negro / Mocha Tierra Adidas Women s Adilette CF + Logo W Sandalia deslizante Ash Blue / Aero Blue / Black Sandalia KEEN Women s Venice H2 Gruta Celeste / Azul Sandalia NIKE Benassi JDI para mujer Blanco / Lobo Gris / Plata Metálica , Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente , Clorts Women s Outdoor Hiking Athletic Sports Sandalia anfibia ligera SD202 Rosado , Sandalias adidas de mujer Adilette Cf + Armad Athletic Slide Negro / Tech Rust Blanco Sandalias planas Eileen Fisher para mujer Sport-Nu Armada Wolky Comfort Jewel Gray Picasso Crash Wolky Women s Rio Sandalia 40-912 Multi , OluKai Kulapa Kai Blanco negro , Sandalia de deslizamiento ultra Comfort NIKE para mujer Blanco negro Teva Terra Sandal para hombre Cross Terra Black Sandalia Drew Angela para mujer Bone Smooth Leather Earth Origins Sophie Escarlata Zapato de agua H2 Go Skechers Performance para mujer gris Sandalia Kea Vent mujer Columbia Marrón oscuro, Hawk , Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 5937 Black Blue OluKai Kulapa Kai Carmesí / negro , FABULICIOUS COCKTAIL-501-5 Tobogán de plataforma con 5 \ Claro / Claro , Sandalia Drew Angela para mujer Bone Smooth Leather KEEN Hombres Newport H2 Sandal Tierra oscura / Acacia , Sandalia NIKE Kawa Slide para mujer Negro / Vivid Pink , Sandalia de mujer KEEN Rose Imán / Gárgola Sandalia de mujer KEEN Rose Aluminio / gris neutro , ALMA NATURAL Sandalia de pescador Beatrice para mujer Rosa ,
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 Chaco Zcloud 2 Sport Sandal para hombre Tri Navy Wolky Comfort Jewel Brandy Smooth Leather Sandalias de encaje Ros Hommerson para mujer Oro Zapatillas de running NIKE para mujer Air Max Motion LW Binario Azul / Blanco Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas para correr Altra Women s Intuition 3.5 Orquídea / Plata Chaco Zcloud 2 Sport Sandal para hombre Negro Zapatillas de trail running Tracerocker para mujer adidas Outdoor Negro / Gris Vista / Utilidad Negro , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Azul marino / rojo / blanco Under Armour Women s HOVR Sonic NC Grafito (105) / Plata Metálica , Zapatillas deportivas NIKE Lunar Apparent para hombre Carbono / Obsidiana ligero , Zapatillas de running NIKE Air Max Zero Premium para hombre Negro, azul marino, blanco Bernie Mev Women s Comfi Flat Negro Sandalia de Gladiador de Asha de mujer Ahnu Oxblood Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón , Skechers Women s Empress-Wide-Awake Gris claro , Jessica Simpson Sandalia de tacón de mujer Mitta Negro , Sam Edelman Geren Sandalia para mujer Negro , NIKE Condición de zoom para mujer 2 Cross Trainer Negro / Blanco - Gunsmoke Nike Tech Xtreme Cadet Glove Negro / Blanco / Gris lobo / Volt , Sneaker de moda con cordones Lynx de DC para hombre Gris / Gris / Rojo , Sam Edelman Women s Mares Slide Sandal Poseidon Blue Satin , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2712- # 254 Gold & Gold Trim Zapatillas Reebok Women s Plus Lite ti Track Ti - Whisper Gray / White / Guava Punch / Sour Melon / Lilac , Sorel Medina IV Premium Boot - Mujer Colegial azul marino / negro , Bota de invierno con cordones D Lites-Short de Skechers para mujer Negro Vionic Women s Tide Ii Pewter Metallic Mocasines y zapatos Slip-Ons Dunbar Hamden de Clarks para mujer Nubuck azul Naughty Monkey Mujeres In lyne Tobillo Botín Índigo TOMS - Slip-On de lienzo clásico para hombre Universidad de la raya negra ,

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 Sandalia Uneek O2-w de mujer KEEN KEEN Aura/ Astral B00CWB0OVK/ Libertad 086d7a7 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 811590
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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.


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