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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Tela o Textil
Importado
Caucho suela
sintético y malla superior
Neopreno forro para mayor comodidad
EVA Entresuela para mayor comodidad
EVA plantilla
Encaje Cierre
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 Sandalia mujer Merrell Tetrex Crest Wrap Sport para mujer Tetrex Vapor 76ffc1e

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 Sandalia mujer Merrell Tetrex Crest Wrap Sport para mujer Tetrex Vapor 76ffc1e

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 original Teva Women s Original Caballos verdes BZees Sandalia Deportiva Jive para Mujer marrón Sandalia Yucatan ECCO Mujer Irritante , Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Core Green / Chalk White / Easy Green , Sirena de mujer Merrell Sandalia atlética Q2 Surf azul , Huracán XLT Mujeres Teva - Avellana / Negro Hazel / Black Drew Womens Cascade Sand Nubuck , Sandalia deportiva Chaco Zvolv X2 para mujer Jengibre anudado , Sandalia de mujer Bellini Nickel Turquesa , Merrell Women s Around Town Print Sandalia atlética Aguanieve Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose Sandalia deportiva Zvolv X de Chaco para mujer Arco caliente , Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Plata Sandalia Savannah de Ryka para mujer Gris blanco Sandalia Yucatan ECCO Mujer Morillo / Puerto / Negro OluKai Ohana Sour Tart / Black Correa de sirena de mujer Merrell Q2 Sandalia de atletismo Turquesa , Earth Origins Sophie Women s Sandal Azul marroquí , Sandalias de Gladiador Mujeres Fretwork Sandalias de Tacón Extrañas Mujeres Zapatos de Cuña de Verano para Mujer Blanco , Sandalias Newport H2 de KEEN para mujer Cuervo / Capri Drew Womens Ariana Dusty Gold , Sandalia Drew Angela para mujer Negro , Sandalia Yucatan ECCO Mujer Orquídea Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Laguna Coral , Sandalias de tirantes Sunstone para mujer Merrell Sodalita Sandalia atlética Salomon Women s Crossamphibian Swift W Mallard azul , Chaco Zcloud 2 Sport Sandal para hombre Dapple Caribou , Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente , Sanda Verra Teva para mujer Atlántico Norte ,
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 ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro Correas de velcro de las mujeres de EpicStep al aire libre que caminan que caminan las sandalias atléticas del pescador de los deportes del trekking Negro Skechers Women s Go Run Forza 3 Teal , Sandalia de cuña Jubilee Propet para mujer Denim Multi , Under Armour Charged CoolSwitch de mujer Glacier Grey (002) / Overcast Grey , Under Armour Women s RailFit 1 Neon Coral (602) / Negro Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Guayaba / Negro Zapatillas de senderismo acuáticas para hombre Astral TR1 Junction Dirt Brown Sandalia Tali Mini Bow de Cole Haan para mujer Oro Specchio J.Renee Womens Mcwayfalls Pastel Multi Sandalia Pisa de mujer amarilla Armada , Brooks Womens Levitate Plata blanca , Sandalia Teagan de mujer Adrianna Papell Medianoche , Zapatillas Mujer PUMA Fierce Satin En Pointe Wn Puma Negro / Plata Sandalias de cuero de Viv Viv de TOMS Pom natural del cáñamo Zapato de skate DC Magnolia TX para mujer Rosa Me Too OLYMPIA1 Zapato plano de cuero para mujer Nappa negro , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón , Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Peacoat / Blanco Sandalia FitFlop Glitzie Slide para mujer Paloma Salvaje , Sandalia Ryana Marilyn Slide para mujer Blanco negro Sandalia de la diapositiva de la Pascua de las mujeres de Sigerson Morrison Siena Sandalia Ryana Marilyn Slide para mujer Azul blanco , Sam Edelman Women s Bay Slide Sandal Sugar Pink Vaquero Saddle Leather Zapatillas de salón de baile de mujer con zapatos muy finos EKS9T56 Signature de 1,6 \ Ellie Shoes Bota vaquera 418 vaquera para mujer Negro Twisted X Boots Women s WDM0017 Driving Moc Azul , Betunia Loafer de mujer Aerosoles Serpiente negra , Betunia Loafer de mujer Aerosoles Black Lv

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 Sandalia mujer Merrell Tetrex Crest Wrap Sport para mujer Tetrex Vapor 76ffc1e 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

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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.