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
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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


Zapatillas mujer de running NIKE Free Run para - mujer Glow - SU16-10 - Blanco Mango/ Sunset Glow 8aa0bdd - movie-store.online

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

Zapatillas mujer de running NIKE Free Run para - mujer Glow - SU16-10 - Blanco Mango/ Sunset Glow 8aa0bdd

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.


Zapatillas mujer de running NIKE Free Run para - mujer Glow - SU16-10 - Blanco Mango/ Sunset Glow 8aa0bdd

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

NIKE Men s Flex 2017 RN Negro / Hematita Metálica / Antracita / Gris Oscuro , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Negro-corredor Rosa-blanco Calzado deportivo New Balance para mujer Vazee Rush V3 Blanco / Vintage Indigo Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima , NIKE Mujeres Air Max Thea , Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout , Under Armour Threadborne Fortis para mujer Negro (001) / Stealth Gray Brooks Womens Hyperion Peacoat / azul marino / cacatúa Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral 361 Zapatillas de running para mujer 361-Enjector Negro / Blanco / 0900 Zapatillas de running adidas Ultra Boost para mujer Azul Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapatillas de entrenamiento Nike Womens Metcon 3 Cool Grey / Solar Red-black-pure Platinum Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-blanco-negro , New Balance Women s w940v3 Running Shoe Plata , Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Azul , nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Hematita Metálica / Negro-gris oscuro , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro , Zapatillas de running adidas Women s Edge Lux W Gris / gris / blanco cristal Zapatillas para correr Mizuno Women s Wave Enigma 6 Púrpura / blanco , Zapatillas de running acolchadas 420v4 para mujer New Balance Gris Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Tierra púrpura / negro Zapatillas de running adidas Alphabounce Beyond para mujer Blanco / Gris / Gris , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Azul , Saucony Women s Cohesion 8 Running Shoe Zapato para correr ASICS Conviction X 2 para mujer Negro / Carbón / Flash Coral ,
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 Calzado informal NIKE Darwin de mujer Negro Negro Blanco Zapatillas de running Mizuno Women s Wave Hitogami 3 Fiery Coral / Blue Depths Zapatillas de deporte de moda Kanye West Desert Rat 500 Zapatillas de correr 3M para hombres Mujeres DB2908 DB2966 Sonrojo , Skechers Performance On The Go 400 Vivacity Flip Flop Azul marino / blanco Sandalia Yucatan ECCO Mujer Poseidón / gris cálido / negro , Under Armour Speedform Velociti para mujeres Glacier Grey (003) / Ultra Blue , Drew Shoe La alegría de las mujeres Combo marrón Saucony Women s Ride 10 zapatillas para correr Gray Berry , Easy Spirit Piers Faded Denim XYD Prom Dancing Shoes Elegant Open Toe Strappy Sandalias de tacón de tobillo Wrap vestido de bombas para las mujeres Plata , Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Lollipop / Peacoat / Bachelor Button / Blanco Naturalizer Samantha Pointed Toe Flat para mujer Oro negro , Zapato de malla Nohea de OLUKAI - Hombre marrón , Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Black Suede / White Mesh K-Swiss Tubos de mujer Infinity CMF Cross Trainer Patriot Blue / Raspberry Wine Zapato de skate Etnies Jameson Vulc para mujer Índigo , Bota de chukka Papalium Baggy L2 para hombres Palladium Negro Zapatillas de golf Sunnylite V2 para mujer de PUMA Blanco-gris Violeta-púrpura Cactus Flor , MOZO Zapatillas para mujer The Maven Sneaker Lunar Rock , Sandalia Birkenstock de cuero Florida para mujer Habana Waxy Leather , Sandalias veganas Arizona Arizona Birkenstock Bordeaux Birko-flor Pull Up , Birkenstock Mujer Mayari Birko-Flor Sandal Moca , Bloch Dance Boost Mesh DRT Split Sole Dance Sneaker Rosado , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7015- Terciopelo beige , Bomba Hazel para mujeres Sam Edelman Midnight Navy Sequin , Grishko 0509/2 2007 Pro-Flex Western Chief Women Impermeable de cuatro ojos con cordones Duck up Púrpura Bota de tobillo Rexford para mujer Piel de Taupe Full Grain Zapatillas de deporte Buckner Fashion de Mark Nason Angeles para mujer borgoña Sam Edelman Women s Dahl Oxford Pata de gallo negro / blanco ,

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.

Zapatillas mujer de running NIKE Free Run para - mujer Glow - SU16-10 - Blanco Mango/ Sunset Glow 8aa0bdd - movie-store.online


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 Zapatillas mujer de running NIKE Free Run para - mujer Glow - SU16-10 - Blanco Mango/ Sunset Glow 8aa0bdd 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

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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.