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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Cuero
Importado
Suela sintética
Amortiguación adicional
Forro transpirable
Suela flexible
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


Sandalia plana para Nanci de plana Naturalizer para mujer Sandalia Mezclilla 3327d97 - 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

Sandalia plana para Nanci de plana Naturalizer para mujer Sandalia Mezclilla 3327d97

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 plana para Nanci de plana Naturalizer para mujer Sandalia Mezclilla 3327d97

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 de cuña Santorini Jambu para mujer Armada , BareTraps Women s Woods Sandal Blanco negro , Zapatilla Ascot de hombre UGG True Navy , Sandalia Vionic Unisex Wave Toe Post negro Sandalias de vestir West Hamptons de Jack Rogers para mujer Oro rosa Sandalia Verona mujer Alegria Cadena de la cadena del carbón Nop Flip Flop de Propet para mujer Gunsmoke / Dusty Taupe , Chancleta Yoga Sling 2 Sanuk para mujer borgoña Sandalia de mujer KEEN Rose Blue Opal / Provincial Blue , Chaco Mujer Zcloud 2 Sport Sandalia Oro metálico , Salvatore Ferragamo Mujer Varina Nerón Sandalia de cuña FLY London Yand709fly para mujer Red / Off White Burrito SOREL Womens Ella T-Strap Paloma Chaco Mujer Zcloud 2 Sport Sandalia Jab Pine Sandalia de Richetta de Vince Camuto para mujer Negro , Sandalia de vestir de tela Leisa de nueve mujeres del oeste Armada , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro , NAOT Kayla Elegant Sandalias de mujer Charol negro Sandalia de vestir Limoade de mujer Aerosoles Serpiente negra , Salvatore Ferragamo Mujer Varina Nero 1 Sandalia de cuña Daytona SoftWalk para mujer Negro , Sandalias ECCO Calzado Womens Bluma Toggle Gladiator Dark Shadow Metallic , Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Ante rojo Sandalia de cuero Terra Fi Lite de Teva para mujer Negro , Sandalia Columbia de Mujer Sandalias Tiburón / Sunset Red , Sandalia Huarache Easy Street Allure para mujer Blanco / Metálico Imprimir Sandalia plana Nanci de Naturalizer para mujer Broncearse , Sandalia Cecily con cuña Dansko para mujer Negro Full Grain , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro
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 adidas Alphabounce Beyond para mujer Gris / gris / naranja claro Sandalia deportiva Tirra para mujer Teva Mar de Bering , Adidas Women s Adilette CF + Logo W Sandalia deslizante Vapor Gray Metallic / Vapor Gray Metallic / Black Sandalia Omnium de mujer Teva Pizarra Zapatillas antideslizantes Skechers for Work para mujer Eldred Negro Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco / Bahía / Plata , Sanda Verra Teva para mujer Negro. , ILHABELA HOLDINGS INC Melissa Womens Beach Diapositiva 3DB Rainbow Sandal Arco iris negro , Zapatillas de running Reebok Women s Club C 85 Blanco / gris claro / chicle Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower Hush Puppies Men s Gus Oxford Marron oscuro , Sandalia de henna de mujer Sam Edelman Ante negro LifeStride Women s Camille Sandal Gris pardo Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Brght Crmsn / Prm Rd / Atmc Pnk / Wh NIKE Women s Tanjun Negro / Negro / Blanco , Rockport Cobb Hill Collection Mujeres Cobb Hill Abbott Curvy T-Strap Cuero caqui Entrenador cruzado New Foam Crush V1 de New Balance para mujer Negro Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Blue Suede / White Mesh , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Gris pardo Entrenador cruzado New Foam Crush V1 de New Balance para mujer Blanco / Plata Visón , Skechers Women-14960 Sneaker gris , Zapatillas de golf Lexington Marc Joseph New York para mujer Patente de Marina Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Negro / Naranja , Cobb Hill Rockport Women s Zahara-CH Flat Gris pardo Zapatillas de deporte lavables Walker Propet para mujer Sr. Brownie Tapsonic para mujer Capezio (K542) Negro Mocasines de vestir de punta estrecha para mujer de Me Too Women Rose Gold Metallic TOMS Clásicos de mujer Tarmac Olive Canvas , Sandalia de gamuza Poppy de TOMS para mujer Sombra de gamuza / lino , TOMS - Slip-On de lienzo clásico para hombre Espiga de carbón

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 plana para Nanci de plana Naturalizer para mujer Sandalia Mezclilla 3327d97 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
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2778363
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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.