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
Importada
Sintética único
Eje mide aproximadamente Low-Top del arco
Sport sandalia
Ancho anchos
Flexible suela
Transpirable malla
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

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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,

Sandalia NIKE Benassi JDI para mujer Habanero Rojo / Habanero Carbono rojo Sandalia KEEN Women s Venice H2 Midnight Navy / Coral caliente , Sandalia deportiva Columbia Montrail para mujer Molokini III Negro, Steam Chaco Women s Z2 Classic Athletic Sandal Bombina Citrus Sandalia Savannah de Ryka para mujer Coral / gris , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso , Sandalia de deslizamiento ultra Comfort NIKE para mujer Mango brillante / negro , Skechers Performance On The Go 400 Tropical Flip Flop Negro / gris , Sandalia deportiva Tirra para mujer Teva Mar de Bering Sandalia de mujer Drew Alana Pewter-bronce , Chancleta Eezay adidas para mujer Vapor Gris Blanco / Amarillo Fácil S , Sandalia Drew Angela para mujer Beige-multi , Skechers Performance On The Go 400 Vivacity Flip Flop Negro Sandalia de susurro de mujer Keen Midnight Navy / French Blue Drew Womens Cascade Denim Blue Nubuck , Chaco Women s ZX2 Classic Athletic Sandal Bloom Peach , OluKai Kulapa Kai Negro plata Earth Origins Sophie Negro , Sandalias de mujer con tirantes Peep Toe de Bordello by Pleaser para mujer Fucsia Earth Origins Sophie Sedona Brown , Sandalia Shannon del Dr. Comfort Women Camello , Sandalia Uneek O2-w de mujer KEEN Blanco / oro de la cosecha , Sandalia original premier universal Teva W de mujer Negro , Sandalia deportiva Marc Jacobs Women s Somewhere Off White / Multi , NIKE Women s WMNS Benassi Duo Ultra Slide, negro / negro Negro Negro Teva Terra-Float Active Lace - Mujeres Paloma Salvaje Chaco Women s ZX2 Classic Athletic Sandal Motif Eclipse , Earth Origins Sophie Women s Sandal Escarlata , Chaco Mujer ZX1 Classic Sport Sandalia Aguacate estático
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 Brooks Women s PureFlow 6 Antracita / Plata / Flor de Cactus Púrpura Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Black Sport , Zapatillas para correr Topo Athletic Runventure 2 - Mujer Piedra / Ciruela Adidas Women s Adilette CF + Logo W Sandalia deslizante Blanco / Negro / Choque Rosa Vibram Furoshiki Phulkari Sneaker para hombre y mujer Flor blanca , Zapatillas para correr Mizuno Women s Wave Sky Acuario-plata Under Armour Fat Tire Shoe - Mujeres Charter Blue / Azul gasolina , Zapatillas de running NIKE Zoom All Out Low para hombre Negro / Plata Metálico / Concord Sandalia Natalia Ryka para mujer Negro Hush Puppies Men s Gus Oxford Marron oscuro Zapatillas para caminar Terrex Voyager CW CP W de adidas Outdoor para mujer Negro / Negro / Blanco Tiza Sandalia Sully de mujer caja amarilla Negro , DAWGS Womens Arch Support Z Sandals Corazones, Diamantes, Picas, Clubes , Sandalia LifeStride para mujer Eden Soft Blush , Naturalizer Samantha Pointed Toe Flat para mujer Brocado , Birkenstock Arizona Birkibuc Sandalia Mocha Birkibuc Sandalia Dominica de Ryka Coral / gris Diadora N.9000 Arrowhead Negro Zapato Dantera, para hombre, L Low Waxed Court Blanco Diadora N9000 Men Toe Snthetic Blue Sneakers True Blue / Poppy Red , Vans Auténtico Kumbu Verde / Blanco Verdadero , BZees Womens Knockout Sandal Multi / Lava , Sandalia de mujer Frisco Slide de Franco Sarto Multi Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Terciopelo gris (pedido personalizado, 4-6 semanas de tiempo de proceso) Trotters para mujer Dea Slingback Pump GOLDWASH / NEGRO Zapatilla de fumadores Classic Stripe Smoking de Soludos Natural / Negro Puma Leadcat Fenty Negro , JOE N JOYCE Mujer London Hammertone con suela suave Sandalias SynSoft Zapatillas estrechas Azul oscuro Propet Women s Tour Walker Strap Sneaker Blanco azul , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Asfalto ,

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 Propet Women s TravelActiv Ss Sandal Ss 12724 Púrpura/ Women negro 2dc3a28 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 2090531
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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.