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 de goma
Comfort tiny pillows insole
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

Sandalias planas B01MYR3NG0 para Sandalias piscina Walking piscina Cradles para mujer Negro 2956dd3

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 para mujer J.Renee Maribeth Slide Gris oscuro , Sandalia Aurora Slide de Marc Jacobs para mujer Plata , Sandalias Jerusalem Sandalia Joanna para mujer rojo Sandalia Maria Slide de Nanette Lepore para mujer Blanquecino , Sandalia Ryana Marilyn Slide para mujer Frapé Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Negro , Sandalias planas de mujer de Easy Street Coñac Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse ALMA NATURAL Sandalia Kiki Slide para mujer Color de malva , Sandalia de mujer Lucienne Suede Slide de Nine West Ante negro L & rsquo; Artiste by Spring Step - Sandalia Sumacah-Slide para mujer Negro / Multi , Sandalia Tilden Slide de Franco Sarto para mujer Tordillo , Sandalia Rockdale Lanea Woven Slide para mujer Tejido blanco , Birkenstock Mujer Mayari Birko-Flor Sandal Negro Sandalia de diapositivas Clara embellecida Marc Jacobs para mujer Salmón La Plume Womens Jessica Broncearse , Birkenstock Mujer Mayari Birko-Flor Sandal Marrón dorado Sandalia de cuña Balen de Charles by Charles David para mujer Oro Sandalias Skechers Bumblers-Bright Start Slide Carbón Sandalia de diapositivas de ocio para mujer de Seychelles Plata Sandalia Mason para mujer Naturalizer Slide Cuero metalizado dorado claro Sandalia Faye Slide de mujer de Eastland Aceituna , La Plume Womens Jessica Hueso , Sam Edelman Sandalia Bernice Slide para mujer Cuero plateado metalizado / pulido Mini Croco de plata suave , Sandalias Emmerly Slide de Vince Camuto para mujer Metal Rose Sam Edelman Women s Bay Slide Sandal Cuero Vaquero Negro Vaquero , Sandalia de diapositivas para mujer LifeStride Remix Lux Navy BareTraps Women s Beverly Slide Sandal Negro Sandalias planas para piscina Walking Cradles para mujer Negro ,
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 NIKE Men s Flex 2017 RN Negro Gris oscuro Gris oscuro NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Dark Obsidian / Black / White / Court Blue , Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Negro / Estaño / Plata Metálica Zapatillas de running NIKE Mens Air Zoom Structure 19 Gris Medio / Brillante Crimson / Brillante Carmesí Sandalia deportiva Nolita para mujer Anne Klein Tela negra OluKai Ohana Tropical / Hielo , Zapato de running 840v4 New Balance para mujer Gris oscuro New Balance Vazee Summit V2 Trail Running Shoe para mujer Negro / verde militar Zapatillas de running adidas Women s Rockadia W Trail Negro / Blanco / Naranja Fácil Zapatillas de running 430v1 New Balance para mujer Gunmetal / Seafoam Saucony Triumph ISO 3 Running Sneaker para mujer Rosa / Berry / Plata Under Armour Women s Drift 2 Negro (001) / Blanco , New Balance Women s w940v3 Running Shoe Plata NIKE Women s Revolution 4 Running Shoe Negro / Pulso carmesí / Antracita Zapatillas para caminar Skechers Performance para mujer Go Step Lite Ultrasock gris Vans Old Skool Unisex Adults Low-Top Trainers Cabernet / True White , Zapatillas Supra Titanium Skate Apagado blanco apagado blanco , Sam Edelman Sandalia Bernice Slide para mujer Nuevo Nude Leopard Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Cuero Sahara , Boigoo Ballroom Latin Salsa Performance Dance Shoes 3.0 Talón con protector de talón Mocasines casuales Alpargata Deconstructed de TOMS para mujer Sudadera de ante azul marino , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Arena , 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) 6030-gold Stardust & Gold Leather , Shimano SHMT3W MultiUse / Touring Shoe Bicicleta de montaña para mujer Verde negro , Under Armour Fusible para mujer FST Apollo Gray (100) / Glacier Grey , Zapatilla de piel de oveja SLPR Fernie Slipper Gris Ahnu Women s Fairfax con cordones Negro La bomba Suzanne de Katy Perry para mujeres Plata Dr. Martens Unisex 1461 W PW 3-Eye Gibson Leather, Moda Oxfords Negro Calzado para caminar CLARKS Women s Wave Andes Sage Nubuck

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 Sandalias planas B01MYR3NG0 para Sandalias piscina Walking piscina Cradles para mujer Negro 2956dd3 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.

Diapositivas
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2823028
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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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