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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100% Textil
Importado
Caucho único
Athletic sandalia con correas en capas con gancho y lazo cierres en el mostrador, tobillo y dedo del pie
Amortiguadores Tecnología ShocPad en el talón
Nylon mango de estabilidad
Microban a base de zinc a prueba de olores tratamiento
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,

Zapato de malla Nohea de OLUKAI - Hombre Rock / Canoa Romika Tahiti 03 Sandalia de vestir para mujer Oceano , Sandalias con plataforma SAS de cuero con punta abierta para mujer Henna Smooth Diapositivas PUMA FURNY x PUMA de piel sintética para mujer Bay / Puma Silver , Drew Womens Ariana Cuero de peltre Chaco Mujer Aubrey Sandal Piña , Sandalia Amara 2 de Carlos Santana para mujer de Carlos Azalea Sandalia Savannah de Ryka para mujer Negro / gris helada Sandalia Rosa mujer TOMS Saffron Suede , adidas Women s Adilette Cloudfoam + Sandalia deslizante Negro / Blanco / Negro , Sam Edelman Sandalia Dariel para mujer Oro fundido / Cuero metalizado Splendid Feodora Sandalia para mujeres Lavanda , Easy Spirit - Sandalias informales para mujer, Everso Armada , Coach Women & iexcl; & macr; s Cassidy Metallic Tumbled Sandals, Style A01627 Lapis Blue , Sandalias FitFlop para mujer H-Bar Slide Coñac , Sandalias KEEN Rose City T Strap para mujer Embarcadero , Flip Flop de Hari Mari Fields - Mujeres Gris claro / azul / rosa Easy Street 30-8320 Sandalia airosa para mujer Blanco , INC International Concepts Womens Landor Fabric Punta abierta plataforma arena. Negro , Sam Edelman Women s Slide Sandalia Pewter / Putty Metallic Boa / Kid Suede Leather Bomba CLARKS Laureti Pearl para mujer Nubuck negro , Sandalia Colette Alegria para mujer Opcionalmente , Sandalias ortopédicas Gabby Adjustable Arch Support de Aetrex para mujer, Negro , CLARKS Sandalia plana Brizo Cady para mujer Textil de arena / blanco Sandalia de cuña Kailynne2 Easy Spirit para mujer Tela Taupe , Bomba de vestido Nika BeautiFeel para mujer Serpiente natural , Bota Deia para mujer de TOMS Broncearse Chancleta Yoga Sling 2 Sanuk para mujer Raya tribal natural / multi SAS Mujer, Pampa Sandals 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 Sandalias KEEN Womenport Newport Hydro-W Acero Gris / Azul Turquesa Sandalia de deslizamiento ultra Comfort NIKE para mujer Obsidiana / azul de cloro Chaco Mujer Zvolv Sport Sandal Eclipse de marfil Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Azul hielo / azul táctil Sandalia Deportiva Jordan Spires de Via Spiga Rico, amapola, gamuza Sandalia Archie Fisherman del Dr. Scholl Moonstone / Pink Nubuck Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Naranja / Ébano NRS Desperado Wetshoe Azul negro , Bota de chukka Bootie Bootie de cuero de mujer Comfort de Cara Comfort para mujer Negro , Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Malla de punto de carbón / gris Sandalia Tallis de mujer VANELi Marina de verano Zapato para caminar Skechers Go Joy para mujer Azul marino / blanco Chancleta Yoga Sling 2 Sanuk para mujer gris Sandalia Sunkiss Cara de Sperry Top-Sider para mujer Armada Zapatillas Skechers for Work Sure Track Trickel Slip Resistant Negro Zapatos de mujer Rockport Ts Wp con cordones Negro Adornos Go Step Lite de Skechers Performance para mujeres Negro , Mocasín de mujer RIALTO Vette C-print negro OLUKAI Alohi - Mujeres Fray Brown / Fray Brown , Vans Unisex Old Skool Zapatillas clásicas de skate Llovizna / True White , Skechers Women s Breathe Easy Calmly Sneaker Negro Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Negro Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Negro , BareTraps Women s Beverly Slide Sandal Negro , Sandalias planas Nori de Easy Street para mujer Pewter Metallic ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ Cuero desnudo de cobre S9216 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) 7017- Serpiente dorada , Zapatillas de ciclismo Pearl Izumi para mujer W Select Road v5 Studio Negro Negro , Abilene Women s Sage by Cowgirl Boot - 3551 Afligido , Bota de mujer Jeffrey Campbell Calhoun-4 Oro negro

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 Lite Terra/ FI Lite de Teva Lights para mujer City Lights Black/ Multi dd7988a 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 6788458
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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.