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
Teñido piel de oveja (piel Origen: Australia, Irlanda, Estados Unidos Reino, o el Estados Unidos)
Sintético único
Eje mide aproximadamente 5 desde el arco
Arranque medidas de apertura aproximadamente 10 alrededor de
Totalmente forrado con pelo
Pretratado para repeler la humedad y manchas
Suela: Treadlite por UGG
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

Bota de de invierno clásica Armada para UI GI para mujer Armada ff33820

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.


Bota de de invierno clásica Armada para UI GI para mujer Armada ff33820

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

Roper 15 \ Bombardero marrón , Ovation Celebrity Slim Secret Knee Patch Mostrar Tan , Bota de moda Clarks Sashlin Fiona para mujer Negro Bota de combate FRYE Julie Hook para mujer Forest Soft Oiled Suede Botín de tobillo Mazin Cayto para mujer Hush Puppies Negro Bota de moda Jessica Simpson para mujer Loring Dark Multi , Aquatalia de Marvin K. Botín de ante Freddi para mujer Ceniza , Vuelo a prueba de agua Belleville 690 Sage Green , Botín de tobillo Yggdrasil Ne23 para mujer El Naturalista Tíbet Bota de tiro Maxfield de Cole Haan para mujer Venda de pelo estampada de ante / negro de Ocelot , Botas Pearlie de Cole Haan para mujer Cuero negro , Lucchese Bootmaker mujeres Catalina-Brown floral impreso shortie tobillo Botín Gris SPYLOVEBUY KIND Botines de tacón alto con tacón alto y punta en punta con cremallera para mujer marrón , Zapato de trabajo Rockport Work Mask Rk6465 para hombre Negro , Bota de lluvia Ap Apx CT para hombre Jiffy Black Ct / Jiffy Black / Natural Rubber , Bella Vita Womens Tex-Italia Tan Suede Themost Womens Pointed Toe Sexy Impreso Stretchy Stiletto High Heels Over The Knee Muslo Botas altas Azul Botas Eric Michael Laguna para mujer marrón , Mocasín con flecos Minnetonka para mujer Dusty Brn , Bota de mujer Tamaris 25115 Negro Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Armada Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Gris oscuro - Arthracite , Bota de Combate Indra UGG para Mujer Negro Botas tobilleras Adaline Bluff de Merrell para mujer Negro Bota de media caña Georgia GB00129 Trail Crazy Horse , Santana Canada Sefora Black - Cocos de Matisse Women s Alabama Ankle Bootie Sillín , Lowa Womens Renegade Gore-TEX Mid Nubuck Boots Espresso Berry Botas Nikki North para mujer nuevas de Clarks Armada ,
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 NIKE Air Zoom Vomero 12 para hombre Lobo Gris Puro / Negro-puro Platino-caliente , Sandalia deportiva ECCO Terra 3S para mujer Dark Shadow / Emerald Brooks Womens Revel Island Blue / Evening Blue / Lime Popsicle KEEN Hombres Newport H2 Sandal Imán / Capuchina , Wolky Comfort Jewel Neutral Multi Color Fantasy , Adidas Women s Adilette CF + Logo W Sandalia deslizante Negro / Blanco / Negro Nfinity Vengeance Cheer Shoe (par) Blanco Zapatillas de running Reebok para mujer Pump Plus Vortex Seaside Grey / Mist / Silver Met , Zapatillas de trail running Salomon para mujer Ellipse 3 Aero W USA Cantera Zapatillas de running adidas Kanadia 8.1 para mujer - AW17 Negro , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Azul marino / rojo , Zapatillas de running Reebok Print 2.0 para mujer Negro / azul polar / peltre Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapato para caminar WW877 para mujer New Balance Plata CLARKS Sandalia plana Brizo Cady para mujer Coral Sandalia de mujer UGG Holly Ocre suave Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón , Sandalias de mujer Merrell, Sandspur Delta Wrap Negro , Sandalia para mujer Alegria Mixie Slip On Gray Harbor Entrenador cruzado Grafik Flow Ryka para mujeres Frost Grey / Bright Chartreuse / Soft Blue New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Silver Mink / Pigment Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Peacoat / Blanco , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro Birkenstock Arizona Sandalia de ante con suela suave Cuero Habana Aceitado , Zapatillas de voleibol adidas Crazyflight X 2 de mujer Blanco / Negro / Poder Rojo , Zapatillas Zeppa en suela francesa para mujer Ola negro Hush Puppies Chowchow Chukka para mujeres Gamuza añil vintage Bomba de vestido Henrika de tela Anne Klein para mujer Pewter Silver Leather , Franco Sarto Quinera Tobillo Botín Negro , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Negro / Negro Camo

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 Bota de de invierno clásica Armada para UI GI para mujer Armada ff33820 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 1873955
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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.