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.

Botas de para lluvia Vintage Emma para Aqua mujer de Helly Hansen Vintage Indigo/ Aqua Blue/ Off White (Mate) 4f17f26




Importado
Superior: Entresuela de goma
: Caucho protector Plantilla entresuela EVA
Suela: Suela de goma
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


Botas de para lluvia Vintage Emma para Aqua mujer de Helly Hansen Vintage Indigo/ Aqua Blue/ Off White (Mate) 4f17f26 - 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

Botas de para lluvia Vintage Emma para Aqua mujer de Helly Hansen Vintage Indigo/ Aqua Blue/ Off White (Mate) 4f17f26

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.


Botas de para lluvia Vintage Emma para Aqua mujer de Helly Hansen Vintage Indigo/ Aqua Blue/ Off White (Mate) 4f17f26

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

Mocasín con flecos Minnetonka para mujer Dusty Brown , Botín de tobillo Ramona para mujer de Bella Vita Gray Burnish Ellie Shoes E-609-Nicky 6 Booty con detalle de purpurina Oro Frye Harness Heath Harness Black Calf Shine Vintage , Botín de tobillo de Dallas para mujeres de Naturalizer Grano de café , 055398210 Timberland PRO - Botas de trabajo TiTAN para mujer - Café marrón Pleaser Adore-1018G Sexy Exotic Dancer Clubwear Botin de pantorrilla / medio pantorrilla de 7 \ Black Glitter / Blk Glitter , Botín Franki Dolce Vita para mujer Antracita Bota de mujer Nina Freida Slouch Negro GOGO, Botas altas elásticas de 3 \ Azul Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 13 Bota Chukka W Coromar para mujer Teva Negro Old Gringo Womens Bianca Vino Timberland Women s Londyn Chelsea Nubuck negro VANELi Womens Reanne Hebilla de Gamuza / Gamuza Negra , Franco Sarto Womens Garfield Bootie Gamuza de Whisky , Bota de tobillo con puntera en punta de cuero concho de Dingo para mujer - Di8940 Broncearse , Bota Chelsea Duckfeet Roskilde Pizarra / cuero rojo , FRYE Samantha Hiker Combat Boot para mujer Coñac , Sandalias con cordones Joanie de SOREL para mujer Verde oliva , Ropa de cama Becca para mujer Tan rústico , Toronjil. Botas de combate Belleville para mujeres con clima cálido y verde caliente - USAF, F600 Sage Green , FRYE Mujeres Naomi Pickstitch mediados de arranque Ceniza FRYE Melissa Pull on Fashion Boot para mujer Negro Botas Pleaser XTREME-1020 para mujer Blk Faux Leather / Blk Matte , Blondo Womens Vivi Waterproof Seta de gamuza , Botas Horze Signature Paddock Negro Polo Ralph Lauren Kids Ranger Hi II Bota de moda Chocolate Nubuck
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 NIKE Flex Adapt TR Cross Trainer para mujer Gris oscuro / Blanco / Negro / Sigilo Saucony Women s Guide 8 Running Shoe Blanco / Crepúsculo / Rosa Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Cool Gray Solar Red Anthracite Chaco Mega Z / Cloud Pyramid Eclipse , Zapatillas para correr NIKE Women s Air Vapormax Flyknit Azul , NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / White Zapatillas de trail running Inov-8 Roclite 290 (W) para mujer Coral rojo Zapato de agua Tetrex Rapid Crest para mujer Merrell Negro Zapatillas de running definitivas Ryka para mujer Gris negro , Zapato para caminar Skechers Go Joy para mujer Negro Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Mezclilla , Zapatillas de running NIKE Women s Sock Dart BR Blanco / Negro-juego Royal , Hush Puppies Mujeres Power Walker Ii Loafer Blanco Sandalia de cuña Karmen de mujer Alegria Diamantes para siempre , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Borgoña / ondulado / estiramiento Tierra Everwood Ladrar Zapatos de caminata SAS Low Me de cuero para mujer Negro Merrell Mujeres Ceylon Sport Casual Mary Jane rojo Zapatillas de deporte Fit Fitness GEL de ASICS para mujer Gris / Corydalis / Negro Sandalias de punta en punta con sandalias de mujer de YDN Sandalias de tacón bajo con cordones de zapatos de diapositivas Azul Zapatillas de skate Converse Unisex JP Pro Ox Negro / Negro / Blanco Sandalia de mujer Lucienne Suede Slide de Nine West Dark Natural Suede Zapatillas de volante adidas Bounce W de Crazyflight para mujer Tinta noble / oro metálico / tiza púrpura , Botas de seguridad Reebok RB874 Stealth SR para mujer - Negro Negro Tangas Birkenstock Negro TOMS Womens Desert Wedge Bootie Taupe Suede Patricia Nash Mujeres Anna Oxford Broncearse Mephisto Valentina Oxford para mujeres Black Bucksoft Drew Shoe Women s Pepper Criss Cross cuero, caucho Casual Oxfords 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.

Botas de para lluvia Vintage Emma para Aqua mujer de Helly Hansen Vintage Indigo/ Aqua Blue/ Off White (Mate) 4f17f26 - movie-store.online


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 Botas de para lluvia Vintage Emma para Aqua mujer de Helly Hansen Vintage Indigo/ Aqua Blue/ Off White (Mate) 4f17f26 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 2080687
2495817
3175225
11341829
13348067
14187854
17327267
18053731
19044371
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.