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.

CLARKS Lexi B01I244PZ6 Myrtle Sandal 1490 para para mujer Negro a102579




Cuero
Importado
Goma suela
Medidas de talón aproximadamente 2
Medidas de la plataforma aproximadamente 0.75
Tobogán de cuña sandalia con cuero de tiras superior con floral apliques y cuentas clúster
Flexible, suela duradera
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


CLARKS Lexi B01I244PZ6 Myrtle Sandal 1490 para para mujer Negro a102579 - 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

CLARKS Lexi B01I244PZ6 Myrtle Sandal 1490 para para mujer Negro a102579

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.


CLARKS Lexi B01I244PZ6 Myrtle Sandal 1490 para para mujer Negro a102579

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

Zapatilla Ascot de hombre UGG Carbón J.Renee Womens Soncino Armada , Splendid Women s BO Sandal Seta , Sandalia Dominica de Ryka Coral / gris , CLARKS Lexi Myrtle Sandal para mujer Negro , Sandalia de tobillo con correa Gomera Finn Comfort para mujer Cognac Plisseelight Sandalia Cyprine Slide de mujer Naturalizer Negro Melissa Women s Diane + Jason Wu Slides Verde , Naot Footwear Mujer Papaki Gris Claro Nubuck / Cuero de Mar Polar , Tecnología Vionic Men s Orthaheel White / Navy Walker ciruela Sandalia Tonya de mujer Trotters Arena , Reggae de Skechers Slim-Forget Web anudada Gore Open Toe Slingback Sandal Armada Sandalia Fernanda de Calvin Klein para mujer Blanco suave , OTBT Nomadic Amarillo dorado Naot Footwear Women s Prescott Hilos de oro cuero / plata Lustre cuero / hilos de plata Leathe SJP de Sarah Jessica Parker Mujer Meteor Sneaker Quilate CLARKS Womens Wynnmere Avah Nubuck negro Sandalia Savannah de Ryka para mujer Beet Red / Hyper Pink KEEN Women s Uneek 8MM Sandal Corazón Púrpura / Niebla de Lavanda , Zapatilla Ascot de hombre UGG castaña , Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro , Sandalias con plataforma SAS de cuero con punta abierta para mujer Ropa web Drew Womens Element Nubuck negro Salvatore Ferragamo Womens Enfola Coral Alexandra Calf Sandalia de mujer Aravon Beaumont Peep Slide Paloma , Nueve West Women s Lynneah Leather Mule Cuero natural oscuro Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Sandalia Patrese de estilo suave para mujer Patente blanca Baile Bella Me de Bernie Mev para mujer Lurex rojo
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 Sandalia de Hollyleaf de las mujeres de Merrell Mar de Bering , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Hyper Cobalt / White Zapatillas de running A7000WX12 para mujer Apex gris , New Balance Women s Summit - Desconocida Trail Running Shoe Rosa / negro Sandalia Jenley de Lucky Brand para mujer Blanco brillante , Zapatillas para correr NIKE Womens Air Presto negras / blancas / negras 878068-001 talla 6 - Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on gris NIKE Men s rsquo; s Zapatos casuales Darwin Zapatillas deportivas ligeras Confort Running Wolf Gray / Cool Gray / Rage Green , L Artiste de Spring Step Womens TUTTIFRUTTI Armada , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gunsmoke / White-atmosphere Grey Linea Paolo Felicia | Ante negro Zapatos de caminata SAS Low Me de cuero para mujer Negro Reebok Zapatillas para caminar de estilo libre para mujer Retro-blanco / Bunker Blue / Patent , Vans Unisex Old Skool Zapatillas clásicas de skate (Lienzo) Negro Vans Old Skool Unisex Adults Low-Top Trainers (35u) (Tablero de ajedrez) Negro / Rojo Zapatilla de pies Cush N de Prop & eacute; t para mujeres Arena , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre (Pop) Gargoyle / Mars Red Under Armour Ua ClutchFit Drive Ii para hombre Negro / Negro / Acueducto , Zapatillas de golf Skechers Women s Go Golf Birdie Blanco negro , Sandalia de la diapositiva Izbremma de la mujer de Lucky Brand Tufo BareTraps Women s Beverly Slide Sandal Negro , Sandalia para mujer Kate Spade New York Coby para mujer Oro Sandalia de mujer de SCHUNA Maduna Slide Summer Red , Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Combo de la Marina , New Balance Sp4040d1 para mujeres Royal / Whit Trotters Lucy Ballet para mujer Negro , Cordani Womens Bree Cuero negro TOMS Womens Desert Wedge Bootie Negro , Aquatalia de Marvin K. Botín de ante Freddi para mujer Ceniza , MICHAEL Michael Kors Mujeres Kendra Peep Toe Botas de cuero de moda ,

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.

CLARKS Lexi B01I244PZ6 Myrtle Sandal 1490 para para mujer Negro a102579 - 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 CLARKS Lexi B01I244PZ6 Myrtle Sandal 1490 para para mujer Negro a102579 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 500950
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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.