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
Sintético suela
Altura del talón 1.79 pulgadas
Ortholite plantilla
Cojín Tecnología suave
Full Cuero de grano
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 CLARKS Wendy Alto Fisherman Sandal B01N56Q89R para mujer Cuero marrón marrón cd6eaa1 - 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 CLARKS Wendy Alto Fisherman Sandal B01N56Q89R para mujer Cuero marrón marrón cd6eaa1

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 CLARKS Wendy Alto Fisherman Sandal B01N56Q89R para mujer Cuero marrón marrón cd6eaa1

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 de mujer de Steve Madden Cuero negro Sandalias Galaxy de BZees para mujer Azul marino , Alpargata Chambray Clara denim de mujer TOMS Chambray / Cuero texturizado negro Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Estaño , XYD Prom Dancing Shoes Elegant Open Toe Strappy Sandalias de tacón de tobillo Wrap vestido de bombas para las mujeres Plata Comfortiva Mujer, Faye Sandalia de tacón medio Negro Sam Edelman Women s Slide Sandalia Saddle Atanado Leather , Chaco Mujer Zcloud X2 Sandalia Deportiva Angora Webb , Mini cuña Huarache para mujer Crocs Multi / Mushroom Slingback de Seleste de las mujeres naturales del alma Tela Taupe Sandalia de cuña Karmen de mujer Alegria Diamantes para siempre , Sandalia de vestir de tela Leisa de nueve mujeres del oeste Blanco / Multi , Sandalia plana Halle de Walking Cradles para mujer Tabaco Agata Cuero JSport by Jambu Women s Newbury-Water Ready Sandalia de pescador Light Taupe / Light Aqua Sandalia Kinsley de Naturalizer para mujer Papaya Imprimir Sandalias planas EVA-Nu de Eileen Fisher para mujer Latte Tumbled Nubuck , Sandalia estampada Benikei JDI de mujer Nike Negro / Púrpura / Rosa Sandalias Joanie II de mujer SOREL Sal marina , G.H. Roma , Sandalia Jenley de Lucky Brand para mujer Vidriado , BareTraps Women s Dasie Platform Sandal Gris pardo , Sam Edelman Sandalia Dariel para mujer Ante negro , KEEN Women s Uneek 8MM Sandal Dahlia Mauve / Camelli , Sandalias con tira en el tobillo Begonia de mujer NAOT Cuero peltre / Cuero dorado / Cuero dorado satinado / Brillo plateado , Sandalia de vestir de mujer Mephisto Francesca Marina Savana , Sandalia FitFlop Mujer Lulu Aztek Stud Back-Strap Caramelo , Corkys Quickstep Women Open-Toe Canvas Black Sport Sandalia Black Silver , Calvin Klein Mujer Sandalia Domenica Negro , Sandalia Saturno de mujer Miz Mooz 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 Short NIKE Tempo para mujer Negro Negro CLARKS Womens Wave Grip Sabio Sandalia Terrell Post II Femenina de Merrell Pizarra , Saucony Women s Guide 10 Zapatillas para correr Azul marino , Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow Bomba Amie Naturalizer para mujeres Negro Dr. Scholl s Womens Granted - Colección original Ante negro Earth Origins Selby Azul marino , Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / negro-blanco-cool gris , Sandalia Gigi de Naturalizer para mujer Negro Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero Taupe CLARKS Sandalia Arla Kaydin para mujer Elástico Azul Marino / Blanco Naturalizer Mujeres Tonya Tender Taupe Leather Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Gris pardo , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Gris pardo , Mocasín de mujer RIALTO Vette Azul Xelero Visalia Comfort Comfort Terapéutico Extra Profundo Casual Shoe Leather Velcro Negro Sandalias de mujer Vionic Perk Carmel Backstrap Sandal Sillín , JSport by Jambu Women s Wendy Flat Gris claro , NIKE Entrenador flexible para mujeres 6 Cool Grey / Hyper Turquoise / Dark Grey , Sandalia de cuero universal deslizante Teva para mujer Negro Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Negro / púrpura , Zapatilla Shadow8 de Easy Spirit para mujer rojo Zapatillas Laurin Moccasin de Tempur-Pedic para mujer Hashbrown , Cole Haan Women s 2.Zerogrand Laser Wing Oxford Ch Argento Metálico Corral antigüedades de mujer Corral Python y botas occidentales de yute indio Azul , Rockport Mujeres Cobb Hill Lizzie Ballet Flat Negro Dansko Women s Loretta Oxford Nubuck negro Twisted X Boots Mens Lace up Impermeable Old Brown Hiker marrón Clásicos de base Slip-on (Tm) para hombres de Vans (Bum de goma) True White / True White ,

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 CLARKS CLARKS Wendy Alto Fisherman Sandal B01N56Q89R para mujer Cuero marrón marrón cd6eaa1 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 3443643
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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.