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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Parte superior de Nuckbuck
Importado
Caucho único
Hermoso cueros dan esto sandalia una prima apariencia
ligero Plantilla moldeada de EVA con funda de cuero para una amortiguación superior y estabilidad superior
A inyección directa Espuma de la media suela de PU proporciona una larga duración amortiguación y excepcional todos los días comodidad para caminar
RECEPTOR tecnología para excelente estabilidad y soporte dinámico desde el suelo hasta despegar
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

Sandalia 12497 Yucatan ECCO ECCO Mujer Polvo Yucatan blanco 8abc4ea

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.


Sandalia 12497 Yucatan ECCO ECCO Mujer Polvo Yucatan blanco 8abc4ea

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 deporte Merrell, para mujer Terran Ari Backstrap Negro , Sandalia Omnium de mujer Teva Pizarra Sandalia deportiva Tirra para mujer Teva Trufa de ciruela , Sanda Verra Teva para mujer Cuerda elástica , Chaco Women s ZX2 Classic Athletic Sandal Negro , Columbia Women s Barraca & shy; Sunlight Athletic Sandal Fósil, Natural , Wolky Comfort Jewel Black Smooth , Correa de sirena de mujer Merrell Q2 Sandalia de atletismo Surf azul Sandalia Yucatan ECCO Mujer Polvo / polvo , Columbia Women s Barraca & shy; Sunlight Athletic Sandal Blanco negro Sandalia de deslizamiento ultra Comfort NIKE para mujer Pure Platinum / Wolf Gray Chaco Mujer Zcloud X2 Sandalia Deportiva Puzzle Poppy , Sandalia de cuña Rheta Sam Edelman para mujer Dusty Rose Suede Sandalias planas Eileen Fisher para mujer Sport-Nu Latté Sandalia Yucatan ECCO Mujer Irritante Chaco Women s Z2 Classic Athletic Sandal Trine en blanco y negro , Sandalia original premier universal Teva W de mujer Negro , Sandalia deportiva Tirra para mujer Teva Simplemente Taupe , Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón OluKai Kulapa Kai Dark Java / Dark Java KEEN Hombres Newport H2 Sandal Azul marino / gris medio Wolky Comfort Jewel Blanco , Sandalias Uneek EXO-W de mujer KEEN Vapor / Paloma , Wolky Comfort Jewel 12800 Blue Nubuckleather Chaco Women s Zx3 Classic Athletic Sandal Festoon azul , KEEN Hombres Newport H2 Sandal Tierra oscura / Acacia , Josef Seibel Mujeres Hailey 33 - Cuero blanco - , Sandalia de mujer Drew Alana Blanco negro Encaje de quimera de Columbia para mujer Negro, Steam ,
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 deporte casual Gazelle de adidas para hombre Negro / Negro / Oro Metálico , Zapatillas Merrell Women s Encore Nova Slip On Black Smooth Zapato de running ASICS Gel-Quantum 360 cm para mujer India Ink / Flash Coral / Mid Grey Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado OluKai Kulapa Kai Dragon Fruit / Black NIKE Women s Tanjun Blanco Blanco Negro , Zapatillas de trail running Olympus 3 para mujer Altra AFW1859F Gris / verde azulado , Sandalia de Hollyleaf de las mujeres de Merrell Helecho , Sandalia Terrell Post II Femenina de Merrell Pizarra , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel , New Balance Women s Summit - Desconocida Trail Running Shoe Verde / negro , Corredor de pista flexible de carga de agilidad Merrell para mujeres Negro , Saco de gladiador Gentle Souls Break My Heart 3 para mujer Oro , Zapatillas de running NIKE Roshe One Print para tobillo y tela para mujer Ember Glow / Black-black-white Bomba de cuña Lansing para mujer Trotters Cazadora de gamuza marrón oscuro Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , Easy Street 30-8320 Sandalia airosa para mujer Negro Sandalia de Richetta de Vince Camuto para mujer Arena de metal Sandalias de mujer Ros Hommerson Mellow rojo , Zapatillas de tenis 696v3 New Balance para mujer Pigmento / coral vivo Zapatillas de deporte Reebok para mujer CL Nylon Mesh M Niebla / tiza , PUMA Womens Mostro Puma Black / Puma Black / Puma White , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Lila congelada , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Rosado Zapatillas para caminar Brooks Women s Addiction Walker Blanco , Betsey Johnson Women s Slide Sandalia Guinga roja , Coach Womens Jessie cerrado tobillo Botines de moda Nappa sedosa negra , Bota Newton del Dr. Martens para hombres Rojo cereza , Arcopedico Mujeres L8 Ante azul marino , FRYE Sacha Moto Shootie tobillera para mujer Estaño ,

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.

Sandalia 12497 Yucatan ECCO ECCO Mujer Polvo Yucatan blanco 8abc4ea - 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 Sandalia 12497 Yucatan ECCO ECCO Mujer Polvo Yucatan blanco 8abc4ea 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 y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

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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.