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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Tela o Textil
Importado
Sintético único
Eje mide aproximadamente no aplica desde el arco
Deslizamiento proporcionando fácil usabilidad
poco detallando la oferta una actualización de moda al holgazán clásico
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

Sam Edelman Loraine Loaine para B01H4XA96A Loaine mujer Navy mujer multi guinga 80a15aa

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.


Sam Edelman Loraine Loaine para B01H4XA96A Loaine mujer Navy mujer multi guinga 80a15aa

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

TOMS - Slip-On de lienzo clásico para hombre Negro / Ciruela Zapato femenino Keli de Alegria Riqueza , Trotters Liz Loafer para mujer Lila , Easy Spirit Women s Traveltime Mule Gamuza Combo Marrón Oscuro Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Blanco , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Lona negra / blanca TOMS - Slip-On de lienzo clásico para hombre Canal Blue Canvas Bocca Loafer de mujer Franco Sarto Negro B.O.C - Mujeres - Peggy Black Tooled , Zapatillas Slip-On Roamer SAS para Mujer Mezclilla , Arcopedico Womens Lolita Slip-On ciruela TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Lana tejida texturizada gris y negra , Sandalia de gamuza Poppy de TOMS para mujer Cuero miel , Skechers Women s Reggae Fest Willows Flat Malla azul marino , Mocasines de vestir de punta estrecha para mujer de Me Too Women Blue Champagne Denim , Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Miel Mocasines casuales Alpargata Deconstructed de TOMS para mujer Raya roja tejida Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer Armada Trask Womens Ali Hueso , Trotters Liz Loafer para mujer Armada Clásicos de base Slip-on (Tm) para hombres de Vans True White TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Diamante tejido susurro Splendid Midi Perf Vionic para mujer Slip-On Gris Nubuck Perfed , Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Melon rojo Zapato femenino Keli de Alegria Slickery Trotters Liz Loafer para mujer Light Tan Tonal , Zapatos OLUKAI Pehuea - Mujer Honu negro / negro , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Negro Negro Zapatillas Slip-On Roamer SAS para Mujer 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 Adidas Adizero Tempo 9 W femenino Blanco / Aero Blue / Core Black Sandalia deportiva Chaco Zvolv X2 para mujer Pino Diamante , NIKE Men s Lunar Fingertrap Cross Trainer Binario Azul Blanco Negro , Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo Zapatillas de running NIKE RN 2017 para mujer gratis Bordeaux / Monarch Purple / Bold Berry Zapatillas de running ASICS Gel-Nimbus 20 para mujer Azul / blanco Tierra Orígenes Shane Sedona Brown Sandalias Birkenstock Unisex Arizona Essentials EVA Khaki - 44 M UE - , Marc Fisher LTD Womens Alida Espadrille Wedge Ante natural , Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron WHITE MOUNTAIN Sandalia Gwinett plana para mujer marrón Birkenstock Unisex Arizona Essentials Sandalias EVA Plata , Zapatillas para caminar Propell Mujer Travellite MJ rojo , Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on Taupe / Coral , Ryka Women s Destiny Cross Trainer Negro / Plata , Zapatillas Adidas Adizero Ubersonic 2w, adidas, para mujer Blanco / Plata Metálico / Naranja Brillante , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Pewter / Calf , Zapatillas de tenis Adidas Performance Barricade Classic Bounce W para mujer Blanco / Oscuro Borgoña / Energy Pink Vans Unisex Old Skool Zapatillas clásicas de skate Negro / Blanco verdadero , Sandalia FLY London Women s Whin176fly Slide Reef Cupido , Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro , Sandalia Dansko para mujer Sophie Negro , TOMS - Slip-On de lienzo clásico para hombre Ceniza TOMS Clásicos de mujer Drizzle Grey Slub Chambray , Zapatilla Amira MUK LUKS para mujer Luz púrpura , Encaje Wander de mujer Palladium Sue Tuxedo Oxford Rio Red Aravon Betty-AR Oxford para mujeres marrón Trotters Liz Loafer para mujer Armada CLARKS Jocolin Vista Boat Shoe para mujer Textil perforado negro CLARKS Jocolin Vista Boat Shoe para mujer Azul / gris sintético

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.

Sam Edelman Loraine Loaine para B01H4XA96A Loaine mujer Navy mujer multi guinga 80a15aa - 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 Sam Edelman Loraine Loaine para B01H4XA96A Loaine mujer Navy mujer multi guinga 80a15aa 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.

Mocasines y Slip-Ons
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 303096
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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.