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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poliéster
Hecho en EE. UU. o importado
con doble correa poliéster jacquard correas superiores alrededor del pie y a través del entresuela para un personalizado ajuste, moldeado por inyección hebilla de bloqueo de la escalera
Toe bucle para seguro ajuste
Ajustable y durable de alta resistencia correas del talón de la correa
Mujeres LUVSEAT específico Entresuela de la PU, antimicrobiano solicitud de control de olores, vegano-amistoso construcción
No marcado Caucho ChacoGrip compuesto, terminal de 3 mm profundidad en suela
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

Chaco Sandal Women B01M0TRGOC s ZX2 Classic Athletic Sandal Palm Classic aguacate 7ab423c

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.


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

Olive Tortuga - Alpargata mujer Aceituna Teva Terra Sandal para hombre Cross Terra Black Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Gris / Negro / Rosa Táctil Sandalia Teva Terra-Float Nova para mujeres Gris , Sandalia deportiva Chaco Zvolv X2 para mujer Azul volcánico , Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7643 Negro Sandalia de mujer W Verra Sport de mujer Teva Atlántico Norte Sandalia Paula Fisherman de Haflinger para mujer Piedra , Skechers Performance On The Go 400 Vivacity Flip Flop Gris pardo , Sandalias KEEN Venice II H2-W para mujer Negro / gris acero , Sandalia atlética Salomon Women s Crossamphibian Swift W Higo Sandalia de susurro de mujer Keen Imán / Sangría Chaco Mujer Zcloud X2 Sandalia Deportiva Runa Teal OluKai Kulapa Kai Negro Negro OLUKAI Haiku Elua Sandal - Mujer Sombra negra / oscura , Chaco Mujer ZX1 Classic Sport Sandalia Estallido de Berry , Sandalia NIKE Benassi JDI para mujer Negro / oro rosa , Chaco Women s Z2 Classic Athletic Sandal Cañón Chaco Mujer Zcloud X2 Sandalia Deportiva Melocotón de tela Sandalia para mujer NIKS Benassi Solarsoft Slide Blanco negro , Sandalia original Teva Women s Original Inca Plum Multi Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre , Sandalia original Teva Women s Original Marled Orchid , Sandalia deportiva Tetra Cloud de Chaco para mujer Jengibre especias , Sandalia NIKE Benassi JDI para mujer Rosa / Plata Teva Terra Sandal para hombre Madang Olive Webbing Zapatillas de viaje Protic W3910 Vista Walker para mujeres Negro , Sandalia de susurro de mujer Keen Sombra Oscura / Cerámica , Wolky Comfort Jewel 12800 Blue Nubuckleather
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 entrenamiento Nike Free Tr 8 Amp para mujer Gris oscuro / Blanco-negro , Zapatillas de running mujer HOKA ONE ONE Clifton 4 Aquifer / Vintage Indigo Sandalia KEEN Clearwater CNX para mujer Negro / Resplandor Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow , Sandalia atlética Salomon Women s Crossamphibian Swift W Deep Peacock Blue , Estilo NIKE Ultra Comfort Slide para mujer: 882695 para mujer 882695-404 Aqua / Cool Gray Zapatillas de running Salomon Women s Speedcross 4 Wide W Trail Poseidón , Salomon Women s Speedcross 4 W Trail Runner Dark Purple / White / Deep Lake , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral , Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow Easy Spirit - Zapatillas Faisal2 para mujer Blanco , Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Plata blanca Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Atlético femenino Zapatos - 910 , Zapatillas de running NIKE Mens Air Zoom Structure 19 Oliva mediana / hierba de mar / garganta verde / bambú Zapatillas clásicas Vionic para mujer Walker Negro , Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata / Lima Skechers Women s Go 4-14918 Calzado para caminar borgoña , Aerosoles Womens High Alert Gamuza rosa Skechers Performance Mujer Go Step Lite Pintoresco Negro Zapatillas Cross-Trainer Ryce para mujer Enhance 3 Gris oscuro , Zapatos de entrenamiento Nike Cheer Scorpion Cross para mujer Blanco / Blanco / Platino puro , Sandalia de gladiador Kleo Alegria para mujer Glimmer Glam Zapatilla deportiva Kailee de mujer Ryka Insignia Blue / Yucca Mint / Summer Grey , Vans Unisex Old Skool Zapatillas clásicas de skate Gum Mono Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Crema / Combo Kenneth Cole REACTION Sandalia de diapositivas con talón deslizado Vikki Slip para mujer Chicle Sandalia de diapositivas Katlee de Dolce Vita para mujer Rose Gold Mesh , Boombah Women s A-Game - Juego de calas intermedias moldeadas - 8 opciones de color - Varios tamaños Negro / púrpura TOMS - Slip-On de lienzo clásico para hombre Blue Marl Drew Shoe Heather Synthetic Oxfords para mujer Cuero 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.

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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 Chaco Sandal Women B01M0TRGOC s ZX2 Classic Athletic Sandal Palm Classic aguacate 7ab423c 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.


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