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.

Zapatillas de deporte Reebok para mujer Reebok CL Nylon tiza 19092 Mesh M Polvo del desierto/ tiza 498c288




Cuero y Textil
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Abrir malla tejida superior
Corte bajo diseño
troquelado Entresuela de EVA
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


Zapatillas de deporte Reebok para mujer Reebok CL Nylon tiza 19092 Mesh M Polvo del desierto/ tiza 498c288 - 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

Zapatillas de deporte Reebok para mujer Reebok CL Nylon tiza 19092 Mesh M Polvo del desierto/ tiza 498c288

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.


Zapatillas de deporte Reebok para mujer Reebok CL Nylon tiza 19092 Mesh M Polvo del desierto/ tiza 498c288

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

Drew Shoe Women s Tulip Comfort Ternero blanco , Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro , Zapato de caminar sin tocar Skechers Performance para mujer Go Step Ante negro Drew Shoe Athena para mujeres Blanco azul , Earth Origins Tova Polvo , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Gris / Franela , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Azul / iridiscente , Sandalia Juno Slide de Vionic Women s Rest Negro , Zapatillas Skechers Performance Go Mini Flex-14006 para mujer Armada , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy , Calzado de mujer de fila Cress Walking Blanco, blanco, Highrise , Dr. Comfort Marla Mujer diabético terapéutico Extra Profundo Calzado Lycra Velcro Negro Zapatillas de deporte Converse Chuck Taylor All Star Coral Ox para mujer Negro / Blanco / Negro , Dyeables, Inc Womens Poppy Dress Pump Satén negro Adidas Sport Performance Mujeres Terrex CC Voyager Sleek Sneakers Negro / Negro / Blanco Tiza , Zapato para caminar Skechers Go Joy para mujer Gris pardo , Skechers Women s Define Sneaker Negro , Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Gris pardo Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Acuarela / Iridiscente , Zapatillas de senderismo Merrell Moab 2 GTX para mujer Negro , Zapatillas de deporte Romy de Easy Spirit para mujer Coral , Zapatillas de deporte Skechers Performance Go Walk 4 Exceed con cordones para mujer Taupe / Coral , Dr. Comfort Marla Mujer diabético terapéutico Extra Profundo Calzado Lycra Velcro Negro , Zapatillas de deporte Skechers Performance Go Walk 4 Exceed con cordones para mujer Negro Zapatillas Reebok para mujer CL Lthr NBK Sneaker Blanco / ópalo Jambu Women s Spain Flat Tierra oscura , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco , Zapatos de tierra Camino Ante metálico plateado , Sandalias de mujer Ros Hommerson Mellow 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 Sandalias Teva Mujer W Terra Fi Lite City Lights Black / Pastel NIKE Women s Tanjun Blanco Blanco Negro Zapatillas para correr Nike Women s Flex Experience Rn 5 Antracita / Rosa explosiva / Negro , Converse Chuck Taylor All Star Seasonal Color Hola Daybreak Pink Zapatillas de running Adidas Performance Ultra s Boost M para hombre Azul / Misterio Azul / Negro , Zapatillas de running New Balance NYC Fresh Foam Zante V3 para mujer Negro , NIKE Women s Wmns Air Max 270, Ocean Bliss / Blanco-Negro-Hot Punch Ocean Bliss / White-black-hot Punch , Sandalias KEEN Venice II H2-W para mujer Skydiver / Blue Opal , Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Gris Zapatillas de running Alphabounce Em W para mujer de adidas Originals Amarillo hielo / Semi Solar amarillo / blanco , Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Gris oscuro / Blanco / Negro / Sigilo Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Blanco / Menta-negro , Dr. Comfort Patty Diabético Terapéutico Mujer Extra Profundo Cordón de Cuero para Zapatos Negro Chaco Mujer Zcloud 2 Sport Sandalia Jab Pine Zapatillas de running ASICS Gel-Nimbus 19 para mujer Flash Coral / Dark Purple / White BareTraps Sandalia Denni Slide para mujer Melón / trigo Sandalias de mujer Bella Vionic Bella Toe en Tortuga - Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Easy Spirit Women s Traveltime Mule Suede claro natural / blanco Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Azul marino / blanco , Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón Zapatillas de viaje Proather para mujer II Malla de plata Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Hueso Ligero / Hueso Ligero Kate Spade New York Women s Isla Slide Sandal Blanco Sandalia de diapositivas de ocio para mujer de Seychelles Blanco Sandalias planas Cammie de Jessica Simpson para mujer Negro Sandalia Pomopom Slide de ALDO para mujer Rosa Varios TOMS Clásicos de mujer Abeto azul Cobb Hill Rockport - Aubrey Dress Pump para mujer Estaño , Steve Madden Women s Gaze tobillo botín Terciopelo de setas ,

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.

Zapatillas de deporte Reebok para mujer Reebok CL Nylon tiza 19092 Mesh M Polvo del desierto/ tiza 498c288 - 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 Zapatillas de deporte Reebok para mujer Reebok CL Nylon tiza 19092 Mesh M Polvo del desierto/ tiza 498c288 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.

Para caminar
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 720837
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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.