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.

Trotters Mujer negro Simmy Mary Jane Jane Flat Ante Mary negro e644513




Cuero
Importado
Sintético suela
Medidas de talón aproximadamente 0.75
Cuero plano ofreciendo contraste costura y asimétrica Correa Mary Jane
Elástico insertar y gancho y lazo cierre en la correa
Flexible Suela anti-impacto
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


Trotters Mujer negro Simmy Mary Jane Jane Flat Ante Mary negro e644513 - 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

Trotters Mujer negro Simmy Mary Jane Jane Flat Ante Mary negro e644513

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.


Trotters Mujer negro Simmy Mary Jane Jane Flat Ante Mary negro e644513

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

Zapatillas Skechers Women You-14966 Gris / azul Trotters Mujer Simmy Mary Jane Flat Ante negro Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Mora , Zapato de caminar WW928HW2 para mujer New Balance Blanco , Drew Shoe Women s Tulip Comfort Hueso Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro SAS Womens Sunny Oro , Linea Paolo Felicia | Cuero perforado negro Bota de mujer Belmore de Wolky Cuero de terciopelo negro Drew Shoe Women s Paradise II Velco Slip-on Gris pardo Zapatillas para caminar Ryka Women s Dash 2 Azul marino / gris / morado Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Gris blanco , Zapatillas para caminar Skechers Performance para mujer Go Step Lite-Persistance Blanco Drew Shoe Women s Tulip Comfort Negro , Zapatillas para caminar Go Walk 4 Super Sock 4 de Skechers Performance para mujer Blanco negro , Sandalias SAS para mujer, Nudu Slide Medianoche , Zapatillas para caminar Lite-Evoke Go Step de Skechers Performance para mujer Negro / gris , Vivien Flat de Bella Vita para mujeres Cuero de gamuza de cabrito negro , Calzado de paseo Elated de S paso de mujeres de Skechers Performance Negro , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Negro Negro Zapatillas para caminar Terrex Voyager CW CP W de adidas Outdoor para mujer Negro / Negro / Blanco Tiza , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul , Dr. Comfort Patty Diabético Terapéutico Mujer Extra Profundo Cordón de Cuero para Zapatos Negro Sandalia Vionic Shelley Slide para mujer Caramelo Zapato de trabajo con cordones Skechers for Work Synergy Sandlot con punta de mujer Negro / rosa Zapatillas para mujer Reebok F / S Hi Fbt Walking Shell Pink / Shell Pink , CLARKS Mujer Gracelin Mara Flat Berenjena , Zapatillas de golf Skechers Performance para mujer Go Golf Elite Canvas Carbón Drew Shoe Women s Tulip Comfort Hueso ,
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 running New Balance 990V3 para mujer Gris / rosa Zapatillas de running New Balance 520v3 para mujer Gris , Zapatillas de running Saucony Women s Triumph ISO 4 Blanco azul adidas Sandalia atlética Anyanda Flex Y W para mujer Easy Mint / Silver Metallic / White , Wolky Comfort Jewel Black Smooth , UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Negro / Naranja Zapatillas para correr Saucony Women s Cohesion 10 Armada Zapatillas de trail running Salomon Speedcross 4 para hombre rojo , Hush Puppies Men s Gus Oxford Broncearse , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue New Balance Women s w990v4 Running Shoe Gray / Castlerock Drew Shoe Women s Tulip Comfort Brown Croc , MBT Mujeres Baridi Mary Jane Malla sintética azul , Stuart Weitzman Womens Carmina Gold / Multi / Nighttime , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro CLARKS Sandalia Saylie Moon para mujer Cuero blanco , Vionic Women s Flex Sierra con cordones Armada , Under Armour Charged All-Around Neutral Nordic Green (903) / Nori Green , New Balance WW411v2 Hook and Loop Walking para mujer Blanco nativo AP Mercury Liteknit Leche Rosa / Shell Blanco / Caucho Natural , Entrenador cruzado de malla 77V2 para mujer New Balance Negro / Plata Nike Tech Xtreme Cadet Glove Pure Platinum / Blanco / Volt / Black , I-RUNNER Eliza Women s Therapeutic Athletic Extra Profundidad Zapatilla de cuero / de malla con cordones Gris azul Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Collard , Sandalias Dior J Pliner Dacia para mujer Piedra , Sandalia tobogán moonstruck de mujer muy volátil Luz de oro Zapatillas Mujer OLUKAI Alaula Dark Java / Trench Blue , Cole Haan Women s Althea Bootie Negro Botín Eli para mujer Bella Vita Blush Kid Suede Bota de montar del latido del corazón de las mujeres muy volátiles Vino

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.

Trotters Mujer negro Simmy Mary Jane Jane Flat Ante Mary negro e644513 - 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 Trotters Mujer negro Simmy Mary Jane Jane Flat Ante Mary negro e644513 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 1562642
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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.