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.

Zapato Porcelana de skate Diadora B.Elite Suede Fiesta/ Zapato Diadora Porcelana Verde d9d16ad




Gamuza
Importada
Eje mide aproximadamente Parte baja del arco
Mantener tu estilo de calle limpio y simple en el B. Elite Gamuza.
Rico parte superior de gamuza con un dedo perforado.
Tradicional cierre con cordones.
Acolchado lengua y cuello.
Suave revestimientos sintéticos. Ligeramente acolchado, con forro textil plantilla. Durable suela de goma.Importado.Medidas: Peso: 1 libra Mediciones del producto fueron tomadas usando tamaño 12 hombres, 13.5 mujeres, ancho Medio. tenga en cuenta que las mediciones puede variar por tamaño
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


Zapato Porcelana de skate Diadora B.Elite Suede Fiesta/ Zapato Diadora Porcelana Verde d9d16ad - 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

Zapato Porcelana de skate Diadora B.Elite Suede Fiesta/ Zapato Diadora Porcelana Verde d9d16ad

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.


Zapato Porcelana de skate Diadora B.Elite Suede Fiesta/ Zapato Diadora Porcelana Verde d9d16ad

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

Supra Vaider LC Sneaker Violeta gris - Blanco Vans Old Skool Unisex Adults Low-Top Trainers Mantequilla de manzana / True White Supra Vaider LC Sneaker Azul marino , Supra Vaider LC Sneaker Blanco oliva Zapato de skate casual Heathrow DC para hombre Negro / Negro / Negro Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco , Supra Skytop Skate Shoe Negro-negro-rojo Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro Supra Vaider LC Sneaker Oliva / Demitasse / Blanco Vans Unisex Old Skool Zapatillas clásicas de skate (Gamuza) Negro / Blanco Vans Unisex Old Skool Zapatillas clásicas de skate (Lienzo) Negro / Blanco , Vans Auténtico Autumn Glaze / True White Supra Skytop Skate Shoe Negro / Gris oscuro / Blanco , Zapatilla de deporte KAANAS Santa Fe Fashion Skate Slip-on Casual Sneaker Serpiente Clásicos de base Slip-on (Tm) para hombres de Vans True White Vans Unisex Old Skool Zapatillas clásicas de skate Sepia Rose / Blanc De Blanc , Vans Unisex Old Skool Zapatillas clásicas de skate Goblin Blue / True White Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Asfalto , Supra Scissor Skate Shoe Caqui / Blanco , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Hueso Ligero / Hueso Ligero Salomon fuerza asalto de velocidad Negro Vans Unisex Old Skool Zapatillas clásicas de skate (Classic Tumble) Negro / True White Clásicos de base Slip-on (Tm) para hombres de Vans Negro , Vans Auténtico Neón Rosa / Negro Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Nautical Blue Vans Auténtico Woodstock / Hueso Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Goma negra / clásica , Supra Scissor Skate Shoe Negro-carbón , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco
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 Saucony Women s Freedom ISO Negro , Zapatillas de running Saucony Women s Triumph ISO 4 Azul marino / menta Zapatillas de running Nike Air Huarache Run Ultra para hombre Negro / Negro / Negro Sandalia KEEN Women s Venice H2 Gasolina / Crepúsculo Sandalia deportiva Marc Jacobs Women s Somewhere Amarillo / Multi , Sandalia de vestir Tillman de SoftWalk para mujer Blanco Supra Martine Run Skate Shoe para hombre Andora-blanco Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Zapatillas Reebok Women s Runner 2.0 MT Track Nube gris / polvo de asteroides / peltre Zapatillas para caminar Ryka Women s Devo Plus 2 Gris / Coral , Propet Women s Fisherman Sandal ciruela , David Tate para mujer Nelly Nubuck negro Sandalia de mujer Yellow Box Amor Oro rosa , Trotters Mujer Simmy Mary Jane Flat Sabio Zapato para caminar Skechers Go Joy para mujer Azul marino / blanco Clásicos de base Slip-on (Tm) para hombres de Vans (Verificador primario) Verdadero azul / blanco Diadora Unisex I.C. Negro / Plata Zapatillas de baloncesto Nike para mujer Air Force 1 Hi Premium Black / Black Gum Med Brown Sail Lila blanqueada / Lila blanqueada Birkenstock Sandalias Madrid Birko-Flor para mujeres Plata , Sandalias AS-Mona Slide de mujer Ash Oro negro , Zapatos rusos de Pointe Rubin Pointe, caña media flexible de U-Cut , Noble Outfitters Womens Muds Stay Cool 6 Mora , Capezio DS24 Rockit Dance Sneaker para mujer Negro , Zapatillas Reebok Women s Plus Lite ti Track Ti - Whisper Gray / White / Guava Punch / Sour Melon / Lilac , Bota de moto Fulton para mujer Roxy Broncearse Zapatillas de señora Jacques Levine 9854 Shearling Negro Mocasín Bae Bailarina SoftMoc para mujer castaña , Sandalias de cuña de cuero de las mujeres ocasionales de Vipic Calypso casquillo del dedo del pie Ante negro Botines de fútbol para tobillo para hombre Mercurial Superfly VI 360 Elite FG Black Easy Spirit Women s Traveltime Mule Marrón oscuro / Multi ,

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.

Zapato Porcelana de skate Diadora B.Elite Suede Fiesta/ Zapato Diadora Porcelana Verde d9d16ad - 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 Zapato Porcelana de skate Diadora B.Elite Suede Fiesta/ Zapato Diadora Porcelana Verde d9d16ad 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.

Skateboarding
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.