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 Slickery femenino Alegria Keli de Alegria femenino Slickery dff8858

Slip poliuretano resistente suelas diseñadas para proporcionar un natural rockero caminando movimiento.
estructurado suelas que también promover perfecto postura reduciendo así estrés en los músculos y articulaciones.
Desmontable (y reemplazable!) plantillas hechas de espuma de memoria de corcho y látex que crea un perfecto encajar formando a lo natural contornos de su pie.
Superior soporte de arco.
Talón altura: 1.5 pulgadas
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 Slickery femenino Alegria Keli de Alegria femenino Slickery dff8858 - 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 Slickery femenino Alegria Keli de Alegria femenino Slickery dff8858

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 Slickery femenino Alegria Keli de Alegria femenino Slickery dff8858

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

Betunia Loafer de mujer Aerosoles Serpiente Roja , Trotters Liz Loafer para mujer Plata blanca , Clásicos de base Slip-on (Tm) para hombres de Vans (Mix Checker) Negro / Blanco verdadero , Zapatillas Slip-On Roamer SAS para Mujer Mezclilla CLARKS Mujer Kinzie Light Loafer Flat Tweed negro , Trotters Liz Loafer para mujer Light Tan Tonal Zapatillas de deporte Slace Sport Sport D Lites Slip-On Mule Rosa Claro / Negro Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Pacífico , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker Emboss) Fallen Rock / Blanc De Blanc Trotters Liz Loafer para mujer marrón Slip clásico de mujer de SAS en Negro Skechers Women s Reggae Fest Willows Flat Malla azul marino , Trotters Liz Loafer para mujer marrón Trotters Liz Loafer para mujer Light Tan Tonal , Zapatillas Slip-On Roamer SAS para Mujer Armada , TOMS Clásicos de mujer Black Cherry Poly Canvas , Propet Women s Tour Walker Strap Sneaker Blanco , Zapatillas Slip-On Roamer SAS para Mujer Armada Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Melon rojo Zapatillas de barco Bluefish Sperry Top-Sider para mujer Avena de lino , Betunia Loafer de mujer Aerosoles Cuero negro Trotters Liz Loafer para mujer Negro Zapato Lites Bates para mujer marrón , Propet Women s Tour Walker Strap Sneaker Blanco Mocasín de gamuza Kilty de Minnetonka para mujer Violeta Azul , Trotters Liz Loafer para mujer Negro Sanuk Women s Pair O Dice Flat Armada , Zapato femenino Keli de Alegria Morning Glory Black , CLARKS Jocolin Vista Boat Shoe para mujer Textil perforado 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 Running Womens Supernova Sequence 9 Vapor Blue F16 / Silver mate / Craft Blue F16 Sandalia Drew Angela para mujer Dusty Multi Metallic , Chaco Women s ZX2 Classic Athletic Sandal Bloom Peach , Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Capri / Noche Azul / Flor de Cactus Púrpura Zapatillas de running ASICS GEL-Foundation 12 para mujer Plata / Pistacho / Pink Glow , NIKE Women s Revolution 4 Running Shoe Gunsmoke / Ocean Bliss / Dark Grey , Zapatillas de running NIKE RN 2017 para mujer gratis Wolf Gray / Off White-pure Platinum-black Sandalias con tira en el tobillo Begonia de mujer NAOT Black Madras , Sandalias de mujer Wolky Caviar negro , Brooks Womens Mazama Flor de Cactus Púrpura / Peacoat / Hot Coral Sandalias de mujer Nina, Georgea Slide Summer Daze Metallic Dyeables, Inc Sandalia de cuña de jazmín de mujer para mujer Silver Shimmer Zapato de caminar WW928HW2 para mujer New Balance Blanco Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra , Zapato de malla Nohea de OLUKAI - Hombre Blanco Mephisto Sandalia mujer Yin Slide Armada Stuart Weitzman Sandalia Rosita para mujer Ballet Taupe BareTraps Women s Rebecca Slide Sandal Blanco , Sandalia de diapositivas Clara embellecida Marc Jacobs para mujer Negro , Birkenstock Mujer Mayari Birko-Flor Sandal Mocca Calvin Klein Dalana Slide Sandalia para mujer Platino blanco Koolaburra de UGG Women s Lezly Slipper Negro Bomba de cuña Farrow para mujeres LifeStride Broncearse , Bos. Marron oscuro , Dr. Martens Unisex 1461 PW 3-Eye Shoe Negro Zapatillas de fútbol adidas Women s Nemeziz, 17,3 cm Fg W Trace Blue / Red Zest / Core Black , Zapatillas de deporte Wisteria para mujer Spring Step Professional Blanco Chaqueta no clasificada del tobillo de Chamonix de las mujeres Crema Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Negro-rojo Trotters Jake para mujer plana Estiramiento negro / Charol ,

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.


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 Slickery femenino Alegria Keli de Alegria femenino Slickery dff8858 - 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 Slickery femenino Alegria Keli de Alegria femenino Slickery dff8858 it out, and I got a chance to try out this new graphics software.


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 447424
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.