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.

Hush Puppies Womens Hush Chowchow B01HSH7MCG WT Oxford Ante Chowchow negro 2a5f6f2

Eje mide aproximadamente Low-Top from arch
¿Quién? dice que no puedes enseña un viejo perro nuevos trucos Hush Cachorros y reg; actualizado a la Chowchow WT Oxford.
Clásico estilo oxford con estilo de zapatilla de deporte suela.
Superior esta construido de gamuza suave.
Cordones cierre asegura pie en su lugar.
Pull-tab en la parte posterior de fácil encendido y apagado. Silencio Cachorros y reg; en el talón el forro proporciona un suave y transpirable vestir. suela proporciona resistencia al deslizamiento y tracción Mediciones: Peso: 6 oz Mediciones del producto fueron tomadas usando tamaño 8, ancho M (SEGUNDO). que las medidas puede variar según el tamaño. Peso de calzado es basado en un solo artículo, no un par
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

Hush Puppies Womens Hush Chowchow B01HSH7MCG WT Oxford Ante Chowchow negro 2a5f6f2 - 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

Hush Puppies Womens Hush Chowchow B01HSH7MCG WT Oxford Ante Chowchow negro 2a5f6f2

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.

Hush Puppies Womens Hush Chowchow B01HSH7MCG WT Oxford Ante Chowchow negro 2a5f6f2

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

Shellys London Mujer Tommy Oxford Armada MONTAÑA BLANCA Womens Saint Closed Toe Oxfords Negro Eileen Fisher Mujeres Fresno Nubuck negro caído , Freja | Cuero negro , CLARKS Everlay Elma Oxford para mujer Cuero negro Zapatillas de deporte Easy Spirit AP1 Sport Negro , Oxford de Drew Amora para mujeres Black Stretch , U-lite Women s Classic Saddle Oxford, Cuero Sadie Zapatos Negro , Zapatillas de deporte Apollo Chukka de hombre nativas Gris Dublin / Blanco Shell / Caucho Blanco Shell Zapatillas de deporte Easy Spirit AP1 Sport Bronce Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Pewter / Calf , Pikolinos Womens Sitges W7J-4665CL Piedra Merrell Solo Aluminio Zapatillas de deporte Trolide Derby Trainer Fashion Rockport para mujer Gris oscuro , Zapatillas Spira Taos con diseño de Oxford para mujer marrón , UGG Mujeres Marji Snake Oxford Negro Cole Haan Women s 2.Zerogrand Laser Wing Oxford Cuero negro , Timberland Heritage s Women s Noreen - Bote con tres ojos para barcos marrón Zapatilla de nebulosa para mujer Geox Negro Georgia GB00157 Oxford Marron oscuro , Ros Hommerson Nadia Oxfords para mujer Desnudo, Bronce Arcopedico Femenino Francesca Oxford Gris Sneaker de moda de rayas de playa lavada campeón de mujer de Keds rojo Drew Shoe Women s Roma Oxfords Cuero negro / microfibra , Easy Spirit Cacia Mary Jane para mujer Beige TOMS Kids 10009918 Alpargata-K Paletas blancas Cobb Hill Rockport Mujeres Nadia Mary Jane Flat Cuero marrón , Propet Travelactive Slip-On Oxford para mujeres Heather negro / rojo , Easy Street Lucky Oxford para mujeres Súper gamuza negra / Patente
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 990v4 para mujer Lago Azul / Lago Azul Columbia Women s Barraca & shy; Sunlight Athletic Sandal Fósil, Natural Calzado casual tejido NIKE NIKE para hombre Azul costero / blanco negro apagado , Sandalia Drew Angela para mujer Dusty Multi Metallic Leather , Encaje de quimera de Columbia para mujer Zuraqua Calzado deportivo de mujer Fila de Statique Dark Shade / Black / Purple Cactus Flower , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Negro / lobo negro gris Calzado de trail running In-X-Talon 212 (S) para hombres Inov-8 Negro / naranja / azul Comfortiva Mujer, Faye Sandalia de tacón medio Whiskey Otago Nubuck Josef Seibel Women s Rosalie 29 Fisherman Sandal Negro Birkenstock Betula Licensed Womens Leo Birko-Flor Basic Navy , Sandalia Drew Cascade para mujer Sport Nubuck blanco Zapatillas Nike Air Max Essential para hombre Grey Mist / Black / Lyon Blue , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Piedra , Zapatillas clásicas Vionic para mujer Walker Negro Easy Spirit Women s Traveltime Mule Patente negra / plateada , Entrenador Ryka Eva Nrg Cross para mujeres Rosado rojo , Los hombres caídos Zapatillas The Easy Skateboard Oxblood Velvet Plataforma CLARKS Leisa Grace para mujeres Cuero azul marino Sandalia Kai Dolce Vita para mujer Cuero dorado , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sandalia plana Dionne de Easy Street para mujer Oro / Metálico Birkenstock Mujer, Madrid Sandalia deslizante Magic Galaxy Silver 3N2 Mofo Turf Trainer Azul real , Vionic Women s Tide Ii rojo , Trotters para mujer Dea Slingback Pump Armada Botín de tobillo Hadley de Bella Vita para mujer Gamuza almendra Zapatilla Lenox de mujer TOMS Azul marino / Slub , Rieker Womens R4373 Gillian 73 Negro / Negro / Negro , Eastland Solstice MLB Women s Oxford New York Yankees-navy

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.

Hush Puppies Womens Hush Chowchow B01HSH7MCG WT Oxford Ante Chowchow negro 2a5f6f2 - 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 Hush Puppies Womens Hush Chowchow B01HSH7MCG WT Oxford Ante Chowchow negro 2a5f6f2 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 347789
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.