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.

Pantuflas Foamtreads Kendale de 19997 mujer B017N3ERRI Foamtreads Tela negra 0b9adeb

Sintético único
Hacer la comodidad una prioridad con la zapatilla 'Kendale' de Foamtreads & reg;.
genuino la parte superior de lana mantén tus pies agradable y cálido a través el invierno.
Ajustable correa de gancho y lazo permite un trato personalizado, ajuste apropiado.
Hecho por el hombre el forro es suave y mantiene los pies sensación de transpirabilidad.
Acolchado plantilla absorbe choque y proporciona amortiguación con cada paso. suela de goma entrega superior tracción para interior / exterior utilizar. con jabón suave; secar solo al aire. Peso: 11 oz Producto las medidas fueron tomado usando tamaño 7, ancho M. Por favor 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

Pantuflas Foamtreads Kendale de 19997 mujer B017N3ERRI Foamtreads Tela negra 0b9adeb - 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

Pantuflas Foamtreads Kendale de 19997 mujer B017N3ERRI Foamtreads Tela negra 0b9adeb

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.

Pantuflas Foamtreads Kendale de 19997 mujer B017N3ERRI Foamtreads Tela negra 0b9adeb

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

Koolaburra de UGG Women s Lezly Slipper Negro Patricia Green Womens Mt. Rosado , Glerups modelo unisex GR Slipper Gris , Bomba de vestido LifeStride Sevyn para mujer Negro 4 , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos melocotón Marc Jacobs Sienna Espadrille Ballet Mujer plana Khaki Multi Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Beige negro , Zapatilla de sombra de mujer KENDALL + KYLIE Negro / níquel , Pantuflas Matilda de Katy Perry para mujer Blush desnuda Bomba Francesca de mujer Adrianna Papell Marfil TOMS Clásicos de mujer Lona azul marino , Mocasín de gamuza Kilty de Minnetonka para mujer Gamuza turquesa , CLARKS Keesha Rosa Dress Pump para mujer Cuero negro , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Leopardo marrón , TOMS Women Classic Slip on Negro Trotters para mujer Dea Slingback Pump Goldwash Mul Calvin Klein Women s Gabrianna Pump Coñac , Zapatilla de calcetín de chalet Woolrich para mujer Zafiro , TOMS - Slip-On de lienzo clásico para hombre Universidad gris oscuro , RJ s Fuzzies Mocasin para mujer Vionic Womens Cosy Juniper Moc Toe Shearling Zapatillas Negro Bomba de vestido asimétrico para mujer Aravon Black Strtch Haflinger Unisex AT Woiled Wool Hard Sole Slipper Negro Rieker Mirjam 11 bomba para mujer Black Multi , Bomba de mujer David Tate Dainty Negro Zapatillas de mujer Foamtreads Coddels champán Zapatillas de deporte con cordones y tacón tobillo de mujer de Calajet para mujeres con tacón en punta 7CM rojo , Trotters Lucy Ballet para mujer Negro , Zapatilla para fumadores de la plataforma Mujer Soludos Broncearse
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 Reebok Women s Runner 2.0 MT Track Negro / violeta violeta / peltre , Sandalia NIKE Kawa Slide para mujer Hyper Turquoise White , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Total Naranja / Foto Azul / Gamma Azul / Blanco , Calzado de running para mujer New Balance Cypher Mulberry / Angora MBT Shoes GT 17 Athletic Athletic Leather / Mesh Lace-up Blanco / morado , Sandalia deportiva NIKE Kawa Slide para hombre Negro / azul brillan / blanco , Good-memories 2018 verano mujer zapatillas plano antideslizante interior informal peludo playa sandalias Color de mapache , Sandalia FitFlop Flare Slide para mujer Guijarro Sandalia de cuero de travesía New Balance para mujer Gris pardo Sandalia Hush Puppies Dachshund Strappy para mujer Cuero Ivory , En la ejecución de calzado de mujer en la nube menta Propet Tour Walker Strap Blanco azul Comfortiva Mujer, Faye Sandalia de tacón medio Negro , Splendid Women s Blake Sandal Níquel , Splendid Women s Brittani Sandal Oro rosa Botín para mujer Pleaser Asp1018 / Bpu / M Blanco Zapatillas de viaje Proather para mujer II Malla roja / blanca ASICS para mujer Gel-Fit Yui 2 SE Brote Verde / Flash Coral / Blanco , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Black-puma White Zapatillas de skate Vans Unisex Old Skool Silver Lining / True White , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Gris Zapatillas de deporte Vans ISO 1.5 para hombre / True White Running Trainers Azul Calzado casual de ante Nike VZTG de mujer NIKE Difuso Azul / Vela de vela Zapatillas de ciclismo híbridas ECM Saddle para mujer ECCO Negro , Así que Danca CH99 3 Heel con T-Strap Character Shoe Negro , Zapatillas Plum Ridge Ii Slipper de Woolrich para mujer Chocolate / Kendall Creek , La bomba Suzanne de Katy Perry para mujeres Plata , School Issue Upper Class 7300 Saddle Shoe (Niño / Niño pequeño / Big Kid) Negro , Clásicos de base Slip-on (Tm) para hombres de Vans Blanco (lienzo) CLARKS Women s CloudSteppers Sillian Jetay Flat Granadina ,

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.

Pantuflas Foamtreads Kendale de 19997 mujer B017N3ERRI Foamtreads Tela negra 0b9adeb - 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 Pantuflas Foamtreads Kendale de 19997 mujer B017N3ERRI Foamtreads Tela negra 0b9adeb 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 1858023
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.