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.

Sandalia Merrell Terrell Convertible II Merrell 7600 para mujer B06ZXXHCGF Dusty Olive 1c61150




Nubuck
Hecho en EE. UU. o importado
Suave, cómodo y listo para la playa, lago o acera, el Merrell & reg; Terran Convertible Las sandalias II son listo para caber forma de tu pie para usar todo el día.
Nubuck parte superior de cuero para fácil limpieza.
Hook-and-loop cierre para seguro agarre.
extraíble correa para el talón.
Transpirable malla de revestimiento. plantilla de microfibra.Recuerde Me y comercio; cunas de la plantilla el pie para durar comodidad. caña de arco de nylon.Merrell amortiguación de aire en el talón absorbe choque impactante y estabilidad.M & reg; Suela SELECT GRIP entrega estabilidad y la tracción sobre terreno húmedo y seco. Importado. Peso: 9 oz Mediciones del producto fueron tomadas usando tamaño 8, ancho M. Tenga en cuenta que mediciones pueden 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


Sandalia Merrell Terrell Convertible II Merrell 7600 para mujer B06ZXXHCGF Dusty Olive 1c61150 - 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

Sandalia Merrell Terrell Convertible II Merrell 7600 para mujer B06ZXXHCGF Dusty Olive 1c61150

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.


Sandalia Merrell Terrell Convertible II Merrell 7600 para mujer B06ZXXHCGF Dusty Olive 1c61150

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

Oasis Women s Zoey Fisherman Shoes marrón Sandalia de mujer Drew Alana Pewter-bronce , Earth Origins Sophie Sedona Brown Sandalia deportiva Coleman 2.Zerogrand Slide de Cole Haan, cuero negro Cuero negro , KEEN Hombres Newport H2 Sandal Azul marino / gris medio , Sandalias Viakix para mujeres y ndash; Cómodo atlético Zapatos con estilo para Senderismo, al aire libre, Caminando, Playa, Deportes acuáticos Gris , Sandalia deportiva Tirra para mujer Teva Negro / Blanco Multi , Sandalia atlética Salomon Women s Crossamphibian Swift W Deep Peacock Blue Sandalia de susurro de mujer Keen Midnight Navy / French Blue Sandalia NIKE Benassi JDI para mujer Blanco / Lobo Gris / Plata Metálica Sandalia de cuña Jubilee Propet para mujer Corteza múltiple , OluKai Kulapa Kai Buganvillas / negro Chaco Women s ZX2 Classic Athletic Sandal Palm aguacate , Earth Origins Sophie Escarlata OLUKAI Pua Sandal - Mujer Deep Red / Bean Skechers Performance On The Go 400 Vivacity Flip Flop Blanco / azul claro Sandalias KEEN Venice II H2-W para mujer Skydiver / Blue Opal , Chaco Women s Wrapsody X Athletic Sandal Picnic Purple , KEEN Sandalias Newport Eco-W para mujer Negro / Imán BZees Sandalia Deportiva Jive para Mujer Gris Sandalia deportiva Nolita para mujer Anne Klein Tela negra , KEEN Hombres Newport H2 Sandal Negro / Gris piedra Wolky Comfort Jewel Bronce , Chancleta Eezay adidas para mujer Vapor Gris Blanco / Amarillo Fácil S Sandalias Newport H2 de KEEN para mujer Cuervo / Capri , Sandalia atlética de Rka Women s Glance Grey / Cool Mist Gray / English Lavender , Sanda Verra Teva para mujer Negro , Sandalias Bali KEEN para mujer Dapple Gray / Vapor Sandalia Uneek O2-w de mujer KEEN Negro 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 Sandalias Bali KEEN para mujer Beso de uva / Hierba Lavendar Chaco Women s Zx3 Classic Athletic Sandal Festoon azul , Chaco Mujer Zcloud Sport Sandal Penny Black , Zapatillas para correr A7200WX085 de Apex para mujeres Rosado Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Blue Fox / Cerulean-college Navy , Zapatillas para correr Inov8 Women s Parkclaw 275 GTX Negro / Coral Zapatillas para trail running Mutant Backcountry de La Sportiva para mujer Plum / Applegreen , Zapato de entrenamiento cruzado deportivo Hydro Sport para mujer RYKA Negro / Berry / Chrome Silver Zapatillas de deporte Skechers Performance Go Walk 4 Exceed con cordones para mujer Carbón de leña / Coral , Skechers Women s Empress-Wide-Awake Negro , Calzado de senderismo Go Joy 15603 de Skechers para mujer Negro / rosa caliente , Chaco Zcloud 2 Sport Sandal para hombre Pirámide Verde Zapatillas de running definitivas Ryka para mujer Gris negro , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede , Calzado para caminar WW812 para mujer New Balance Blanco Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Blanco apagado / blanco Sandalia Birkenstock de cuero Florida para mujer Sfb Black Birko-flor Sandalia para mujer Layne Slide de Pour La Victoire Amapola , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2829- Oro Sparkle & Snake , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2824- Brillo marrón satinado y dorado , Bota de campo soberano de Mountain Horse Negro Puf lindo de Patricia Green para mujer Negro , CLARKS Adriel Phyliss Pump para mujer Sintético de patente negro , Sansha Hi-Step Dance Sneaker Negro Zapatillas de deporte Reebok Women s Studio Basics Tiza / Sand Stone / Sour Melon / White , Bota tobillo Luna de Miz Mooz para mujer rojo , Cole Haan Women s 2.Zerogrand Laser Wing Oxford Tela a rayas gris , Zapatillas para caminar Rockport para mujer, expresiones, acera Chranson Ebano Zapato femenino Keli de Alegria Blue Croco

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.

Sandalia Merrell Terrell Convertible II Merrell 7600 para mujer B06ZXXHCGF Dusty Olive 1c61150 - 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 Sandalia Merrell Terrell Convertible II Merrell 7600 para mujer B06ZXXHCGF Dusty Olive 1c61150 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 1741915
3805523
4704769
5826591
6204610
9214602
10139260
14921790
18797494
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.