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.

Zapatillas para de caminar Go Flex 2-14992 de Zapatillas Skechers mujer Performance para mujer Carbón 979917f




Textil
Importado
Caucho único
Único diseño de suela para la máxima flexibilidad y comodidad
Ligero y receptivo amortiguación
Goga máximo alto rebote plantilla energiza cada paso
Transpirable malla superior
suave forro de tela
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


Zapatillas para de caminar Go Flex 2-14992 de Zapatillas Skechers mujer Performance para mujer Carbón 979917f - 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

Zapatillas para de caminar Go Flex 2-14992 de Zapatillas Skechers mujer Performance para mujer Carbón 979917f

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.


Zapatillas para de caminar Go Flex 2-14992 de Zapatillas Skechers mujer Performance para mujer Carbón 979917f

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

Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Chocolate Zapatillas para caminar Lite-Agile Go Step de Skechers Performance para mujer Teal Vionic Women s Flex Sierra con cordones Blanco negro , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata / Cuero Zapato para caminar Metrolyte para mujer ASICS Negro / Regatta Blue / Regatta Blue , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Armada Easy Spirit - Zapatillas Faisal2 para mujer Azul oscuro-azul marino , Zapatillas para caminar adidas Outdoor Terrex Voyager DLX W para mujer Gray Two / Gray Four / Chalk White Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Negro Zapatillas de deporte Romy de Easy Spirit para mujer Gris claro , Prop & eacute; t Olivia Oxford para mujeres Bronco Brown , Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Malla negra de punto / carbón de leña Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco , Calzado de senderismo Skechers Go s 15601 para mujer Blanco negro Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata blanca , Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Cuero gris , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Acuarela floral , Zapatillas Skechers Go s Lite Lite-Mystic para caminar Negro , Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Super Sock 2 Slip-On Viejo negro Zapato de caminar WW1300v1 para mujer New Balance Gris , Zapato de caminata Skechers Performance para mujer Go Walk 4 Pursuit Negro Negro , Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Gray Marina CLARKS Zapatillas CloudSteppers para mujer Sillian Tino Zapatillas con cordones Tela azul marino oscuro Hush Puppies Men s Gus Oxford Marron oscuro Zapatilla de deporte Dansko para mujeres Negro / negro gamuza , Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Negro , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Negro , Skechers Women s Go 4-14918 Calzado para caminar Gris pardo , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Negro / Estiramiento
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 para correr Nike Zoom All Out Flyknit para mujer Negro / Cool Gray / Volt / White , Calzado de running casual para mujer New Balance WL555, para mujeres Menta de plata Sandalias Drew de mujer de Drew Cuero liso de color salmón , Sandalia KEEN Women s Venice H2 Negro / Gris neutro Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Dark Raisin / Light Thistle Clorts Women & Men s Water Shoe Sandalia para caminar de secado rápido WT05 Rosado Calzado de agua Columbia Drainmaker IV para mujer Negro, hielo gris Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Binario Azul / Gris Oscuro / Oscuro Obsidiana Zapato de lona KEEN Sienna MJ para mujer Pluma gris Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Lentejuelas negro / blanco , Sandalias Harper Huarache de SoftWalk para mujer Rojo / Tostado Bernie Mev Women s Comfi Flat Pantalones , Zapatillas de running adidas Cloudfoam QT Racer para mujer Negro / Negro / Blanco Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Miel / Coral Plano original de la mujer del Dr. Scholl s Collection Collection Kimberned Pointed Toe Vestido azul nubuck , Skechers Women s Microburst Showdown Walking Sneaker Negro Calzado de trabajo resistente al deslizamiento Memory Flux de Fila para mujeres Negro / Negro / Negro Entrenador cruzado Minimus 20v7 de New Balance para mujeres Negro Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Carbón Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Blanco / rojo / azul , Sandalia Harley-Davidson Ames Slide para mujer Negro ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ Cuero desnudo de cobre S9216 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) 1700- Beige Nubuck / Gold Trim Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ marrón Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Rosado , Converse Chuck Taylor All Star Core Ox Armada , Trotters Liz Loafer para mujer Rosado TOMS - Slip-On de lienzo clásico para hombre Negro / Ciruela , Mocasín con cordones ocasionales de cuero KEESKY para mujer Zapatos sin cordones planos Negro

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.

Zapatillas para de caminar Go Flex 2-14992 de Zapatillas Skechers mujer Performance para mujer Carbón 979917f - 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 Zapatillas para de caminar Go Flex 2-14992 de Zapatillas Skechers mujer Performance para mujer Carbón 979917f 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 45319
3894862
4959868
5115751
6753370
6825867
13288736
13707843
14156712
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.