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.

Jacques 11500 Levine Pyramid B07CR4LDND Stud - Zapatilla de con cuero con cuña de mujer Metálico Multi da0ea1a




Cuero
Importado
gamuza suela
Medidas de talón aproximadamente 1.5 pulgadas
Disponible en cuero mate o cuero metalizado superior con cuero recortar. parte superior de cuero
Cuero recortar con pirámide adornos de espárragos
Cuero revestimiento
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


Jacques 11500 Levine Pyramid B07CR4LDND Stud - Zapatilla de con cuero con cuña de mujer Metálico Multi da0ea1a - 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

Jacques 11500 Levine Pyramid B07CR4LDND Stud - Zapatilla de con cuero con cuña de mujer Metálico Multi da0ea1a

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.


Jacques 11500 Levine Pyramid B07CR4LDND Stud - Zapatilla de con cuero con cuña de mujer Metálico Multi da0ea1a

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

Pantuflas Kendale de mujer Foamtreads Dusty Rose Zapatilla Dakota de las mujeres de Australia UGG Girasol , Baabuk Gus Slipper Gris claro , Bomba de Angel II para mujeres estilo suave Black Peau De Soire , Sandalia Wakataua N413 para mujer El Naturalista Curry Zapatillas de casa Mulise Unisex - Zapatillas de interior y exterior de lana - Made in Europe Fuchsia Multi Zapatilla Camper Wabi 20889 para mujer Gris Sandalias de verano XYD Fringe Slide Slip On Comfortable Slippers Zapatillas de hombre sólido holgazán Easy Walking marrón Bomba de cuña Farrow para mujeres LifeStride Negro 2 , Birkenstock Mules para mujeres Blanco CLARKS Mujer Poppy Pump Flores Cuero crema Zapatos OLUKAI Pehuea - Mujer Dusty Olive / Palm Trotters para mujer Dea Slingback Pump Patente perlada multi acolchada / negra , Bomba de Angel II para mujeres estilo suave Putty Elegance TOMS - Slip-On de lienzo clásico para hombre Microstripe azul marino / recorte , TOMS Women Classic Slip on Ceniza Bomba de vestido Cindie Easy Spirit para mujer Nubuck natural UGG Mujer Ansley mocasín Chocolate Trotters para mujer Dea Slingback Pump Armada , Zapatilla Glamour de mujer verde Daniel Estaño , Trotters Lucy Ballet para mujer Negro , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Leopardo negro Mocasín Sabrina Moc II Lamo para mujer Rosado Orthaheel by Vionic Indulge Gemma Mujeres punta redonda lona gris Slipper Broncearse , Zapatilla Viajero N5272 El Naturalista para mujer Grafito , Predicciones Comfort Plus Women s Karmen Pump Vino de gamuza , Zapatilla de tela Henrii de Nine West para mujer Negro Cobb Hill Rockport - Aubrey Dress Pump para mujer Caqui , Jacques Levine Pyramid Stud - Zapatilla de cuero con cuña de mujer Metálico Multi
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 Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7643 Naranja Azul Marino , NIKE Free RN - 831509601 Fuego rosa / azul resplandor / luz violeta / rosa explosión , Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Saúco nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Plata Metálica 002 , Zapatillas de running Flex Ride 630v5 para mujer New Balance Azul marino / plateado , Zapatillas para correr Saucony Progrid Echelon 3 para mujer Blanco / verde / cidra , Zapato para correr ASICS Conviction X 2 para mujer Negro / Carbón / Flash Coral Sandalia KEEN Women s Venice H2 Sweet Grape / Regal Orchid Zapatillas de running Skechers Performance para mujer Go Run 400 Action Azul marino / blanco , Zapatillas de senderismo adidas outdoor Terrex AX2R - Mujer Gris crudo / negro / amarillo semi-congelado Zapatillas de agua adidas Slip-On liso para mujer, adidas outdoor, para mujer Gris oscuro Heather / Chalk Blanco / Negro New Balance WW577 Hook and Loop Walking para mujer Blanco , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Estaño , Zapatillas de running Reebok Women s Club C 85 Blanco / gris claro , Zapato de tenis ASICS Gel-Dedicate 5 para mujer Mónaco Azul / Blanco Zapatillas de deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Coastal Blue 400 New Balance WW813 Hook and Loop Walking para mujer Blanco , Skechers Women s Summits Slip On Bungee Lace espuma de memoria negro / negro Negro Diadora N9000 Men Toe Snthetic Blue Sneakers Rojo / Ferrari Italia / Alaska Gray Under Armour Ua ClutchFit Drive Ii para hombre Púrpura / blanco , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker rojo , Sandalias de golf Oregon Mudders WCS400 para mujeres con suela de niple de césped , Sandalia Sloan Slide de Naturalizer para mujer Negro , Zapatillas de voleibol adidas Crazyflight X 2 de mujer Choque amarillo / ceniza plateado / blanco , Sandalias sintéticas Pleaser, mujer, cielo 301MR Claro, peltre negro Minnetonka Womens Chrissy Slipper Bootie Canela , Bota vaquera para mujer de Sebago Negro , Botas Tumbling para mujer de Dr. Scholl s Shoes Estuco de microfibra , Bota para mujer Apolo Earthies Terciopelo burdeos Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Blanco ,

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.

Jacques 11500 Levine Pyramid B07CR4LDND Stud - Zapatilla de con cuero con cuña de mujer Metálico Multi da0ea1a - 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 Jacques 11500 Levine Pyramid B07CR4LDND Stud - Zapatilla de con cuero con cuña de mujer Metálico Multi da0ea1a 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 818192
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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.