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.

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Cuero
Importado
Goma suela
Cuero revestimientos para todo el día comodidad
extraíble, contorneado de triple densidad plantilla con arco soporte y amortiguación
Tuckboard con mango de acero proporciona estabilidad y soporte
Rubber suela con cuero pila de talón tiene sido tratado con Protector scotchgard de 3 m
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


Dansko Women s Louise B075V64FQW Oxford Dansko Vino Nappa Women Pulido d7a2914 - 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

Dansko Women s Louise B075V64FQW Oxford Dansko Vino Nappa Women Pulido d7a2914

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.


Dansko Women s Louise B075V64FQW Oxford Dansko Vino Nappa Women Pulido d7a2914

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

Mordenmiss Nuevo verano de primavera y verano de flores hechas a mano zapatos planos de cuero Mocasines Estilo 2 de color caqui , Zapatillas para mujer Del Rey TOMS (Explosión de luz azul marino) , Prop & eacute; t Nessie Oxford para mujeres Negro Zapatillas de skate DC Women s Villain SE Slip-on Oro rosa Bomba de vestido caliente de mujer Rose Petals Negro / Bronce Aravon Betty-AR Oxford para mujeres Negro , Drew Shoe Women s Paradise II Velco Slip-on Ternero negro , Geox Kookean 3 Oxford para mujer Plata , Moda mujer Skechers-Slide Thru Blanquecino , Zapatillas de moda Native Unisex Apollo Moc. Snapper Red / Cabo Blue / Shell White , CLARKS Tilmont Ivy Oxford para mujeres Black Combi TOMS Kids 10009918 Alpargata-K Peacock apenado hoja , Zapatillas de deporte Buckner Fashion de Mark Nason Angeles para mujer Oro Rockport Mujeres Ayva Oxford Gris claro Wolverine 1883 by Women s Alice Slip-On Shoe Marrón floral Ballet de Cole Haan Leah Skimmer para mujeres Cuero blanco , Castine Oxford de mujeres de Eastland marrón Zapatillas de moda Native Unisex Apollo Moc. Regatta Blue / Shell White / Regatta Rubber , Propet Travelactive Slip-On Oxford para mujeres Heather negro / azul marino CLARKS Womens Un Cruise Lace (Wave) Sneaker Combi de cuero blanco / arena nubuck , Easy Street Lucky Oxford para mujeres Súper gamuza negra / Patente Drew Shoe Women s Hope cuero terapéutico moda Oxfords Negro Easy Spirit Cacia Mary Jane para mujer Gris MaxMuxun Mujer Zapatos Perforated Lace UP Brogues Wingtip Comfort Oxfords Negro , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Gris Mapa de Easy Spirit Women s Explore Light Natural Multi Zapatillas Loraty Easy Spirit para mujer marrón Sandalia de lana Anica de Cole Haan para mujer Estampado de estrellas de plata / metalizado , Zapatillas Skechers Sport Mujer Imperio Moda Negro / 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 Sandalia de vestir Tillman de SoftWalk para mujer rojo , Sandalia Shaye Sport mujer Donald J Pliner Platino Calzado de running Reebok Print Run Prime Ultk para mujer Carbón / Negro / Violeta picante / Ponche de guayaba / Blanco Brooks Womens Adrenaline GTS 18 Púrpura / negro / champán Skechers Women s Go 4-14918 Calzado para caminar Carbón , Zapatillas para caminar Skechers Gostyle 15606 para mujer Rosado Zapatillas para caminar Go Walk 4 Super Sock 4 de Skechers Performance para mujer Negro Skechers Women s Define Sneaker Azul marino Skechers Go Walk Lite-15350 Wide Sneaker para mujer Negro , New Balance WW813 Hook and Loop Walking para mujer Gris Apis Mt. Negro , Nike Tech Xtreme Cadet Glove Negro / Hyper Crimson / Hot Punch / Volt Ryka Women s Destiny Cross Trainer Gris / Rosa Vans Unisex Old Skool Zapatillas clásicas de skate Blank De Blanc Black , Zapatillas de deporte Fenty X High Top Trainer de mujer PUMA Negro Negro Vince Camuto Sandalias Ellanna Slide para mujer Moody Blues La Plume Jen de las mujeres Serpiente de plata Imprimir , Vince Camuto Sanella deslizante de mujer Rubor radiante , La Plume Womens Jessica Bronce , Sam Edelman Women s Olivie 3 Slide Sandalia Negro / Negro / Multi , Sandalia Embellecida Sage de Marc Jacobs para Mujer Marfil 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 7039 - Escala de plata , Bota del oeste del pájaro azul del vintage de las mujeres de Dan Post Sanded Chocolate / Teal Blue Bird Trotters Lucy Ballet para mujer Armada , Glerups Unisex G-01 - Botas de fieltro Naturebrown , Wedge del desierto de las mujeres de TOMS Suede en cuero crudo / Kiltie , Bomba de vestido Kalila Total Motion Kalport para mujer Patente Gray Perla Gris Asolo Womens Greenwood GV ML Grapeade , Zapato Lites Bates para mujer marrón TOMS Clásicos de mujer Lienzo rojo ,

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.

Dansko Women s Louise B075V64FQW Oxford Dansko Vino Nappa Women Pulido d7a2914 - 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 Dansko Women s Louise B075V64FQW Oxford Dansko Vino Nappa Women Pulido d7a2914 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.

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

Geometry Construction Reference

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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.