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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Hecho por el hombre
Importado
Suela de goma
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


Bomba 2398 Sadie Wedge 40MM de Patente Cole negra Haan para mujer Patente negra e11a783 - 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

Bomba 2398 Sadie Wedge 40MM de Patente Cole negra Haan para mujer Patente negra e11a783

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.


Bomba 2398 Sadie Wedge 40MM de Patente Cole negra Haan para mujer Patente negra e11a783

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

Zapatillas Iris Easy Spirit para mujer marrón Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Amarillo negro , Bomba de vestido Daisie de mujer Steve Madden Cuero blanco , Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Patente fucsia francesa / Nude Trim / Gold Studs , Vionic Bliss - Sandalias con sandalias ortopédicas para mujer Tan Leopard , Vionic Womens Kari Plaid rosa , Clarks Artisan Caswell Génova Natural Bomba de mujer David Tate Dainty Armada Pantuflas The Joy de Katy Perry para mujer Negro , Bomba de vestido Reva de Naturalizer para mujeres Negro Mephisto Helen Tropic Cuero rosa con estampado de flores - TOMS Women Classic Slip on Negro / Ciruela Pantuflas Wilma de Katy Perry para mujer Carbón , CLARKS Keesha Raine Dress Pump para mujer Charol de cocodrilo negro Bomba de vestido Daisie de mujer Steve Madden Peltre negro , TOMS Clásicos de mujer Gris claro , Nuevas zapatillas de punta redonda de mujer Mordenmiss Estilo 4 Café Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Beige , Trotters Lucy Ballet para mujer Desnudo Bomba de vestido de pasión Easy Street para mujer Patente de arruga de color topo , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Denim azul claro Propet Cush N Foot Slipper & Oxy Cleaner Bundle Corduroy negro Bomba Astoria Nine West para mujeres Ante natural , Sandalia de vestir Carrson de mujer Steve Madden Azul marino Orthofeet más Cómodo Plantar Fascitis plana Pies Diabetic Arch Soporte ortopédico Zapatillas de cuero para Mujeres - Charlotte , Trotters para mujer Dea Slingback Pump Negro Acolchado , Vionic Womens Gemma Luxe Slipper Tan Leopard , Trotters para mujer Dea Slingback Pump Negro Negro Zapatillas de piel de oveja australiana de piel genuina de LAMB para mujer Rosado
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 de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Sandalia KEEN Women s Uneek Slide-w Dapple Gray / Vapor Zapatillas NIKE Women s Revolution 3 para correr Wolf Gray / Lava Glow / Blanco / Pure Platinum adidas Women s Edge Lux W, NEGRO / GRIS Blanco-negro-gris , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Fiyi Calvin Klein Mujer Sandalia Domenica Agrietado Metálico / Plata , Zapatillas de running Kinvara 5 Saucony para mujer Blanco / Crepúsculo / Puesta de sol , Atletismo femenino de Apex Negro , Sandalia Gabby de mujer Cole Haan Ante negro , Sandalia de vestir Rubie de mujer chino de lavandería Oro metálico Sandalia Harman Bandolino para mujer Oro suave Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Negro , Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover , Sandalia plana Nanci de Naturalizer para mujer Mezclilla , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Blanco Zapatillas para caminar Skechers Performance para mujer Go Flex Train Heather gris , PUMA Womens Fenty Trainer Cantera New Balance 711v3 Cush + Cross Trainer para mujeres Blanco Easy Spirit - Zapatillas Faisal2 para mujer Negro , Zapatillas Skechers Women You-14966 Blanco Zapatillas de tenis ASICS Gel-Solution Speed ​​3 para mujer Negro / Plata / Rosa Diva New Balance Wx577v4 Cross Trainer para mujeres Gris / Heather , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Blanco óptico Zapatillas Adidas W Tech Response Ftwwht / Ft Golf Shoe, para mujer Claro / Gris Sandalia Rockdale Lanea Woven Slide para mujer Latte tejido Salón de baile de mujeres Zapatillas Dance Salsa Práctica latina Zapatos 1670cEB Cómodo-Muy Fino 1.5 \ de 5] Satén marrón Ariat Womens Heritage Contour Field Zip Tall Riding Negro , CLARKS Keesha Raine Dress Pump para mujer Nude Croc Patent Leather , Mocasines con suela de goma con cordones de cuero Twisted X para mujer - Bombardero / Rosa Broncearse Zapato de uniforme de cuero Bates para mujer 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.

Bomba 2398 Sadie Wedge 40MM de Patente Cole negra Haan para mujer Patente negra e11a783 - 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 Bomba 2398 Sadie Wedge 40MM de Patente Cole negra Haan para mujer Patente negra e11a783 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

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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.