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.

BareTraps Women s Rebecca BareTraps Slide Slide Sandal s Negro Negro 291b5f8




Hecho por el hombre
Importado
Sintético suela
Plataforma mide aproximadamente 0.75 pulgadas pulgadas pulgadas
Flexible Suela
Punta Abierta
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


BareTraps Women s Rebecca BareTraps Slide Slide Sandal s Negro Negro 291b5f8 - 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

BareTraps Women s Rebecca BareTraps Slide Slide Sandal s Negro Negro 291b5f8

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.


BareTraps Women s Rebecca BareTraps Slide Slide Sandal s Negro Negro 291b5f8

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

Sandalia tobogán elástica bebe para mujer Negro , ALMA NATURAL Sandalia Adalia Slide para mujer Puesta de sol Sam Edelman Women s Weslee Slide Sandal Neutro Multi Sandalia de mujer Veronica Slide de Nanette Lepore Rosa tropical Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Cuero negro de acción Sandalia Dolce Vita Hazle Slide para mujer Cuero natural Sandalia tobogán X-Band con plataforma Mikenzie de Gentle Souls para mujer Dark Natural Sandalia de mujer Frisco Slide de Franco Sarto Oro caliente Sandalia de cuero Merrell Dwaykair Seaway para mujer Merrell Oak , Marc Jacobs Women s Aurora Mule Plata , Romika Fidschi 22 Sandalia de vestir para mujer Black Kombi , Sandalia Sloan Slide de Naturalizer para mujer Gunmetal Birkenstock Mujer, Madrid Sandalia deslizante Caramelo Sandalia de tobogán plano de piscina Iona de Gentle Souls Negro , Sandalia Sloan Slide de Naturalizer para mujer Negro , CLARKS Leisa Cacti Slide Sandalia para mujer Cuero negro Sandalia Juno Slide de Vionic Women s Rest Oscuridad Vince Camuto Sandalias Ellanna Slide para mujer Urban Lux Sandalias con tachuelas Gale para mujer Donald J Pliner Adular , Sandalia de mujer Lidia Mephisto Slide Platino , Sandalia de viaje Via Spiga Women s Porter Pavo real gamuza , ALMA NATURAL Sandalia Kiki Slide para mujer Pulir Sandalia de mujer de SCHUNA Maduna Slide Pez vela Birkenstock Arizona Sandalia de ante con suela suave Cuero marrón de Amalfi , Plataforma de campo Leisa para mujeres de CLARKS Cuero Multi Metálico , Sandalia Mason para mujer Naturalizer Slide Cuero metalizado lila Lauren by Ralph Lauren Sandalia mujer Davan Slide Rosa empolvado , Prop & eacute; t Womens Aurora Slide Negro , CLARKS Mujer Flip Flop Arla Glison Tela estampada en 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 Sandalia Yucatan ECCO Mujer Polvo blanco , Zapato de running ASICS Gel-Quantum 360 cm para mujer Flash Coral / Negro / Plata , Earth Origins Sophie Azul marino , NIKE Women s Tanjun Partícula rosa / blanco , Calzado casual Slip-On de Bernie Mev para mujer Black Multi , Sandalia Drew Angela para mujer Beige-multi Zapatillas de running New Balance para mujer 590v3 Reflection / Vintage Indio Guantes Trail Trail Merrell Women s Glove 4 Negro , Zapato de correr ideal para rejilla de mujer Saucony Gris blanco SAS Womens Sunny Teal , Sandalias de peonía Easy Spirit para mujer Antik Silver / Antik Silver / Antik Silver , Propet Women s Fisherman Sandal Mezclilla Under Armour Men s Speedform Intake Rhino gris / azul infinito / blanco , Zapato de running GEL-Excite 3 para mujer ASICS Silver / Hot Pink / Lime Punch JSport by Jambu Women s Wendy Flat Negro , Plano original de la mujer del Dr. Scholl s Collection Collection Kimberned Pointed Toe Cuero Greige Maribel Easy Spirit Women s Traveltime Mule Azul , Naturalizer Samantha Pointed Toe Flat para mujer Caramelo Zapatillas adidas Originals Mujer Superstar 80S 3D MT W Ftwwht / Ftwwht / Owhite , Supra Vaider LC Sneaker Andorra blanco Clásicos de base Slip-on (Tm) para hombres de Vans (Vintage Aloha) blk / Tr Wht Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Tela natural / negra , ~ 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 \ 2702 Copper Sparkle & Ultra Copper , Ariat Womens Devon Pro VX Paddock Chocolate encerado Calzas de la dinastía Púrpura / blanco CLARKS Enfield Senya Tobillera para mujeres Gamuza marrón oscuro , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Blue Denim Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Leopardo negro , Botas de agua para mujer Chooka Whidbey Negro , ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Negro caucho / Marrón G Nubuck

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.

BareTraps Women s Rebecca BareTraps Slide Slide Sandal s Negro Negro 291b5f8 - 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 BareTraps Women s Rebecca BareTraps Slide Slide Sandal s Negro Negro 291b5f8 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.

Diapositivas
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.