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.

Calvin Negro s Klein Oxford Women s Della Oxford Negro 0a34014




90% cuero / 10% elastico
Importado
Suela de goma
Seccion Gore
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


Calvin Negro s Klein Oxford Women s Della Oxford Negro 0a34014 - 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

Calvin Negro s Klein Oxford Women s Della Oxford Negro 0a34014

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.


Calvin Negro s Klein Oxford Women s Della Oxford Negro 0a34014

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

Nurse Mates Womens Fleet Blanco Propet Travelactive Slip-On Oxford para mujeres Heather negro / blanco Zapatillas de deporte lavables Walker Propet para mujer Sr Pewter , Zapato de punta de ala 3989 de Dr. Martens para mujer Negro , Mujer Alegria Belle Mary Jane Flat Desierto Negro , Steve Madden Womens Kraze Negro Zapatillas de moda Native Unisex Apollo Moc. Antorcha roja / concha blanca / caucho natural , Wolverine 1883 de Women s Elsie Oxford Gris pardo Mephisto Women s Lenza Oxford Gray Bucksoft / Patent; , Fulinken Leather Women Oxford Flats Zapatos con cordones Boyfriends Botines Oro Pikolinos Womens Santorini W7G-3517 brandy ED Ellen DeGeneres Mujeres Phoebe Dark Brindle Split Suede / Waxed Cotton CLARKS Mujeres Raisie Bloom Oxford Borgoña nobuck , Rockport Cobb Hill Women s Paulette Flat Armada Zapato casual mujer TOMS Cabrillo Mezcla de chambray azul marino Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue marrón , Finn Comfort Prevention Line Oxford Kaffee , Sperry Top-Sider Mujeres Pier View Core Azul Zapatilla de deporte Worksong de Caterpillar Aria Mephisto Mens Cooper Black Carnaby Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Negro / Combo Bomba de cuña Tali Grand Lac WDG40 para mujer Cole Haan Coral Haze perforado Zapato para caminar con láser Mephisto para mujer Indigo Bucksoft / azul Cole Loan de Conductor de borla de rodeo de mujer de Cole Haan Nuevo Mineral Nubuck Rojo / Marfil Zapato Durashocks de cuero Bates para mujer Negro Aerosoles Mujeres East Village Oxford Cuero Taupe , Keds Womens Champion Pineapple Chambray Azul , Mordenmiss Nuevo verano de primavera y verano de flores hechas a mano zapatos planos de cuero Mocasines Estilo 2 Verde Timberland Womens Lakeville Oxford Marrón medio gamuza ,
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 Savannah de Ryka para mujer Gris / morado , Sandalia Unisex Slide de Telic Blanco , Under Armour Threadborne Slingflex para mujer Marlin Azul / Blanco / Cristal Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Oro negro / metálico Sandalia Drew Angela para mujer Beige-multi , Sandalias planas Eileen Fisher para mujer Sport-Nu Latté Sandalia atlética de Rka Women s Glance Azul / verde azulado , Zapatillas de running NIKE Air Max Zero SI para mujer Negro / Ligero Bone-cool Gris-total , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Gris oscuro , J.Renee Womens Soncino Ante negro , Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Garza / Blanco / Luces púrpuras , SoftWalk Women s Topeka Flat Sabio , Bomba Jude para mujer Adrianna Papell Plata esterlina metalizada Sandalia de las mujeres de Madden Girl Brando Slide-On Tela Taupe Easy Spirit Twist2 Sneaker para mujer Natural , David Tate Wish Sandalia para mujer Negro , NIKE Condición de zoom para mujer 2 Cross Trainer Negro / Blanco - Gunsmoke , Sandalia con tachuelas Tali Mini Bow de Cole Haan para mujer Habanero Gold , Dr. Zen Jordan Comfort Profundidad extra terapéutica para mujer Cuero para calzado / malla con cordones Negro / Plata Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers CHOCOLATE / CASI NEGRO / BLANCO Under Armour Women s Performance Spikeless Blanco (100) / Brillo , Sandalia para mujer Odosi City Slide de Soludos Denim mediano , Dance Class Mujer CS401 Split Sole Jazz / Obstrucción Oxford Blanco , Salón de baile de mujeres Zapatillas Dance Salsa Práctica latina Zapatos 1670cEB Cómodo-Muy Fino 1.5 \ de 5] Cuero marrón UGG Women s Bailey Bow Tall Negro , Sandalia Birkenstock de cuero Florida para mujer Sfb Blue Birko-flor , Botas Trixie Joe Rocket para mujer (negro, talla 9) - Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro Funtasma de Pleaser Women s Lust-3000 Boot Black Str Patent , Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Neon Blush / 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.

Calvin Negro s Klein Oxford Women s Della Oxford Negro 0a34014 - 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 Calvin Negro s Klein Oxford Women s Della Oxford Negro 0a34014 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.