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.

Sandalia Enroelia Slide de Blanco ALDO B0051WYTPO para mujer de Blanco 4b0fc16




Cuero
Importado
Suela sintética
Plantilla sintética
Sin forro
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


Sandalia Enroelia Slide de Blanco ALDO B0051WYTPO para mujer de Blanco 4b0fc16 - 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

Sandalia Enroelia Slide de Blanco ALDO B0051WYTPO para mujer de Blanco 4b0fc16

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.


Sandalia Enroelia Slide de Blanco ALDO B0051WYTPO para mujer de Blanco 4b0fc16

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 Sonia Slide de Dolce Vita para mujer Elástico marino , Sandalia plana de lujo de satén Ivarene de Nine West para mujer Luz natural , J Slides Sandalia de mujer Naomi Slide rojo Sandalia de mujer Hana Slide de Via Spiga Cuero de arena Sam Edelman Women s Slide Sandalia Denim Blue Sandalia Dolce Vita Delana Slide para mujer Lt Gold Leather , Sandalia con tacón Florence Block de Via Spiga para mujer Patente de arena , Sandalia Shanny Slide de Jessica Simpson para mujer Blanco BareTraps Women s Cherilyn Slide Sandal Blanco Sandalia Donald J Pliner Gess Slide para mujer Mango , BareTraps Women s Cherilyn Slide Sandal champán Sandalia Tilden Slide de Franco Sarto para mujer Negro J Slides Sandalia de mujer Naomi Slide Gris pardo , Aveline Slide de Ted Baker para mujeres Blue Harmony , Sandalias de mujer Skechers Pop-up Summer Rush-Rhinestone Shower Slide Sandal Negro Negro , Sandalia Dolce Vita con diadema para mujer Tejido negro , Skechers para mujeres Reggae Fest-Marlin-Fisherman Open Back Mule Ajuste relajado y espuma de memoria a / C Zapato de agua Negro Sandalia Pour La Victoire Mujer Mallory Slide Tropic Trip , BareTraps Women s Cherilyn Slide Sandal Blanco , Sam Edelman - Sandalia de mujer Liliana Slide Denim Blue , Sandalia Maria Slide de Nanette Lepore para mujer Negro , Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Platino , Sandalia de cuero Nirveli de Nine West para mujer Cuero blanco , Sandalias Donald J Pliner para mujer Rose Multi Skechers Rumblers-Flashies Sandalia deslizante para mujer Gris pardo , Kea de las mujeres de Flexus Blanco AYA de VINTAGE HAVANA WOMENS PLATFORM ESPADRILLE DIAPOSITIVA Floral azul , Charles by Charles David Sandalia mujer Kadia Slide Negro , Sandalia para mujer Kate Spade New York Coby para mujer Blanco
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 deportiva Tirra para mujer Teva Atlántico Norte Sandalia Terrell Ivy Post Sport Merrell para mujer Sodalita Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Fila Navy / Wedgewood / Deep Pink Brooks Womens PureFlow 7 Blanco negro Wolky Comfort Jewel 70500 canales rojos , Wolky Comfort Jewel Beige , Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón Under Armour Threadborne Slingflex para mujer Apollo Gray (962) / Penta Pink Zapatillas de running Kinvara 8 Saucony para mujer Rojo blanco Chancleta Yoga Sling 2 Sanuk para mujer Nudo natural de la línea , KEEN Kids Seacamp Ii CNX Water Shoe Very Berry / Lilac Chiffon Chancleta Yoga Sling 2 Sanuk para mujer Rojo brillante , Zapatillas de running adidas Ultraboost W para mujer Blanco / Blanco / Blanco Kenneth Cole New York Mujeres Sumner 2 Multi-color con cordones Jogger Sneaker Blanco , Walking Cradles Womens Henna Taupe Nubu estampado claro , Earth Origins Tova Azul cobalto Calzado para caminar CLARKS Un Voltra para mujer Cuero negro Trotters Mujer Simmy Mary Jane Flat Cuero caído Veg negro , Vans Auténtico Bay / True White , Vans Unisex Old Skool Zapatillas clásicas de skate (Lienzo) Negro / Blanco verdadero , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Gris oscuro Botas de paddock con cordones de la Fundación Dublin Childs Negro VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Negro Negro Trotters para mujer Dea Slingback Pump Negro Negro Trotters para mujer Dea Slingback Pump Patente perlada multi acolchada / negra Botas de cuero Neevah de UGG Australia para mujer Cuero negro , Sandalia Birkenstock de cuero Florida para mujer Habana Waxy Leather , Botas cortas de cuero Mordenmiss para mujer Zapatos nuevos Estilo 1-marrón Botas de seguridad Reebok RB874 Stealth SR para mujer - Negro Negro , Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer borgoña ,

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.

Sandalia Enroelia Slide de Blanco ALDO B0051WYTPO para mujer de Blanco 4b0fc16 - 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 Sandalia Enroelia Slide de Blanco ALDO B0051WYTPO para mujer de Blanco 4b0fc16 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

Thirteen elementary straightedge 573297
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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.