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 sintético
Goma suela
Retro inspirada zapatilla deportiva Formstrips de la firma a los lados y lengüeta exagerada suela de tracción
T-toe construcción
Acolchado entresuela con arco soporte
Cordones vamp
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


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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

PUMA Sneaker Men s/ Roma Basic Sneaker Blanco/ Peacoat s/ avena 0d90fd1

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.


PUMA Sneaker Men s/ Roma Basic Sneaker Blanco/ Peacoat s/ avena 0d90fd1

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 de running NIKE Lunar Skyelux para mujer Golpe caliente / Marrón nocturno PUMA Men s Roma Basic Sneaker Blanco / gris claro , NIKE Men s Roshe Two Running Shoe Midnight Navy / Black-sail-volt NIKE Mens N98 Tribute Track Jacket Blanco negro , Zapatillas NIKE para mujer Air Zoom Mariah FK Racer PRM Running , Short NIKE Tempo para mujer Dinastía púrpura / Turf Naranja / Gris lobo Zapatillas de running adidas Women s Edge Rc W Gris / Granito / Coral Fácil , Zapatillas de running Saucony PowerGrid para mujeres Gris / Coral , Zapatillas de running ASICS Gel-Contend 4 para mujer Gris Medio / Mar Glaciar / Plata Skechers Performance Women s Go Run 400 Obstruct Gris / azul Zapatillas de golf adidas W CC Ballerina II para mujer Clear Aqua / Running blanco / plata mate , Zapatillas para correr ASICS Gel-DS Trainer 22 para mujer Acuario / Aqua Splash / Flash Coral Short NIKE Tempo para mujer Blk / Blk / Vivid Sky / Wolf Gr Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird PUMA Men s Roma Basic Sneaker Negro / Blanco / Plata , Under Armour Women s Drift 2 Sigilo gris (008) / rosa islandesa Brooks Womens Hyperion Negro / Dewberry / Aruba Blue , Zapatillas de running adidas Women s Cf Racer Tr W, mujer Gray Two / Gray Two / Black , Brooks Womens Ghost 9 Silver / Blue Atoll / Lime Punch Zapato deportivo para acolchado 635v2 New Balance para mujer Gris , Zapatillas de running TOPO FLI-Lyte 2 para mujer Amarillo negro Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Brght Crmsn / Prm Rd-atmc Pnk-wh , Brooks Women s Adrenaline GTS 17 Carbón metálico / negro / oro rosa , Short NIKE Tempo para mujer Armada , Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal , Zapatillas de running Kinvara 8 Saucony para mujer Azul marino Púrpura Zapatillas de running NIKE RN 2017 para mujer gratis Binario azul / oscuro cielo azul-obsidiana ,
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 Drew Element para mujer Denim Blue Nubuck , Sandalias KEEN Clearwater CNX-W para mujer Higo de verano / Crabapple , Sandalia Belmar Athletic de Ryka para mujer Gris oscuro , Zapatillas de running adidas Duramo 8 para mujer Turquesa Zapatillas de running AMPLICA para mujer ASICS Negro / Negro / Blanco , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Carbon / Black / Hot Orange , Zapatillas de trail running W620v2 para mujer New Balance Negro / Plata Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker W para mujer Gray Five / Black / Utility Black Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco-negro-corredor Azul Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Vino tinto / blanco Sandalias planas de mujer Grasshoppers Estaño Zapatillas de viaje Proather para mujer II rojo , Sandalia Rosa mujer TOMS Cuero miel Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Wolf Grey / Ghost Green / Chlorine Blue , Zapatillas Mujer PUMA Fierce Satin En Pointe Wn Puma Negro / Plata Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / blanco , Lakai Limited Calzado para hombre Daly Lienzo blanco , Clásicos de base Slip-on (Tm) para hombres de Vans (Denim C & l) Negro / Goma , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Apenas Verde / Negro , Sandalia de viaje Via Spiga Women s Porter Patente naranja caliente Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1670- Black Pu , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Mustard Slub Chambray , Zapatillas Adidas Performance Adizero Track de Adidas Blanco / Infrarrojo / Metálico / Plata , Botas de suela de cuero antideslizante de 6 \ Ls60 negro Zapatillas NIKE Roshe Two Hi Flyknit para mujer 861708 Sneakers Boots Negro Negro Gris oscuro 001 Bota Darcy para mujer Easy Street Negro / Gore Rieker womens Slipper gris Gris PIC / PAY Jonas - Cordones de las mujeres Oxford - Clásico Ropa de hombre de cuero plana Zapatos estilo holgazán (Nueva caída) Gamuza burdeos , Muck Boot Tremont Wellie Tall Snow para mujeres Prados negros CLARKS Women s Candra Blush Flat Berenjena ,

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.

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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 PUMA Sneaker Men s/ Roma Basic Sneaker Blanco/ Peacoat s/ avena 0d90fd1 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3471180
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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.