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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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


Zapatillas Plata de running NIKE para mujer Zapatillas Lunarepic/ Low Flyknit 2 Plata negra/ metálica 2a5f767 - 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

Zapatillas Plata de running NIKE para mujer Zapatillas Lunarepic/ Low Flyknit 2 Plata negra/ metálica 2a5f767

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.


Zapatillas Plata de running NIKE para mujer Zapatillas Lunarepic/ Low Flyknit 2 Plata negra/ metálica 2a5f767

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 Reebok Hexaffect 5.0 MTM para mujer Ash Gray / Black / Sour Melon / Pewter / White / Alloy Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Gris oscuro / Azul Racer / Negro Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Negro / Aluminio , Zapatillas para correr Nike Women s Free Rn Distance 2 Hyper Violet / White Dark Iris , Zapatillas de montaña de montaña Akyra La Sportiva para mujer Slate / Jadegreen Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Ash Gray / Black / Sour Melon / Pewter / White / Alloy Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral , Zapatillas de running NIKE Air Max Thea para mujer Gunsmoke / Particle Rose-black Under Armour Women s Drift 2 Sigilo gris (008) / rosa islandesa Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea , Zapatillas de running Mizuno para mujer Synchro MX Rosa / negro Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris claro Nike Men s Air Presto Essential Velvet Brown / Light Bone Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro Zapatillas de running NIKE Air Max Thea para mujer Blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 TRABAJA AZUL / OSCURO OSCURO-CLORO AZUL , ASICS La sacudida de las mujeres Glacier Grey / Aqua Splash / Blanco , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Blue Print / Blue Print NIKE Women s Tanjun Iced Jade Aurora verde blanco Calzado de entrenamiento para correr NIKE Flyknit Lunar2 para mujer NIKE Blanco / Blanco-lsr Orng-tm Orng , Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Blanco Aurora Verde 104 Zapatillas de trail running Inov-8 para mujer Road Claw 275 Plata / azul / rosa , Camiseta adidas Swift Run W para mujer Gris / Plateado Metálico / Blanco , Zapatillas NIKE Kids Revolution 3 (GS) para correr Gamma Blue / Pink Blast / Black / White , Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello Zapatillas de running Adidas Women s Energy Cloud WTC W Gris oscuro Zapatillas de running definitivas Ryka para mujer Gris 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 Zapatillas de running A7100w Apex para mujer Armada Zapatillas de running New Balance Coast V3 para mujer Blanco , Sandalia deportiva Marc Jacobs Women s Somewhere Crema Multi , Chaco Women s ZX2 Classic Athletic Sandal Varsity Pine Teva Terra Sandal para hombre Madang Blue , Converse Chuck Taylor All Star Seasonal Color Hola Beach Glass , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Acero , Nike Hombres Air Zoom Pegasus 33 Blanco , Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel , El corredor de trail impermeable Xa Pro 3D Cs para mujer de Salomon Sombra Salomon Women s X-Mission 3 W-w Atlántico Norte Sandalia Savannah de Ryka para mujer Gris blanco Zapato para caminar WW1865v1 New Balance para mujer Estratos , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Verde , Zapatillas para caminar Skechers Performance para mujer Go Step Lite Ultrasock Gris pardo , Zapatillas para caminar Reebok Skyscape Revolution para mujer Negro / Blanco / Ultima Púrpura / Verde cargado New Balance 711v3 Heather Cross Trainer para mujeres Pigmento / cobalto vivo , Sandalia Vanessa de mujer Alegria Chrysalis Black , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Negro / ondulado / estiramiento Phenma femenino de PUMA EP Puma Blanco / Beige Metálico Supra Vaider LC Sneaker Carbón blanco , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco / negro , Golf de mujeres y rsquo; s Peso ligero genuino Cuero hecho en Brasil Wall Street Rendimiento de golf Marc Joseph NY Zapatos de moda Tan Napa , Sandalia para mujer bien descansada Seychelles Negro , Sandalia Maria Slide de Nanette Lepore para mujer Rosa empolvado , Sam Edelman Sandalia Bernice Slide para mujer Saddle Vaquero Saddle Leather Bota Ecuestre Xandy LifeStride para mujer Negro , Skechers - Bikers holgados para mujeres, Traffic Loafer Autopista de carbón , Bota de combate FRYE Julie Hook para mujer Forest Soft Oiled Suede , Sandalia Birkenstock de cuero Florida para mujer Negro Birko-flor

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.

Zapatillas Plata de running NIKE para mujer Zapatillas Lunarepic/ Low Flyknit 2 Plata negra/ metálica 2a5f767 - 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 Zapatillas Plata de running NIKE para mujer Zapatillas Lunarepic/ Low Flyknit 2 Plata negra/ metálica 2a5f767 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 287904
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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.