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 cuero o textil
Importado
Espuma único
Completamente superior plegable y talón para fácil embalaje
Airy malla superior para la comodidad del clima cálido y estilo
En forma de panal almohadillas de espuma de la malla superior para mayor comodidad
Longitud completa entresuela es durable suficiente para servir como suela
Caucho suela con inspiración de Waffle diseño para la tracción y durabilidad
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

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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.


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

Converse Chuck Taylor All Star Seasonal Color Hola Blanco / Vapor Rosa / Blanco Converse Chuck Taylor All Star Seasonal Color Hola Carbón Zapatillas de baloncesto Jordan Kids Jumpman Team II Negro / infrarrojo 23-cool gris-infrarrojo 23 , Brooks Womens PureGrit 6 Negro / Ebony / Diva Pink Converse Chuck Taylor All Star Seasonal Color Hola Ratón NIKE W Zoom All Out Low 2 Womens Aj0036-200 Dusty Peach / Mtlc Red Bronze , Zapatillas de running Spira Phoenix para mujer con resortes Royal / Black / White Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Mono negro Converse Chuck Taylor All Star Seasonal Color Hola Azul / blanco / rojo NIKE Mujeres Air Zoom Pegasus 33 Platino / Negro / Gris / Rosa explosión Zapatillas de running NIKE Unisex Flyknit Racer Negro / gris oscuro / blanco , Calzado de running rápido para mujer Reebok, tejido flexible Cool Shadow / Porcelain / Blue Lagoon / White / Blue , Zapatillas de running Adizero Prime Sp de adidas Originals Núcleo Negro, Ftwr Blanco, Naranja , Calzado de running rápido para mujer Reebok, tejido flexible Blanco / gris ceniza / negro / polvo del desierto , Skechers Women s Go Ride 7 Running Shoe Blanco negro Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco negro , Zapato de running ASICS GEL-Kayano 22 para mujer Onyx / Plata / Carbón de leña , Sandalia deportiva NIKE Kawa Slide para hombre Lobo gris / antracita / gris lobo , Zapatillas Nike Women s Flex Trainer 5 Antracita / Blanco / Gris Oscuro / Hipervioleta , Zapatillas de running NIKE Roshe Two para mujer Blanco / Blanco-Puro Platino , Converse Chuck Taylor All Star Seasonal Color Hola Brillo de sol , Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , Zapatillas NIKE Women s Revolution 3 para correr Negro / Metálico Gris Oscuro / Antracita , Zapato de senderismo KEEN para mujer Versago Dahlia roja / coral profundo , Zapatillas de running adidas Alphabounce Cr W, mujer Core Black / Core Black / Core Black Zapatillas de running New Balance Coast V3 para mujer Silver Mink , Zapatillas de running ASICS Gel-Scram 4 para mujer Negro / gris oscuro , Zapatillas de running Spira Phoenix para mujer con resortes Salmón / 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 Ahnu para mujer - Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Malla negra , Sandalia de susurro de mujer Keen Beet Red / Honeysuckle Zapatillas para correr Nike Women s Flex Experience Rn 5 Antracita / Rosa explosiva / Negro adidas Womens Pureboost 2.0 Clima Ash Pearl , Sandalia Dusty Sport mujer muy volátil Negro , Sandalias Under Armour UA Ignite VII SL para mujer Negro / Cerise NIKE Women s Tanjun Sport Fuchsia White Tart , Columbia Montrail Women s Fluidflex X.S.R. Camelia Roja, Melonade , Zapatillas para correr Mizuno Women s Wave Rider 21 Estate Blue / High Risk Red (Peachtree) , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco , Adidas Sport Performance Mujeres Terrex CC Voyager Sleek Sneakers Aero Blue / Chalk Blue / Aero Pink , OLUKAI Alohi - Mujeres Trench Blue / Trench Blue , Zapatillas NIKE Kids Revolution 3 (GS) para correr Juego Royal / Black / Wolf Gray / White , Vivien Flat de Bella Vita para mujeres Cuero de gamuza de cabrito negro Easy Spirit - Zapatillas Faisal2 para mujer gris , NIKE Men s in-Season 7 Cross Trainer Plata blanca , Vans Old Skool Unisex Adults Low-Top Trainers Control primario negro / blanco Etnies Womens Scout XT Azul claro , Zapatillas de golf Lexington Marc Joseph New York para mujer Serpiente Metálica Sandalia Mason para mujer Naturalizer Slide Negro , UGG Mujer W Joni Negro , Noble Outfitters Mujeres en todo alrededor de las botas de flores Ladrar Botas Nikki North para mujer nuevas de Clarks Armada , Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 20 , Bomba de Angel II para mujeres estilo suave Elegancia de la Armada Zapato de seguridad deportivo Soyay RB191 Reebok Work para mujer Negro Trask Womens Lena Gamuza metalizada azul marino , Mephisto Valentina Oxford para mujeres Gray Bucksoft Betunia Loafer de mujer Aerosoles Broncearse ,

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 Zapatillas de running NIKE Juvenate NIKE/ para/ mujer Orquídea/ Orquídea/ Midnight Navy White 0acb822 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 1203233
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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.