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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Textil
Importado
Sintético suela
Plataforma mide aproximadamente 1 pulgada
lavable cinta de poliéster Compresión superior
entresuela de Eva moldeada
Non-marking suela de goma
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

KEEN Sandalias Newport Eco-W Sandalias para mujer Rose Garden/ Newport mujer Tawny Port 5aa32db

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,

Propet Women s TravelActiv Ss Sandal Azul negro Sandalia Columbia Sunbreeze Vent con dos correas Black Platinum Chaco Women s Zx2 Classic Athletic Sandal Tunnel Tango Wolky Comfort Jewel White Stone Print , Sanda Verra Teva para mujer Cuerda elástica , Sandalia Juniper Mujer Chaco Moho Skechers sandalia ancha de deporte 600-brilliancy para mujer Armada , Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Oro OluKai Ohana Pokeberry / Dark Java Sandalia Jacardia Merrell para mujer Gris pardo Sandalia Terrell Ivy Post Sport Merrell para mujer Negro Sandalias de cuero KEEN Clearwater CNX para mujer Mineral azul / amarillo Sandalias KEEN Womenport Newport Hydro-W Imán / Coral KEEN sandalia de Venecia para mujer Mineral azul / ámbar Sandalia original Teva Women s Original Negro , Teva Terra-Float Active Lace - Mujeres Negro Sandalias planas Eileen Fisher para mujer Sport-Nu Armada , OluKai Ohana Pewter / Black , Sandalia de susurro de mujer Keen Negro / Gárgola , Sandalia de susurro de mujer Keen Poseidon / Blue Danube Sandalia de cuña Cape May de mujer Jambu Medianoche Imprimir , KEEN Sandalias Newport Eco-W para mujer Rose Garden / Tawny Port , Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo , Sandalia Terrell Convertible II Merrell para mujer Armada , Sandalia deportiva Tetra Cloud de Chaco para mujer Jengibre especias , Chaco Women s Z2 Classic Athletic Sandal Florista Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia gris Sandalia Drew Angela para mujer Bone Smooth Leather , Earth Origins Sophie 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 volante Nike Women s Zoom Hyperface 2 Blanco negro Brooks Womens PureGrit 6 Negro / Ebony / Diva Pink NIKE Kids gratis Rn (grande) Aurora Verde / Plata Metálica / Jade Claro Fila Women s Memory Deltaspeed 2 Mesh, hecho por el hombre, Rubber Running Sneakers Negro, Dewberry, Plata Oscura Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Negro / Plata , Zapatillas para correr ASICS Gel-Venture 6 para mujer Pavo real de carbono / negro / rosa , Zapatillas para correr New Balance Rushv3 para mujer Trueno Negro Zapatillas para correr Mizuno Women s Wave Rider 21 High Rise / Gray Stone , Short NIKE Tempo para mujer Negro / Negro / Noble Rojo / Gris lobo , Zapatillas de agua Merrell para mujer All Out Blaze Aero Sport Negro / Aventurina , ASICS Womens Gel-Fortitude 8 Gris medio / blanco / azul de porcelana , Tierra Libra Alpaca Soft Leather , Sandalias Chardon de Hush Puppies Chardon para mujer Negro Zapatillas de deporte Arizona Trotters para mujer Desnudo , Sandalia Drew Cascade para mujer Nubuck marrón Deflector de gravedad Alivio del dolor comprobado Desafío G de las mujeres Mighty Walk - Mejor Zapatos para el talón Dolor, dolor en el pie y fascitis plantar gris , PUMA Mujeres Fierce Varsity Knit Wn Sneaker Puma Black-puma Black Zapato de entrenamiento transformado Vazee 77v1 para mujer New Balance Espectral / Acuario Drew Shoe Women s Tulip Comfort Brown Print , Zapatillas DC Beach Trase TX Skate Azul claro Sandalia Rockdale Lanea Woven Slide para mujer Tejido blanco , Zapatillas de skate DC SE Pure High-Top TX SE de mujer Gris / gris / gris Charles by Charles David Sandalia para mujer Nicki Slide Negro , Trotters para mujer Dea Slingback Pump Negro Negro Lamo Womens s Sheepskin Scuff Slipper & Toe Warmer Bundle Negro , Zapatillas de ciclismo lanzadera X-Alp Pearl iZUMi para mujer Perla negra / ahumada Sandalias de vestir Patti de Sam Edelman para mujer Yute Glitzy Fabric Bota Deia para mujer de TOMS Negro Negro , Botas a la rodilla con cordones Wild Diva Timberly-65 Coñac , ED Ellen DeGeneres Mujeres Antona Azul Medio Denim Tictac

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 KEEN Sandalias Newport Eco-W Sandalias para mujer Rose Garden/ Newport mujer Tawny Port 5aa32db 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.

Sandalias y toboganes deportivos
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.