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
Importado
Goma único
Eje mide aproximadamente Tapa baja del arco
Tacón mide aproximadamente 1 1/2
Plataforma mide aproximadamente 3/4
Medicare / HCPCS código A5500 diabético zapato
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,

Sandalia deportiva Tirra para mujer Teva Mar de Bering Sandalia de mujer Drew Alana Pewter-bronce Chancleta Eezay adidas para mujer Vapor Gris Blanco / Amarillo Fácil S , Sandalia Drew Angela para mujer Beige-multi , Skechers Performance On The Go 400 Vivacity Flip Flop Negro Sandalia de susurro de mujer Keen Midnight Navy / French Blue , Drew Womens Cascade Denim Blue Nubuck , Chaco Women s ZX2 Classic Athletic Sandal Bloom Peach OluKai Kulapa Kai Negro plata Earth Origins Sophie Negro , Sandalias de mujer con tirantes Peep Toe de Bordello by Pleaser para mujer Fucsia , Earth Origins Sophie Sedona Brown Sandalia Shannon del Dr. Comfort Women Camello , Sandalia Uneek O2-w de mujer KEEN Blanco / oro de la cosecha Sandalia original premier universal Teva W de mujer Negro , Sandalia deportiva Marc Jacobs Women s Somewhere Off White / Multi NIKE Women s WMNS Benassi Duo Ultra Slide, negro / negro Negro Negro , Teva Terra-Float Active Lace - Mujeres Paloma Salvaje , Chaco Women s ZX2 Classic Athletic Sandal Motif Eclipse , Earth Origins Sophie Women s Sandal Escarlata , Chaco Mujer ZX1 Classic Sport Sandalia Aguacate estático Chaco Mujer Zcloud X2 Sandalia Deportiva Popline Boa Earth Origins Sophie Frost Grey , Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad , Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Resquicio de esperanza Wolky Rio Sandal Red Patent Metal Sandalia de deporte Merrell, para mujer Terran Ari Backstrap Sodalita , Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Laguna Azul , OluKai Ohana Pewter / Black ,
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 Calzado deportivo New Balance para mujer Vazee Rush V3 Blanco / Vintage Indigo Converse Chuck Taylor All Star Seasonal Color Hola Delfín Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Deep Lake / Lime Punch , Zapatillas Nike Women s Flex Trainer 5 Cl Gray / Sprt Fchs / Pnk Pw / Stlth New Balance Women s w940v3 Running Shoe Azul , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosado , Zapatillas de running Kineta Relay para mujer Saucony Púrpura / rosa , Astral Mary Jay - Mujeres Negro Sandalia ecológica ECCO Women s Offroad Lite Slide Caliza Zapato de running COMUTORA para mujer ASICS Glacier Grey / Glacier Grey / White , Calzado de senderismo Go Joy 15602 de Skechers para mujer Negro / rosa caliente , Propet Women s Fisherman Sandal ciruela , Sandalias Paulie T-Strap de mujer Vionic Broncearse Sandalia de cuña Delphina de Carlos Santana para mujer de Carlos Azul Zapatilla de deporte ECM Biom Fjuel Tie Sneaker para caminar Negro Negro Zapatillas de entrenamiento Reebok Crossfit Lite Lo TR para mujer Chalk / Weathered White / Trek Gray / Purple Shadow Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Coñac , Vans Auténtico Cómics / Blk / Tru , Supra Scissor Skate Shoe Azul marino , Sandalia Sloan Slide de Naturalizer para mujer Gunmetal , Sandalia de cuña Ollie Wedge Chinese Laundry para mujer Rich Brown Smooth Sam Edelman Women s Bay 2 Slide Sandalia Moderno Ivory Vaquero Saddle Leather , Bota de vaquero western Ariat Women s Heritage Roper Marrón angustiado , Calzas de la dinastía Púrpura / blanco Pantuflas Aquazzura Womens Powder Puff Flat Marrón Zapatillas de deporte Reebok Esoterra DMX Lite para mujer Blanco negro 055398210 Timberland PRO - Botas de trabajo TiTAN para mujer - Café marrón , Zapato de uniforme de cuero Bates para mujer Negro , T.U.K. Negro , Zapatillas de deporte de algodón Denim Magic Debs de Rocket Dog para mujer Black Canvas / Glitter Wrap ,

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 viaje Protic de W3910 Vista Walker viaje para mujeres para Taupe liso fa5bfe5 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

Thirteen elementary straightedge 4521812
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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.