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.

Zapatillas s de running Saucony Women s Mirage Mirage 19994 4 Verde negro fcc61b1




Sintético y Nylon
Importado
Goma suela
4 mm de talón a punta soltar
FlexFilm construcción
Longitud completa Amortiguación PowerGrid
HydraMAX Revestimiento
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 s de running Saucony Women s Mirage Mirage 19994 4 Verde negro fcc61b1 - 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 s de running Saucony Women s Mirage Mirage 19994 4 Verde negro fcc61b1

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 s de running Saucony Women s Mirage Mirage 19994 4 Verde negro fcc61b1

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

Calzado de running casual para mujer New Balance WL555, para mujeres Menta de plata , Zapatillas de running NIKE Lunar Skyelux para mujer Aluminio / Binario Azul / Azul Mediano , Zapatillas de running Wstro para mujer New Balance Visón plateado Lanzamiento de Brooks Womens 4 Antracita / cerámica / sacador de lima Zapatillas de running ASICS Gel-Kayano 25 para mujer, blanco / blanco, 6.5 B (M) US - Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow Zapatillas para correr NIKE Kaishi 2.0 para mujer Wolf Gray / Pink Black / White Skechers Sport Skech Air Run High Fashion Sneaker Adornos de malla negra / rosa fuerte , Amortiguación 635v2 New Balance para mujer Gris , Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink Black Bright Mango 600 , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , NIKE Men s Roshe Two Running Shoe University Red / White / Track Red , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel Zapato de running ASICS Alpine XT para mujer Gris Pizarra / Gris Pizarra Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver Skechers Performance Women s Go Run 400 Obstruct Azul marino / Aqua Zapatillas de running adidas Ultraboost W para mujer Misterio Rojo / Misterio Rojo / Rosa Táctil Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento NIKE Women s Tanjun Iced Jade Aurora verde blanco Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo , NIKE Men s Benassi Just Do It Athletic Sandal Midnight Navy / Windchill Nike Men s Air Presto Essential Universidad / Rojo / Negro / Negro , Calzado de carretera Inov-8 Road-X-Treme 250 Gris / Negro / Rojo , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Binario Azul / Blanco-azul Luna , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Negro , NIKE Women s Revolution 4 Running Shoe Gunsmoke / Ocean Bliss / Dark Grey
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 ASICS La sacudida de las mujeres Ónix negro Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , Zapatillas de deporte casual Gazelle de adidas para hombre Arcilla marrón / arcilla marrón-goma , Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Volt / Blue Lagoon / Blanco , Zapatillas de caminar de yoga minimalista para mujer de ala Vivobarefoot Brezo Zapatillas de trail running Salomon Speedcross 4 para hombre Poseidón , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blanco Blanco Zapatillas de running Gore-TEX 910v4 New Balance para mujer Pigmento / Porcelana azul Frances Valentine Womens Lizzie Camel Patent Zapato deportivo ASICS Gel-Excite 4 para mujer Lapis / rosa fuerte / amarillo de seguridad , Zapato de running ASICS GT-1000 6 para mujer Negro / Negro / Plata , Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Desierto de color topo , Skechers Women s Go Step Lite-Indulge Loafer Flat Negro , Easy Spirit Women s Traveltime Mule Chispa de plata , Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Blanco / Tricolor Entrenador campeón Gusto Cross Trainer Luz Coral Mesh , Supra mujer Skytop Sneaker Black-fade / Black Vans Auténtico Chambray Coral / True White Zapato de skate Diadora B.Elite Suede Tuareg azul marino , Zapatillas de golf adidas W Adistar Lite Boa Ftwwht de adidas para mujer Blanco / Plateado Metálico / Choque Rosa Alpargata FRYE Femenina Ally Deco Stud Criss Cross Sandalia Slide Dusty Rose , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla Zapatillas de ante de las mujeres Laurentian Chief con mocasines con cuello de piel de conejo Zapatillas Shimano SHRP9 Flagship Road Performance Shoe para hombre Negro Zapatillas de running New Spike 900 v3 para pista New Balance para mujer Rosado morado Bota impermeable Timberland Womens Velvet-Accent Premium borgoña Zapatillas de deporte y construcción Caterpillar Superstat Waterproof Nano Toe / Tater para mujer Tater , Jeffrey Campbell Womens Leona Boot Taupe Suede , Aquatalia de Marvin K. Botín de ante perforado mujer Shari para mujer Negro , Tommy Hilfiger Women s Jouston Oxford borgoña

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 s de running Saucony Women s Mirage Mirage 19994 4 Verde negro fcc61b1 - 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 s de running Saucony Women s Mirage Mirage 19994 4 Verde negro fcc61b1 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 1604339
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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.


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