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 de running Reebok Women s running Runner Mt Negro 19595 Calavera gris/ Asteroide Polvo/ Niebla/ Blanco/ Negro/ Peltre dd2f92c




Sintético
Importado
Caucho suela
Tejido abierto malla superior
Corte bajo diseño y ancho 4e dimensionamiento
Cmeva entresuela
Antepié ranuras flexibles
Alta abrasión caucho y carbono 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


Zapatillas de running Reebok Women s running Runner Mt Negro 19595 Calavera gris/ Asteroide Polvo/ Niebla/ Blanco/ Negro/ Peltre dd2f92c - 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 de running Reebok Women s running Runner Mt Negro 19595 Calavera gris/ Asteroide Polvo/ Niebla/ Blanco/ Negro/ Peltre dd2f92c

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 de running Reebok Women s running Runner Mt Negro 19595 Calavera gris/ Asteroide Polvo/ Niebla/ Blanco/ Negro/ Peltre dd2f92c

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 Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle WMNS Zoom Fly Vasto gris , Zapatillas de running NIKE Huarache Run SE para mujer Azul Zapatillas de running A7100w Apex para mujer Armada Zapatillas de running ASICS Gel-Nimbus 19 para mujer Flash Coral / Dark Purple / White , Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Azul Indigo / Plata / Rosa Brillo , NIKE Women s WMNS Air Max 270, Blanco / Blanco-Blanco Blanco / Blanco-blanco , Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino , Calzado de running NIKE RN 2017 2017 para mujer Solar Red / Black / University Red Calzado deportivo New Balance para mujer Vazee FuelCell V1 Saúco / Plata Zapatillas para correr New Balance Rushv3 para mujer Trueno Negro , Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Negro Zapatillas de running Wstro para mujer New Balance Negro / gris / Lime Glow Calzado de running Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Blanco Zapatillas de running Skechers Go Run Ultra Road 2 para mujer Carbón Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Microchip / Blanco / Carbón metálico Zapatillas de running Saucony para mujer Hurricane Iso 4 Gris / Aqua , Brooks Womens Asteria Knockout Pink / Clemantis / Black Rieker womens Slipper adria / jeans Adria / Jeans NIKE Men s Flex 2017 RN Negro Gris oscuro Gris oscuro NIKE Women s Revolution 4 Running Shoe Negro / Pulso carmesí / Antracita 857661-012 MUJERES W AIR MAX ZERO NIKE NEGRO OSCURO GRIS BLANCO Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Blanco-Gris Oscuro-Gris Fresco Supra Martine Run Skate Shoe para hombre Caqui Vintage / Caqui Vintage , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Trabajo azul / gris Supra Martine Run Skate Shoe para hombre Gris blanco , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul Mediano / Negro-aluminio-electro Verde
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 Nike Hombres Air Zoom Pegasus 33 Estrella Azul / Blanco / Azul Costero Zapatillas náuticas para mujer Saucony Originals Broncearse , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Rojo de alto riesgo , Zapatos OLUKAI Pehuea - Mujer Vintage Indigo / Palm Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño CLARKS Sandalia Saylie Moon para mujer Cuero azul marino , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina , Zapato para correr Saucony Women s Cohesion 9 Azul marino / Coral / Rosa , Zapatillas para caminar RYKA Women s Devotion Plus Blanco / plata cromo / almendra helada , Sandalias Abbott Gladiator de Cobb Hill para mujer Caqui claro , Zapato deportivo ASICS Gel-Kayano 24 para mujer Concha Rosa / Blanco / Begonia Rosa , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Black / Cosmo Pink / Winter Bloom , Zapatillas para caminar Brooks Women s Addiction Walker Blanco Sandalias de mujer de Britton Leather Open Toe con punta abierta marrón Skechers Sport Skech Air 2.0 City Love Fashion Sneaker Negro / rosa Zapatilla de deporte de alta calidad para mujer ECCO 8 Sombra oscura , Zapato de skate Diadora B.Elite Suede Beige Zapatillas para caminar WW1400v1 New Balance para mujer Marron oscuro APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Gunmetal / White / Gum , Zapato de skate casual Heathrow DC para hombre Gris , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Negro Birkenstock Women s Lola Sandal Cuero Cognac , Jig Foo Latin Salsa Rumba Chacha Practice Ballroom Zapatos de baile para mujeres Negro Zapatillas Ixia de Easy Spirit para mujer Marfil Vionic Womens Kari Tela escocesa gris , Tangas Birkenstock Negro mate , Rocket Dog para mujer Jetson Grey Summer Jersey Dr. Comfort Vigor Mujer diabético terapéutico Extra profundidad senderismo botas de cuero de encaje Negro , Drew Shoe Women s Paradise II Velco Slip-on Negro , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Ceniza ,

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 de running Reebok Women s running Runner Mt Negro 19595 Calavera gris/ Asteroide Polvo/ Niebla/ Blanco/ Negro/ Peltre dd2f92c - 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 de running Reebok Women s running Runner Mt Negro 19595 Calavera gris/ Asteroide Polvo/ Niebla/ Blanco/ Negro/ Peltre dd2f92c 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 1210403
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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.