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.

Calzado Rosa deportivo New Balance Vazee Urge/ v1 negro para mujer Rosa/ negro 976b87f




Hecho por el hombre
Importado
Sintético único
Rápido entresuela de rebote
moldeada forro de calcetines
Botín construcción
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


Calzado Rosa deportivo New Balance Vazee Urge/ v1 negro para mujer Rosa/ negro 976b87f - 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

Calzado Rosa deportivo New Balance Vazee Urge/ v1 negro para mujer Rosa/ negro 976b87f

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.


Calzado Rosa deportivo New Balance Vazee Urge/ v1 negro para mujer Rosa/ negro 976b87f

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

HOKA ONE ONE Zapatillas para mujer Arahi Azul Medieval / Lavanda Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Melón Blanco / Brillante Zapatillas para correr NIKE Women s Air Vapormax Flyknit Gris oscuro / Reflejar órbita azul plateado , Zapatillas de running Nike Kids Air Max Zero Essential GS Olive-violet / Work Blue / Armory Navy , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue , Calzado de entrenamiento para correr NIKE Flyknit Lunar2 para mujer NIKE naranja , Zapato deportivo ASICS Gel-Excite 4 para mujer Gris medio / blanco / verde hielo , Zapato de running ASICS GT-1000 6 para mujer Negro / Blanco / Atlantis Nike Men s Air Presto Essential Binary Blue 400 , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Negro / Fantasma , Short NIKE Tempo para mujer Deporte Rojo / Blanco / Negro / Blanco Zapatillas de running Gel-Moya para mujer de ASICS Azul negro , Nike Hombres Air Zoom Pegasus 33 Equipo Naranja / Blanco-negro-puro Platnum Zapatillas de running NIKE Air Max Flair para hombre Blanco negro Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Gunmetal / Thunder NIKE Men s Roshe Two Running Shoe Industrial Blue White 402 , Zapatillas para correr NIKE Women s Air Zoom Structure 21 Gris oscuro / lobo blanco gris , Saucony Triumph ISO 3 Running Sneaker para mujer Blanco / verde azulado / Citron , Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco , Under Armour Women s HOVR Sonic NC Gris Nublado (101) / Blanco Nike Men s Air Presto Essential Hueso claro / gris oscuro Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / Begonia rosa Amortiguación 460v2 New Balance para mujeres Gris , Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Blanco , Zapatillas para correr Hoka One One Stinson 3 ATR Fuchsia / Fusion Coral para mujer - Zapatillas de running adidas Alphabounce Em W para mujer Misterio Ruby / Bahia Magenta / Gray Five
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 running Fresh Life Arishi V1 para mujer New Balance Plata / rosa , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Azul hielo / azul táctil , Sandalia universal W Sanborn de Teva para mujer Nuez Converse Chuck Taylor All Star Seasonal Color Hola Negro / Garza / Negro , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial azul marino / colegiado azul marino / azul marino nocturno Colchoneta de amortiguación 620v2 New Balance para mujer Silver Mink / Gunmetal Corredor de pista flexible de carga de agilidad Merrell para mujeres Aguanieve Dr. Comfort Annie-X Mujer diabética terapéutica Doble Profundidad Zapato Lycra Velcro Negro Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer White Wolf Gray Hyper Grape 103 , Zapatillas para caminar Travelite para mujer Propet Gris pardo Converse Chuck Taylor All Star Core Ox Entintado / entintado , Vans Auténtico Duke / Black / Gold Foil , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker azul , Zapatos de golf FootJoy para mujer Enjoy Negro / púrpura , Zapatillas de golf adidas W Adipure DC de adidas para mujer Ftwr Blanco / Plata Met./ Plata Met. Sandalia Ryana Marilyn Slide para mujer Frapé , Lauren by Ralph Lauren Sandalia mujer Davan Slide Deep Saddle Tan Super Soft Leather Vince Camuto Sandalias Ellanna Slide para mujer Rojo cereza Diapositivas de Sam Edelman Gertrude para mujeres entrecruzadas Red Kid Suede Leather , Zapato de voleibol Gel-Rocket 8 femenino de ASICS Negro / Negro / Blanco , VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Port Royale Red / Black , Zapatilla Sandalia de la Unión de Mujeres Sol Sana Negro Negro , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1620- Tan Satin Nueve West Women s Lynneah Leather Mule Cuero negro Botas de parche de textura Claremont de Ann Creek para mujer Borgoña / Marfil Bomba Astoria Nine West para mujeres Tela púrpura oscuro , XYD Casual Botines Slouchy Botas de tacón alto con punta redonda de gamuza para las mujeres Saddle Brown Bota a la rodilla Pleaser Seduce-2000 para mujer Patente negra Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro Zapatos De Bowling Dexter Kerrie Sarga gris

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.

Calzado Rosa deportivo New Balance Vazee Urge/ v1 negro para mujer Rosa/ negro 976b87f - 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 Calzado Rosa deportivo New Balance Vazee Urge/ v1 negro para mujer Rosa/ negro 976b87f 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 142366
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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.