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 Berry running Saucony Women Women s/ Freedom ISO Berry/ rosa 7e0186f




Estiramiento de ingeniería malla
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Primero siempre de cuerpo entero Entresuela EVERUN
Engineered malla elástica superior
ISOFIT para un ajuste dinámico
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 Berry running Saucony Women Women s/ Freedom ISO Berry/ rosa 7e0186f - 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 Berry running Saucony Women Women s/ Freedom ISO Berry/ rosa 7e0186f

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 Berry running Saucony Women Women s/ Freedom ISO Berry/ rosa 7e0186f

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

Zapatillas de running NIKE Mens Air Zoom Structure 19 Blanco / negro / verde eléctrico Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Blanco , Zapatillas de running Cortana 4 Saucony para mujer Azul / rosa / negro Skechers Sport Skech Air Run High Fashion Sneaker Ajuste de malla de carbón / turquesa , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Violeta plata / persa negra / metálica , Short NIKE Tempo para mujer Juego Royal / Game Royal / Wolf Gray , New Balance Women s w990v4 Running Shoe Gray / Castlerock Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla oceánica / Blanco-azul Glow-hyper Jade , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Cobre rosa / sal marina / chicle New Balance Women s w990v4 Running Shoe Armada , Zapatillas de running Newton Kismet II para mujer Rosa / Blanco , Zapatillas para correr Altra Women s Torin 3.0 Azul / Coral Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Berenjena / Sal marina , Zapatillas de running Pearl NZUMi W Em Road N 2 V3 para mujer Plata / Aqua Mint , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Black / White-hyper Punch Zapatillas de running Kineta Relay para mujer Saucony Fuschia / cerveza , Zapatillas de running Saucony Women s Breakthru 3 Azul / menta Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Azul / Verde Fantasma , Zapatillas para correr Mizuno Women s Wave Rider 21 High Rise / Gray Stone Zapatillas de running Questr Cc W de mujer adidas Aero Pink S, Aero Pink S, Carbono S , NIKE Women s Wmns Air Max 270, Coral Stardust / Black-Summit White Rosado , Camiseta adidas Swift Run W para mujer Trace rosa / gris tres / cristal blanco Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Negro En la ejecución de calzado de mujer en la nube Denim / Blanco Calzado de running NIKE RN 2017 2017 para mujer Negro / blanco / antracita / gris lobo Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / gris antracita-cool Zapatillas de running NIKE Lunar Skyelux para mujer Celadon-azul , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono ,
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 Sandalia deportiva Big Water para mujer Columbia Titanio Mhw, blanco Sandalia Drew Angela para mujer Dusty Multi Metallic Leather Zapatillas de running adidas Alphabounce Beyond w para mujer Calzado blanco / calzado blanco / Core Black Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Multi Color 999 Zapatillas para correr Saucony Women s Cohesion 10 Gris azul Zapatillas de mujer Salomon X Ultra 3 W para excursionismo Sombra , Zapato de entrenamiento cruzado deportivo Hydro Sport para mujer RYKA Detox Blue / Twinkle Blue / Chrome Silver , Zapatillas de deporte para correr Nike Air Max Ivo Fashion Running Gym para mujer Pure Platinum / Blanco I-RUNNER Maria Puntera terapéutica para mujer con profundidad extra Calzado de cuero / malla con cordones Gris / Salmón , Sandalia Ahnu Malini Mary Jane para mujer Piedra roja Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono , Sandalia de cuña Daytona SoftWalk para mujer Broncearse , Zapatillas de deporte adidas Originals SL Loop Racer W Lifestyle para mujer Blanco / Blush Green / Vista Pink Zapatillas para correr Mizuno Women s Wave Enigma 6 Deslumbrante Azul-ardiente Coral-plata Skechers Women s Go Flex Max Loafer Armada , MOZO Zapatillas para mujer The Maven Sneaker Verde , CLARKS Mujer Gracelin Art Mary Jane Flat Negro Zapatillas de mujer Clarks Un.Hinton Oxfords Nubuck negro , Zapatillas para caminar Nell Ryka para mujer Azul marino / rosa Atletismo femenino de Apex Negro Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro , Sandalia de vestir Lauren de Jack Rogers para mujer Oro plateado Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Blanco / rosa , Sandalias Birkenstock de mujer con aplicaciones suaves de Granada Cuero Habana Aceitado AYA de VINTAGE HAVANA WOMENS PLATFORM ESPADRILLE DIAPOSITIVA Floral azul , Kea de las mujeres de Flexus Ante rojo Sandalias planas de mujer de Easy Street Cocodrilo rojo , New Balance Sm4040r1 para mujeres Rojo blanco Bomba de vestido de consuelo para mujer LifeStride Negro , Splendid Midi Perf Vionic para mujer Slip-On Cuero perfilado blanco

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 Berry running Saucony Women Women s/ Freedom ISO Berry/ rosa 7e0186f - 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 Berry running Saucony Women Women s/ Freedom ISO Berry/ rosa 7e0186f 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 5754282
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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.