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 para correr Brooks pasión/ Cascadia 12 para mujer Bluebird Peacoat/ Flor de la pasión/ Bluebird 3cb15fa




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Importado
Patada hasta los talones y agitar el polvo como llevas a el salvaje al aire libre en Brooks & reg; Cascadia 12 trail-running zapato!
Predecesor: Cascadia 11.
Soporte Tipo: Neutro.
Amortiguación: Ligero, flexible respuesta.
Superficie: Sendero. 10 mm. Tejido grueso malla superior con sintético y microfibra superposiciones. sillín de medio pie. Cierre con cordones. Heel pull loop.Padded lengua y cuello Tela de elemento suave forro ofrece una Gran sensación en el zapato. Extraíble plantilla de perfil. De longitud completa S-257 Cushsole entresuela compuesta utiliza una plantilla tablero para aumentar felpa y flexibilidad.Trabajo específico Pivote de cuatro puntos Sistema de publicación suministros de tecnología suspensión estabilizadora en ambos lados o
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 para correr Brooks pasión/ Cascadia 12 para mujer Bluebird Peacoat/ Flor de la pasión/ Bluebird 3cb15fa - 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 para correr Brooks pasión/ Cascadia 12 para mujer Bluebird Peacoat/ Flor de la pasión/ Bluebird 3cb15fa

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 para correr Brooks pasión/ Cascadia 12 para mujer Bluebird Peacoat/ Flor de la pasión/ Bluebird 3cb15fa

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

Nike Men s Air Presto Essential Antracita / Negro / Gris Fresco , Zapatillas de running NIKE para mujer Epic React Flyknit Negro / gris oscuro Calzado de running NIKE Kids Roshe One (GS) Deep Royal Blue / Midnight Navy / Wolf Gray , Zapatillas de trail running Inov-8 Inov8 para mujer X-Talon 230 Gris / verde azulado Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul , Zapatillas de running NIKE RN 2017 para mujer gratis Antracita / Plata Metálica / Negro , Zapatillas de correr de gel y corrido para mujer de ASICS Pale Blue / Silver / Aquarium , adidas Performance Ultraboost X para mujer Blanco / gris perla / blanco cristal , Zapato de running Gel-Torrance para mujer ASICS Aluminio / Bahía / Gris Medio , New Balance Women s w990v4 Running Shoe Gris / rosa , Zapatillas de running Mizuno Wave Rider 20 para mujer Negro / Blanco / Rosa Glo , Amortiguación 635v2 New Balance para mujer Gris , Zapatillas de running Adidas Supernova Sequence 9 para mujer Rosado , NIKE Mujeres Downshifter 7 Negro / Polarizado Azul-blanco Zapatillas de running Reebok para mujer ZigWild TR ll Carbón / solar rosa / negro , Zapatillas de running Alphabounce W para mujer adidas Performance Easy Orange / White / Light Onix , Zapatillas para correr NIKE Womens Sock Dart Blanco / puro platino Zapatillas de running ASICS Gel-Nimbus 19 para mujer Glaciar gris / plata / blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Turq / Black-volt-rio NIKE New Air Max 90 Ultra 2.0 Running Sneaker para mujer Negro , Zapatillas de running adidas Alphabounce Em W para mujer Easy Coral / Sun Glow / Gray One , Zapatillas para correr NIKE Air Huarache Run Ultra SE para mujer Polvo / Peltre metálico / Negro NIKE Mujeres WMNS Air Huarache City bajo, obsidiana / blanco-gran gris Obsidiana / Blanco-vasta gris , Zapatillas de running Mizuno Wave para mujer con sombra Fucsia Púrpura / Plateado / Azulejo Azul , Zapatillas de running Under Armour Speedform Intake 2 para mujer Azul , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Mora , Zapatillas de running Mizuno Wave para mujer Inspire 10 Blanco , Amortiguación 635v2 New Balance para mujer Plata New Balance Women s w940v3 Running Shoe Plata ,
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 Negro / blanco / antracita / gris fresco , Sandalia KEEN Women s Venice H2 Cascade / Misty Jade , Zapatillas para correr Mizuno Women s Wave Rider 21 Azul pavo real / blanco , Zapatillas para correr Altra Women s Intuition 3.5 Hemlock / Peltre , Zapatillas deportivas 574S Sport New Balance para hombre Varsity naranja / blanco Sandalia Teva para mujer Terra-Float Livia Lux Armada Zapatillas de running New Balance para mujer Fresh Foam Zante V3 Performance Añada Indigo / Sal Marina , Zapatillas de running Classic-M Vibram para hombre Negro Negro , Zapatillas de entrenamiento Primus Lite para entrenamiento de mujer Vivobarefoot Iffley Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Gris / Coral Prop & eacute; t Olivia Oxford para mujeres Bronco Brown , Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Carbón , Zapatillas de running NIKE Air Max Torch 4 para mujer Blanco / menta Sandalia Alegria Mujer Anah Strap Pera , Sandalia Alegria Mujer Anah Strap Dusty Blue , BareTraps Sandalia para mujer Octavia Gris , Sandalia Amara 2 de Carlos Santana para mujer de Carlos Azalea Zapatilla de deporte S-Inspira de Skechers para mujer Negro , Cole Haan Studiogrand Knit Cross Strap Sneaker Black Knit Entrenador campeón Gusto Cross Trainer Malla blanca Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Pigmento , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker rojo , Sandalia de tobillo de mujer Steve Madden Natural 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2824- Brillo marrón satinado y dorado , Ringor Dynasty Turf Shoe- Lanzamiento Azul marino / blanco Under Armour Fusible para mujer FST Negro (002) / Glacier Grey Bota final femenina Jeffrey Campbell Tela escocesa amarilla , Freewaters, mujer, chloe, casa, resbalón, en, zapatilla borgoña , Sandalia Birkenstock de cuero Florida para mujer Sfb Blue Birko-flor , Corral Women s Patch Western Snip Toe Boots Marrón, coñac ,

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 para correr Brooks pasión/ Cascadia 12 para mujer Bluebird Peacoat/ Flor de la pasión/ Bluebird 3cb15fa - 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 para correr Brooks pasión/ Cascadia 12 para mujer Bluebird Peacoat/ Flor de la pasión/ Bluebird 3cb15fa 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 150424
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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.