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.

Entrenador negro Jemma Jemma femenino Cuero 20000 negro 7a6a735




Cuero
Importado
Talón mide aproximadamente 3.5 pulgadas
Por favor Nota: COACH & trade; los artículos no pueden ser enviado a militares direcciones (APO o FPO) y direcciones en Hawaii, el Islas Virgenes, Guam o cualquier otro lugares fuera del continental EE. UU.
Líneas elegantes y cuero premium, el entrenador y el comercio; Jemma bootie lo hará ¡no decepcionar!
Premium parte superior de cuero.
Lado cierre de cremallera para fácil y apagado.
Tire de la pestaña en el talón. Suave y cuero suave forro. Ligeramente plantilla acolchada. Envuelto tacón. Cuero liso y suela sintética.Importado.Medidas: Altura del tacón: 3 1 & frasl; 2 in Peso: 10 Circunferencia de oz 9 1 & frasl; 2 in Eje: 6 3 & frasl; 4 en mediciones de productos fueron tomadas usando tamaño 8, ancho M. Tenga en cuenta que mediciones pueden variar según el tamaño. Peso de calzado es basado en un solo artículo, no un par
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


Entrenador negro Jemma Jemma femenino Cuero 20000 negro 7a6a735 - 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

Entrenador negro Jemma Jemma femenino Cuero 20000 negro 7a6a735

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.


Entrenador negro Jemma Jemma femenino Cuero 20000 negro 7a6a735

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

Baile Bella Me de Bernie Mev para mujer Jeans azul marino KEEN Kids Seacamp Ii CNX Water Shoe Racing Red / Gargoyle , Sandalias de mujer Wolky Cuero naranja de Vegi , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Lino de Camo , Sandalias de cuero de Viv Viv de TOMS Yute metálico natural , Sandalias con plataforma SAS de cuero con punta abierta para mujer Negro Sandalia Huarache Easy Street Allure para mujer Armada , Bomba de cuña Dreamstride Women s Dreams Negro Earth Origins Selby Negro , Birkenstock Arizona Sandalia de ante con suela suave Cuero plateado metalizado , Sandalia de mujer Columbia Kyra Vent II con correa Sabio de plata / Guijarro XYD Women Open Toe Low Heel Strappy Rhinestone Sandalias Correa de tobillo vestido de novia Bombas con cremallera Oro Sandalia de gladiador Samina de BareTraps para mujer Gris pardo , JBU by Jambu Women s Wildflower Sandal Carbón Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Silver Specchio / Hemp Victoria s Secret PINK Strap Sport Sandalias con correas Negro , Vince Camuto Women s Prilla Ballet Flat Negro Sam Edelman Sandalia Dariel para mujer Satén gris claro CLARKS Lexi Myrtle Sandal para mujer marrón , Sam Edelman sandalia Gio Slide para mujer Ante negro Sandalia plana Halle de Walking Cradles para mujer Cuero Agata Negro , Sandalia LifeStride para mujer Eden Moongold Tanga Kahikolu de mujer OLUKAI Melocotón / Tostado , NAOT Womens Tucson Cuero gris vintage Sandalias Joanie II de mujer SOREL CAMELLO , Sandalia Mia Mooz para mujer Avellana , New Balance Women s Renew Thong Sandal menta , Sandalia Tonya de mujer Trotters rojo Cuña con cordones Tansy Noble para mujeres Vionic Ante negro
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 Zapato de running ASICS GT-1000 6 para mujer Mid Gray / Carbon / Flash Coral Skechers Women s Go Ride 7 Running Shoe Turquesa New Balance Wrl247fc para mujeres Silver Min Sandalia deportiva Tirra para mujer Teva Mar de Bering , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Binario Azul / Blanco , Zapatillas de running adidas Alphabounce Beyond W, para mujer Núcleo negro, núcleo negro, tela gris cinco Calzado de trail running SCARPA Neutron 2 GTX para mujer Negro / Cerámica Zapatilla Skechers para mujer Go Run 600-Obtain Frambuesa Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Choque Verde / Negro , Corredor de trail Speed-Wak W-W de mujeres Salomon Nube Oscura / Luz Onix / Azul Burbuja , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro Negro , Zapatillas para caminar WW1400v1 New Balance para mujer Marron oscuro , Zapatillas de running Nalu para mujer Ryka Rosa / Plata Chaco Mujer ZX3 Classic Sport Sandalia Shiver Pine , Sandalias con hebilla de botón Ridgeport para mujer Rockport Plata Dr. Comfort Annie-X Mujer diabética terapéutica Doble Profundidad Zapato Lycra Velcro Negro , Sandalia universal W Ysidro de mujer Teva Gris pardo Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer gris , Zapatillas VI-S para fitness y yoga Vibram para mujer Negro , K-Swiss St329 CMF Sneaker para mujer Blanco azul , Zapato de skate DC Magnolia TX para mujer Azul blanco , Sandalia Beau Slide Kate Spade New York para mujer Oro rosa , Sandalia de diapositivas de ocio para mujer de Seychelles rojo , Roper Women s Liza Southwestern Liza - Mocasines ocasionales para conducir Gris , Botín de ante TOMS Sevilla para mujer Honey Suede School Issue Upper Class 7300 Saddle Shoe (Niño / Niño pequeño / Big Kid) Blanco negro , Bota de invierno con cordones D Lites-Short de Skechers para mujer Negro Easy Spirit Women s Traveltime Mule Gamuza morado oscuro / púrpura oscuro , Zapatillas Skechers Women s Microburst One hasta Fashion Sneaker Pizarra / 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.

Entrenador negro Jemma Jemma femenino Cuero 20000 negro 7a6a735 - 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 Entrenador negro Jemma Jemma femenino Cuero 20000 negro 7a6a735 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.

Sandalias
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

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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.