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.

Sandalia Zvolv deportiva Chaco Zvolv mujer X2 para mujer deportiva Monte Mint fe2ae26




Poliéster
Importado
Goma suela
con doble correa poliéster jacquard correas superiores alrededor del pie y a través del entresuela para un personalizado ajuste
Ajustable y durable de alta resistencia correas del talón de las correas
moldeado por inyección hebilla de bloqueo de escalera
topográfico surcos texturizados en plantilla para tracción
No marcado ecotread 25% reciclado compuesto de goma con 3 mm z-inspirado asas
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


Sandalia Zvolv deportiva Chaco Zvolv mujer X2 para mujer deportiva Monte Mint fe2ae26 - 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

Sandalia Zvolv deportiva Chaco Zvolv mujer X2 para mujer deportiva Monte Mint fe2ae26

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.


Sandalia Zvolv deportiva Chaco Zvolv mujer X2 para mujer deportiva Monte Mint fe2ae26

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

Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Negro , Sandalia deportiva Teva W Hurricane 4 para mujer Negro , Chaco Zcloud 2 Sport Sandal para hombre Pirámide Verde Sandalia Teva Mujer Terra-Float Livia Lux Negro , Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Helecho Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo , Sandalia original premier universal Teva W de mujer Aceituna Sandalia Yucatan ECCO Mujer Shadow White , Chaco Women s ZX2 Classic Athletic Sandal Palm aguacate Propet Women s TravelActiv Ss Sandal Negro rojo Sandalia atlética Columbia Slide Caprizee de cuero para mujer Negro / grafito Sandalia Savannah de Ryka para mujer Moonrock / Snowline Ecru KEEN sandalia de Venecia para mujer Spray de oliva / surf negro , Sandalia Terra-Float Livia de Teva para mujer Lima / Blanco Sandalia Teva Women s W Hurricane Xlt2 Sport Negro , Earth Origins Sophie Women s Sandal Frost Grey Tacones altos Mujeres Chanclas Sandalias de verano Plataforma Cuñas Zapatillas Chica39; s Fashion Beach Shoes rojo , Sandalia deportiva ECCO Terra 3S para mujer Negro / Teaberry , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso , Sandalias de deporte Nike Kawa Sport Shower Slide, negro / blanco Blanco negro Sandalia atlética de la correa de Barraca de las mujeres de Colombia Gris oscuro, salmón salvaje Zapato de agua H2 Go Skechers Performance para mujer Azul marino / gris , Teva Terra-Float Active Lace - Mujeres Chispas de chocolate , Sandalia de cuña Rheta Sam Edelman para mujer Ante negro Teva ALP Premier - Mujeres Negro , Zapato de agua Helmo Hansen Watermoc 2 para mujer Azul marino / Sorbete / Noche azul , Sandalias Bali KEEN para mujer Higo de verano / Crabapple Correas de velcro de las mujeres de EpicStep al aire libre que caminan que caminan las sandalias atléticas del pescador de los deportes del trekking Armada
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 Atlética Merrell para Mujeres, Alrededor de la Ciudad Vertical La sandalia Switchback del campamento base de mujer de North Face Tnf Black / Vintage White , Sanda Verra Teva para mujer Cuerda elástica Carro de mujer Brooks Fucsia Zapatillas de running Kineta Relay para mujer Saucony Coral / Mint Salomon Women s Speedcross 4 W Trail Runner Rosa virtual Salomon Women s Speedcross 4 W Trail Runner Surf hawaiano , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Espuma de mar / cielo cubierto Zapatillas de running adidas Women s Edge Lux W Gris / Blanco / Metálico / Plata , Zapatillas de trail running Inov-8 Trailtalon 235 (W) para mujer Negro / púrpura Zapatillas de running Icebug para mujer Acceleritas4 RB9X Traction Orquídea / Veneno Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Yo tiempo negro / blanco , Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Negro / Microdot , Splendid Feodora Sandalia para mujeres Gris pardo , CLARKS Sandalia plana Brizo Cady para mujer Puntos blancos fucsia Zapatos de caminata SAS Low Me de cuero para mujer Moca Easy Spirit Women s Traveltime Mule Azul marino-azul marino oscuro , Easy Spirit Women s Traveltime Mule Stone-medium Blue-leaf Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja Supra mujer Skytop Sneaker Marrón / Brogue / Hueso , Lakai Limited Footwear Mens Griffin Negro / Reflective Suede , Zapatillas de skate Vans Unisex Authentic Platform Tablero de ajedrez / Globos oculares 7236 , Vans Old Skool Unisex Adults Low-Top Trainers Blanco negro , Zapatillas de baloncesto PUMA Women s Sky II HI Futur Minimal WN Cereza de Barbados Zapatillas de deporte adidas Climacool Ballerina para mujer Solar Azul / Blanco / Plata Metálica Sandalia Ryana Marilyn Slide para mujer Blanco negro Sandalias planas para piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural 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 8881- Brown Nubuck & Beige Trim Zapatilla Sandalia de la Unión de Mujeres Sol Sana Ecru Bella Vita Womens Tex-Italia Grey Suede ,

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.

Sandalia Zvolv deportiva Chaco Zvolv mujer X2 para mujer deportiva Monte Mint fe2ae26 - 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 Sandalia Zvolv deportiva Chaco Zvolv mujer X2 para mujer deportiva Monte Mint fe2ae26 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 y toboganes deportivos
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.