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.

New Negro Balance Women s w990v4 Running/ Shoe Negro 14871/ Plata 29d6e08




75% piel de cerdo de gamuza / 25% Malla
Hecho en Estados Unidos
Caucho único
ENCAP tecnología de suela intermedia proporciona soporte y durabilidad máxima
ENCAP anillo de poliuretano con etileno vinilo núcleo de acetato en talón
Doble densidad espuma de cuello
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


New Negro Balance Women s w990v4 Running/ Shoe Negro 14871/ Plata 29d6e08 - 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

New Negro Balance Women s w990v4 Running/ Shoe Negro 14871/ Plata 29d6e08

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.


New Negro Balance Women s w990v4 Running/ Shoe Negro 14871/ Plata 29d6e08

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

Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima Under Armour Women s HOVR Sonic NC Formación Azul / Brillo / Blanco Zapatillas de running Mizuno Women s Wave Catalyst 2 Turquesa / Coral Fiery , Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Dragon Fruit / Sea Salt / Phantom Zapatillas de running Nike Air Zoom Elite 8 para mujer Pink Blast / Black / White / Elctrc Grn Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas Lobo / Gris / Negro / Fresco / Gris / Voltio Zapatillas de running Under Armour Micro G Pursuit D para mujer Antracita (100) / Negro , adidas Performance Ultraboost X para mujer Core Black / Core Black / Utility Black , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Zapatillas para correr Nike Zoom All Out Flyknit para mujer Bluecap / White-deep Royal Blue , Zapatillas de running Under Armour Speedform Intake 2 para mujer Negro (002) / Acero Zapatillas de running NIKE para mujer Epic React Flyknit Gris blanco Calzado de running Reebok Print Run Prime Ultk para mujer Nube gris / gris meteorito / azul fresco Zapato de running Indigo Ryka para mujer Gris / menta Zapatillas de running adidas Alphabounce Beyond W, para mujer Gris dos grises tres grises uno , Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco negro Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Púrpura Calzado TouchUp Cheer para mujer de Kaepa Blanco , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Abismo / llovizna , Short NIKE Tempo para mujer Vivid Sky Zapatillas de running ASICS Gel-Nimbus 19 para mujer Diva Blue / Flash Coral / Aqua Splash , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco , Nike Men s Air Presto Essential Wolf Gray / Wolf Gray-white , Zapatillas para correr WRL247WM New Balance para mujer Estratos , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Borrar Jade / White-midnight Turq-volt Zapatillas de running New Balance Foam Zantev2 para mujer New Balance Coral / gris Calzado deportivo New Balance Vazee Urge v1 para mujer Gris / rosa , Calzado deportivo NIKE CK Racer para mujer Cerúleo / cerúleo / vela ,
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 Gel-Torrance para mujer ASICS Aluminio / Bahía / Gris Medio Converse Chuck Taylor All Star Seasonal Color Hola Roadtrip Blue / White / Black Converse Chuck Taylor All Star Seasonal Color Hola Pasta de chile , Zapatillas de trail running Pearl NZUMi W 2012 Trail N 1 para mujer Scuba Blue / Shadow Gray , Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Negro Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul humo / azul oscuro / Cantalupo Splendid Sandalia de mujer Farley Blanco Sandalias de mujer Wolky Neutral Multi Color Fantasy Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Verdadero rojo / amarillo vibrante Sandalias Harper Huarache de SoftWalk para mujer Negro / peltre Hush Puppies Men s Gus Oxford Cuero blanco deportivo APL: Athletic Propulsion Labs Zapatillas de running mujer Techloom Phantom Brillar / Brillar / Brillar , Zapatillas de deporte WN Slip-on Farece para mujer de PUMA Puma Black-puma Black Clásicos de base Slip-on (Tm) para hombres de Vans (Bum de goma) True White / True White , Vans Unisex Old Skool Zapatillas clásicas de skate Goblin Blue / True White , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla Sandalia Kace para mujer Calvin Klein Negro , Sandalia con tacón Florence Block de Via Spiga para mujer Patente negra , Zapatillas de voleibol Crazyflight X 2 de adidas Originals para mujer Blanco / Negro / Poder Rojo , Salón de baile de mujeres Zapatillas Dance Salsa Práctica latina Zapatos 9627EB Cómodo-Muy Fino 1.3 \ de 5] Nubuck negro Botas con punta de tacón de mujer Ferrini Ladies Teju Lizard Negro , Calzado de pista WMD800V5 para mujer New Balance Blanco / morado Ariat Unisex Scout Chap Half Chaps Negro , Zapatillas de ante negras de Laurentian Chief para mujer con mocasines con cuello de piel de conejo Sandalias de la diapositiva del dedo del pie abierto de la moda de las mujeres de XYD Deslizamiento de cuero en las zapatillas de la zapatilla del vestido de la puerta Saddle Brown Bomba de vestido exquisita Aerosoles mujeres Black Sparkle , Zapatillas de deporte Amherst Fashion Timberland para mujer Vintage Indigo , Nine Seven Patent Cuero de las mujeres Puntera puntiaguda Heel Pearl decorado Hecho a mano con clase Bombas Zapatos Negro Botines cortos de señorita Macie Women s Brook Brook Short Booties - U2010-02 marrón STEVEN de Steve Madden Women s Adrianna Flat Terciopelo negro

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.

New Negro Balance Women s w990v4 Running/ Shoe Negro 14871/ Plata 29d6e08 - 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 New Negro Balance Women s w990v4 Running/ Shoe Negro 14871/ Plata 29d6e08 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 3260951
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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.