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.

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Textil
Importado
Caucho único
Eje mide aproximadamente parte superior del arco
Fresco Entresuela de espuma
extraíble insertar
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


Calzado/ de New 19280 running para mujer New Balance Cypher Castlerock/ White c02755b - 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

Calzado/ de New 19280 running para mujer New Balance Cypher Castlerock/ White c02755b

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.


Calzado/ de New 19280 running para mujer New Balance Cypher Castlerock/ White c02755b

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

Under Armour Threadborne Fortis para mujer Gris Nublado (103) / Blanco Zapatillas de entrenamiento NIKE Metcon Dsx Flyknit para mujer Negro / Cool Gray-hyper Punch , Zapatillas para caminar Skechers Go Run-Mojo para mujer Negro Zapatillas para correr Pacev2 para mujer New Balance Silver Mink / Black , Brooks Women s Dyad 9 Primer gris / Capri Breeze / Plata , Zapatillas de running adidas Cloudfoam Super Flex W para mujer Misterio Azul / Blanco / Naranja Brillante , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Blanco / Negro - Pure Platinum Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue , NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / Summit White-lagoon Pulse Zapato deportivo ASICS Gel-Kayano 24 para mujer Gris medio / negro / Atlantis Pantalón largo clásico NIKE Golf para mujer Tech Essentials Cool Grey / Court Purple / Dark Grey , Zapatillas de running Pure Cloudfoam de adidas para mujer Blanco / Blanco / Negro , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Lobo Gris Puro / Negro-puro Platino-caliente Zapatillas de running NIKE Mens Air Zoom Structure 19 Luz retro / blanco-verde azulado , Zapatillas de running Reebok Women s Supreme 3.0 MT Nube gris / Caribe trullo / polvo de asteroides NIKE Air Max 90 Womens Style: 325213 Mujeres 325213-048 Lobo gris / blanco-gris oscuro adidas outdoor para mujer Terrex Swift R2 GTX Grey Three / Gray Two / Chalk Coral , Nike Men s Air Presto Essential negro Zapatillas de running cross cross para mujer ASICS Cross Faire 2 para mujer Polvo azul / blanco / azul marino , Calzado de running NIKE RN 2017 2017 para mujer Blanco / Metálico / Plata , Zapatillas de running Nike Kaishi para hombre Negro NIKE Women s Tanjun Iced Jade Aurora verde blanco Zapatillas de running ASICS Gel-Contend 4 para mujer Rose azulada / helada Zapatillas Reebok Women s Runner 2.0 MT Track Lila Shadow / Deep Cobalt / Electric Flash / Pewter , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Indigo Blue / Paradise Green / Pink Glow , Zapatillas de running adidas Originals Women s Tubular Dawn Originals Lbrown / Lbrown / Crywht , Saucony Women s Guide 8 Running Shoe Azul / azul marino / amarillo Zapatillas de running Nike Air Huarache Run Ultra Br para mujer Platino Blanco / Blanco-puro Zapato de running GEL-Excite 3 para mujer ASICS Rosa fuerte / Blanco / Amarillo intermitente ,
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 Sandalias Bali KEEN para mujer Red Violet / Boysenberry Gel ASICS para mujer-Kayano 25 Carbon / Mid Grey 12 D US - , Zapatillas de running NIKE Runallday para mujer Negro / lobo blanco gris , Zapato de running femenino Stinson 3 de HOKA ONE ONE Blanco , Sanda Verra Teva para mujer Cuerda elástica Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Yo tiempo negro / blanco Calzado de espuma fresca New Balance para mujer 1165v1 Gris blanco , CLARKS Sandalia plana Brizo Sammie para mujer Microfibra perforada de arena , Zapatilla de deporte de cobre de la cuerda feroz de las mujeres de PUMA Cobre rosa / Puma negro Zapato de skate casual Heathrow DC para hombre Natural Zapatillas altas con cordones All Star de cuero Converse Chuck Taylor Trueno / Negro / Papiro Sandalia de cuña OAHU de China Laundry Women Camel Suede , Nacido - Mujeres - Tepati Marrón claro de grano completo , Sandalias Skechers para mujer Rumbler Wave-Ibiza Summer Slide Blanco Sam Edelman Women s Bay Slide Sandal Cuero de silla de montar Vaquero blanco brillante , Zapato Tap-Flex para mujer, de Bloch Dance Negro , Sailor Moon Chibi Usa Cosplay Shoes Boots por encargo , mejor sentido Unisex Lana clásica Zapatilla para Adultos - Toda la lana natural - Ultra cómodo - Muchos tamaños y Colores Gris Pantuflas de mujer de lujo de piel de oveja zapatilla / Sandalia / Thong Slipper by Bushga ciruela , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2707- Satén blanco y adornos de plata # 119 Zapatillas de ciclismo Pearl XZUMi W X-ALP Drift IV para mujer Negro / Anochecer Azul , Bomba de vestido LifeStride Sevyn para mujer Luxe Navy 4 Zapatilla deslizante Nakiska para mujer SOREL Redwood, Dark Stone , Sandalia de vestir Taimi de Naturalizer para mujer Brillo negro , Botín de ante TOMS Sevilla para mujer Black Suede / Honey Leather , Zapatilla de cuello de garganta 4640 de Jacques Levine para mujer Plata blanca , Rockport Mujeres Truwalk Zero Moreza Mesh Mary Jane Flat Lavable negro , Wolky Comfort Rio 40820 Denim Craquelé Cuero , Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse Mocasines casuales Alpargata Deconstructed de TOMS para mujer Peltre Metálico Cuero ,

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.

Calzado/ de New 19280 running para mujer New Balance Cypher Castlerock/ White c02755b - 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 Calzado/ de New 19280 running para mujer New Balance Cypher Castlerock/ White c02755b 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 1895964
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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.