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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Imitación de cuero
Importado
Su apariencia de primavera siempre estar en flor con el Corinne sandal por LifeStride & reg;!
Superior esta construido de imitación de cuero.
Floral diseño recortado adorna las correas.
Fácil diseño sin cordones con estilo de puntera abierta.
Elastico-gore dentro del slingback correa proporciona un ajustable y desgaste ajustado Plantilla de espuma de memoria asegura una gran capacidad de respuesta y amortiguación. Soft System & reg; paquete de comodidad proporciona todo el día soporte, flexión, y amortiguación. suela de tracción construcción.Importado. Mediciones: Altura del tacón: 1 en peso: Producto de 7 oz las medidas fueron tomado usando tamaño 7.5, ancho W (C). Tenga en cuenta que mediciones pueden variar según el tamaño. Peso de calzado es basado en un solo
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


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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

LifeStride Womens Womens Armada Corinne 12229 Armada 7a8a7eb

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.


LifeStride Womens Womens Armada Corinne 12229 Armada 7a8a7eb

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 de cuña Daytona SoftWalk para mujer Azul marino W Tan , Sandalia del Atlas Bayton para mujer naranja , Vionic Splendid Lou- Sandalia con correa para mujer Gris , CLARKS Sillian Stork Fisherman Sandal para mujer Nubuck rojo sintético Propet Women s Fisherman Sandal ciruela Sandalia Savannah de Ryka para mujer Beet Red / Hyper Pink , Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Gold-a , Merrell Women s Around Town Sunvue Post Sandalia Textura negra , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Coñac Sandalia Gigi de Naturalizer para mujer Arce claro , Naot Footwear Mujer Papaki Terciopelo / Canela de Terciopelo Negro , Tecnología Vionic Men s Orthaheel White / Navy Walker Gris oscuro , Sandalia casual destello de las mujeres de ECCO Negro , Sandalia Wendy Huarache de Naturalizer para mujer Blanco , New Balance Women s Renew Thong Sandal Cafe Blanco , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Bomba natural Nora para mujeres Negro Chaco Z / Cloud X2 Remix Cottage Poppy , Sandalias EVA Gizeh de mujer Birkenstock Negro , ALMA NATURAL Ballet femenino femenino Tela de microfibra negra , Chaco Women s Rowan Sandal Tierra de siena Sandalia Cyprine Slide de mujer Naturalizer Rosa , Sandalia Relax de Steve Madden para mujer Terciopelo azul Sandalia de gladiador ECCO Womens Flash T-Strap, Warm Gray Metallic / Moon Rock CLARKS Haley Stork Loafer para mujer Mushroom Nubuck Sandalia de mujer Corkys Float Bronce Marc Fisher LTD Mujeres Glenna Gris claro gamuza OLUKAI Womens Eheu Thong Sandal Negro Negro , Bomba de espejos Zahn de Naturalizer para mujeres 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 NIKE Women s Air Force 1 Mid 07 Le - Converse Chuck Taylor All Star Seasonal Color Hola rojo , Zapato NIKE Roshe One femenino Negro / antracita / negro Zapatillas NIKE Femenil Flex 2016 Rn Running Negro / Metálico Cool Gray / Pink Blast / Blanco Zapatillas de running ISO para mujeres de la guía Saucony Negro / Denim NIKE Men s Roshe Two Running Shoe Iguana / Black-sail-volt , La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Fjord / Malibu Blue Merrell Siren Edge Hiker para mujer Aluminio , Zapatillas de trail running Salomon Speedcross 4 para hombre Azul Zapato deportivo ASICS Gel-Excite 4 para mujer Diva Azul / Blanco / Sol Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Blanco invierno , Calzado de running NIKE Kids Roshe One (GS) Coastal Blue / White-blue Spark , Zapatillas de running NIKE Womens Huarache Run Ultra PRM para mujer EQUIPO NARANJA / NEGRO / NEGRO / BLANCO , Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Azul marino / gris , Brooks Women s Ravenna 8 Negro / Diva Pink / Plum Caspia Sandalia de mujer Venus de la tierra Sonrojo , CLARKS Sandalia Corsio Amelia de tiras de tobillo para mujer Ante negro CLARKS Mujer Gracelin Art Mary Jane Flat Negro Zapatillas de tenis adidas Adizero Ubersonic 3 Clay para mujer Negro / plata / amarillo , Zapatillas Adidas Performance Mujer Adipure 360.3 W Flash rojo / oscuro sólido gris / plata Zapatillas de skate casual unisex Heathrow Prestige para hombre de DC Negro / Marrón / Blanco , JORDAN 12 Zapatillas niños RETRO BP 151186-400 Burdeos / Vela-Plata Metálica FootJoy Emerge Golf Shoes Liquidación 2017 Mujeres Blanco-m , Sandalia Janissah Slide de Nine West para mujer Cuero negro Bota de trabajo con abertura para mujer Ariat, estampado de hipopótamo de chocolate, 10 B US Impresión de chocolate hipopótamo , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Raya tejida negra Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Violet White , Botas de montar para mujeres Midnight Dream, sin calificación Negro , Sandalias de peonía Easy Spirit para mujer Gris pardo , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada

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.

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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 LifeStride Womens Womens Armada Corinne 12229 Armada 7a8a7eb 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

Thirteen elementary straightedge 857299
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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.